The complexity of Boolean surjective general-valued CSPs

TL;DR

This paper classifies the computational complexity of Boolean surjective VCSPs, revealing new tractable classes and providing algorithms for near-optimal solutions, thus advancing understanding of these optimization problems.

Contribution

It establishes a complete complexity classification for Boolean surjective VCSPs, including a novel tractable class related to downsets and upsets, and introduces a polynomial-time enumeration algorithm.

Findings

Complete complexity dichotomy for Boolean surjective VCSPs

Identification of a new tractable class related to downsets and upsets

Polynomial-time algorithm for enumerating near-optimal solutions

Abstract

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from $D=\{0,1\}$ and an optimal assignment is required to use both labels from $D$. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for $\{0,\infty\}$-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and H\'ebrard. For the maximisation problem of $\mathbb{Q}_{\geq 0}$-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.

Equations

$${\varphi }_I(x_1,\dots ,x_n)=\sum _{i=1}^q{w}_i\cdot {\gamma }_i({\mathbf{x}}_i),$$

$$\sum _{(u,v)\in E,u\in C,v\notin C}w(u,v).$$

$${\varphi }_I(x_1,\dots ,x_n)={\rho }_0(s)+{\rho }_1(t)+\sum _{(u,v)\in E}w(u,v)\cdot \gamma (u,v).$$

$$\sum _{\{u,v\}\in E,|\{u,v\}\cap C|=1}w(u,v).$$

$${\varphi }_I(x_1,\dots ,x_n)=\sum _{\{u,v\}\in E}w(u,v)\cdot \gamma (u,v).$$

$$\mathrm{Feas}(\gamma )=\left\{\mathbf{x}\in D^r|\gamma (\mathbf{x})<\infty \right\}.$$

$$\mathrm{Opt}(\gamma )=\left\{\mathbf{x}\in \mathrm{Feas}(\gamma )|\gamma (\mathbf{x})\le \gamma (\mathbf{y})\text{ for every }\mathbf{y}\in D^r\right\}.$$

$$\mathrm{VCSP}(\Gamma ){\le }_p\mathrm{VCS}{\mathrm{P}}_{\mathrm{s}}(\Gamma ).$$

$$\mathrm{VCS}{\mathrm{P}}_{\mathrm{s}}(\Gamma ){\le }_p\mathrm{VCSP}(\Gamma \cup {\mathcal{C}}_D).$$

$$\mathrm{VCS}{\mathrm{P}}_{\mathrm{s}}({\Gamma }^{\ast }){\le }_p\mathrm{VCS}{\mathrm{P}}_{\mathrm{s}}(\Gamma ).$$

$$h\left({\mathbf{x}}_1,\dots ,{\mathbf{x}}_k\right)=\left(h\left(x_{1,1},\dots ,x_{k,1}\right),\dots ,h\left(x_{1,r},\dots ,x_{k,r}\right)\right).$$

$$\sum _{i=1}^k\gamma (h_i({\mathbf{x}}_1,\dots ,{\mathbf{x}}_k))\le \sum _{i=1}^k\gamma ({\mathbf{x}}_i).$$

$$\alpha \cdot (\gamma (\mathbf{x})+\gamma (\mathbf{y})-2\cdot \gamma (0^r))\ge \gamma (\mathrm{sub}(\mathbf{x},\mathbf{y}))-\gamma (0^r).$$

$$\alpha \cdot (2\cdot \gamma (\mathbf{x})-2\cdot \gamma (0^r))\ge \gamma (\mathrm{sub}(\mathbf{x},\mathbf{x}))-\gamma (0^r)=0$$

$$\infty >\alpha \cdot (\gamma (\mathbf{x})+\gamma (\mathbf{y})-2\cdot \gamma (0^r))\ge \gamma (\mathrm{sub}(\mathbf{x},\mathbf{y}))-\gamma (0^r),$$

$$0=\alpha \cdot (\gamma (\mathbf{x})+\gamma (\mathbf{y})-2\cdot \gamma (0^r))\ge \gamma (\mathrm{sub}(\mathbf{x},\mathbf{y}))-\gamma (0^r);$$

$$\rho (x_1,\dots ,x_r)={\rho }^{\prime }\left(x_{\pi (1)},\dots ,x_{\pi (r^{\prime })}\right)+\sum _{i=r^{\prime }+1}^r{\rho }_{=}\left(x_{\pi (i)},x_{a_i}\right).$$

$${\mu }_w(x,y,z)=\{\begin{array}{ll}2& \text{if z=1 and x=y,}\\ 1& \text{if z=1 and x=y,}\\ 0& \text{if z=0 and x=y=0,}\\ w& \text{otherwise.}\end{array}$$

$${\varphi }_I(x_1,\dots ,x_n)=\sum _{i=1}^q{w}_i\cdot {\gamma }_i({\mathbf{x}}_i),$$

$$c(x)=\sum _{i\in B_x}{w}_i\cdot {\gamma }_i(s({\mathbf{x}}_i)).$$

$${\varphi }_I(s^{\prime })$$

$$\ge {\varphi }_I(s)-\frac{|D|}{n}\cdot a_{\text{max}}\cdot {\varphi }_I(s)$$

$$\ge \left(1-\frac{|D|}{n}\cdot a_{\text{max}}\right)\cdot r\cdot {\mathrm{opt}}_I$$

$$\ge (r-ϵ)\cdot {s-opt}_I.$$

$${\mathrm{Round}}_{\alpha }(\gamma )(\mathbf{x})=\{\begin{array}{ll}0& \text{if γ(x)∈[0,1],}\\ \infty & \text{if γ(x)∈(α,∞].}\end{array}$$

$$0\le {\alpha }^{17}\cdot (\gamma (\mathbf{u})+\gamma (\mathbf{v}))<\gamma (\mathrm{sub}(\mathbf{u},\mathbf{v})).$$

$$A_r=\left\{(x_1,\dots ,x_r)\in \{0,1{\}}^r|\sum _{i=1}^r{x}_i\equiv 0(\text{mbox}mod2)\right\}.$$

$$\mathrm{sub}(\mathbf{u},\mathbf{v})$$

$$=(0,0,0,1)\oplus \mathbf{u}$$

$$=(0,1,0,0)\oplus (\mathbf{u}\oplus \mathbf{v})$$

$$=(1,0,0,0)\oplus \neg (\mathbf{v}),$$

$$\gamma (X)+\gamma (Y)\le \gamma (X\cup Y)$$

$$\gamma (X)+\gamma (Y)\ge \gamma (X\setminus Y)+\gamma (Y\setminus X)$$

Full text

We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for $\{0,\infty\}$-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hébrard. For the maximisation problem of $\mathbb{Q}_{\geq 0}$-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability.

Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.

1 Introduction

The framework of valued constraint satisfaction problems (VCSPs) captures many fundamental discrete optimisation problems. A VCSP instance $I=(V,D,\phi_{I})$ is given by a finite set of variables $V=\{x_{1},\dots,x_{n}\}$, a finite set of labels $D$ called the domain, and an objective function $\phi_{I}:D^{n}\to\overline{\mathbb{Q}}$, where $\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}$ denotes the set of extended rationals. The objective function is expressed by a weighted sum of valued constraints

[TABLE]

where $\gamma_{i}:D^{\operatorname{ar}(\gamma_{i})}\to\overline{\mathbb{Q}}$ is a weighted relation of arity $\operatorname{ar}(\gamma_{i})\in\mathbb{Z}_{\geq 1}$, $w_{i}\in\mathbb{Q}_{\geq 0}$ is the weight and $\mathbf{x}_{i}\in V^{\operatorname{ar}(\gamma_{i})}$ the scope of the $i$th valued constraint. (Note that zero weights are allowed; we define $0\cdot\infty=\infty$.) The value of an assignment of domain labels to variables $s:V\to D$ equals $\phi_{I}(s)=\phi_{I}(s(x_{1}),\dots,s(x_{n}))$. An assignment $s$ is feasible if $\phi_{I}(s)<\infty$, and it is optimal if it is feasible and $\phi_{I}(s)\leq\phi_{I}(s^{\prime})$ for all assignments $s^{\prime}$. Given an instance $I$, the goal is to find an optimal assignment, i.e. one that minimises $\phi_{I}$. A valued constraint language (or just a language) $\Gamma$ is set of weighted relations over a domain $D$. We denote by $\operatorname{VCSP}(\Gamma)$ the class of all VCSP instances that use only weighted relations from a language $\Gamma$ in their objective function. VCSPs are also called general-valued CSPs [31] to emphasise the fact that (decision) CSPs are a special case of VCSPs in which weighted relations only assign values [math] and $\infty$. (However, $\mathbb{Q}$-valued VCSPs [42] do not include CSPs as a special case.)

For an example of a VCSP, consider the $(s,t)$-Min-Cut problem [40]. Given a digraph $G=(V,E)$ with a source $s\in V$, sink $t\in V$, and edge weights $w:E\to\mathbb{Q}_{>0}$, the goal is to find a set $C\subseteq V$ with $s\in C$ and $t\not\in C$ that minimises

[TABLE]

We show how the $(s,t)$-Min-Cut problem can be expressed as a VCSP over a domain $D=\{0,1\}$ (a domain of size $2$ such as this one is called Boolean). We define a language $\Gamma_{\sf cut}=\{\rho_{0},\rho_{1},\gamma\}$ as follows: For $d\in D$, $\rho_{d}:D\to\overline{\mathbb{Q}}$ is defined by $\rho_{d}(x)=0$ if $x=d$ and $\rho_{d}(x)=\infty$ if $x\neq d$. Weighted relation $\gamma:D^{2}\to\overline{\mathbb{Q}}$ is defined by $\gamma(x,y)=1$ if $x=0$ and $y=1$, and $\gamma(x,y)=0$ otherwise. Given an $(s,t)$-Min-Cut instance on a digraph $G=(V,E)$, the problem of finding an optimal $(s,t)$-Min-Cut in $G$ is equivalent to solving an instance $I=(V,D,\phi_{I})$ of $\operatorname{VCSP}(\Gamma_{\sf cut})$ such that

[TABLE]

It is well known that the $(s,t)$-Min-Cut problem is solvable in polynomial time. Since every instance $I$ of $\operatorname{VCSP}(\Gamma_{\sf cut})$ can be reduced to an instance of the $(s,t)$-Min-Cut problem, $\operatorname{VCSP}(\Gamma_{\sf cut})$ is solvable in polynomial time.

A language $\Gamma$ is called tractable if, for every finite $\Gamma^{\prime}\subseteq\Gamma$, $\operatorname{VCSP}(\Gamma^{\prime})$ is solvable in polynomial time. If there exists a finite $\Gamma^{\prime}\subseteq\Gamma$ such that $\operatorname{VCSP}(\Gamma^{\prime})$ is NP-hard, then $\Gamma$ is called intractable.111Defining tractability in terms of finite subsets ensures that the tractability of a language is independent of whether the weighted relations are represented explicitly (by tables of values) or implicitly (by oracles). For example, language $\Gamma_{\sf cut}$ is tractable. It is natural to ask about the complexity of $\operatorname{VCSP}(\Gamma)$ for a fixed language $\Gamma$. Cohen et al. [12] obtained a dichotomy classification of Boolean languages: They identified eight tractable classes (one of which correspons to submodularity [40] and includes $\Gamma_{\sf cut}$) and showed that the remaining languages are intractable. The dichotomy classification from [12] is an extension of Schaefer’s celebrated result [39], which gave a dichotomy for Boolean $\{0,\infty\}$-valued constraint languages, and the work of Creignou [13], which established a dichotomy classification for Boolean $\{0,1\}$-valued constraint languages.

The surjective variant of VCSPs further requires that assignments of domain labels to variables be surjective (an assignment $s:V\to D$ is surjective if, for every $d\in D$, there exists $x\in V$ such that $s(x)=d$). Thus, the goal is to find an assignment that is optimal among surjective assignments. For Boolean VCSPs with $D=\{0,1\}$, this simply means that the all-zero and all-one assignments are disregarded. We define $\operatorname{VCSP}(\Gamma)$, tractability, and intractability in the surjective setting analogously with regular VCSPs, and refer to them as $\operatorname{VCSP_{s}}(\Gamma)$, s-tractability, and s-intractability.

For an example of a surjective VCSP, consider the (global) Min-Cut problem [40]. Given a graph $G=(V,E)$ and edge weights $w:E\to\mathbb{Q}_{>0}$, the goal is to find a set $C\subseteq V$ with $\emptyset\subsetneq C\subsetneq V$ that minimises

[TABLE]

Again, this problem can be expressed over a Boolean domain $D=\{0,1\}$. We define a weighted relation $\gamma:D^{2}\to\overline{\mathbb{Q}}$ by $\gamma(x,y)=0$ if $x=y$ and $\gamma(x,y)=1$ if $x\neq y$. Then the problem of finding an optimal Min-Cut in a graph $G=(V,E)$ is equivalent to solving an instance $I=(V,D,\phi_{I})$ of $\operatorname{VCSP_{s}}(\{\gamma\})$ such that

[TABLE]

Note that the two non-surjective assignments to $I$ correspond to sets $\emptyset$ and $V$, which are not admissible solutions to the Min-Cut problem. Since every instance of $\operatorname{VCSP_{s}}(\{\gamma\})$ can be straightforwardly translated to a Min-Cut instance, and the Min-Cut problem is solvable in polynomial time (say, by a reduction to the $(s,t)$-Min-Cut problem, though other algorithms exist [41]), language $\{\gamma\}$ is s-tractable.

The computational complexity of $\operatorname{VCSP}(\Gamma)$ and $\operatorname{VCSP_{s}}(\Gamma)$ is closely related. Namely, $\operatorname{VCSP}(\Gamma)$ is polynomial-time reducible to $\operatorname{VCSP_{s}}(\Gamma)$ (see Lemma 5), i.e., any intractable language is also s-intractable. Let $\mathcal{C}_{D}=\{\rho_{d}~{}|~{}d\in D\}$, where we define $\rho_{d}:D\to\overline{\mathbb{Q}}$ by $\rho_{d}(x)=0$ if $x=d$ and $\rho_{d}(x)=\infty$ if $x\neq d$; these unary weighted relations are called constants. Conversely, $\operatorname{VCSP_{s}}(\Gamma)$ is polynomial-time reducible to $\operatorname{VCSP}(\Gamma\cup\mathcal{C}_{D})$ (see Lemma 6), i.e., any tractable language containing constants $\mathcal{C}_{D}$ is also s-tractable. In the case of Boolean $\{0,\infty\}$-valued languages, Schaefer’s dichotomy involves six tractable classes. Four of them include constants, and hence they are s-tractable. Creignou and Hébrard [14] showed that the remaining two classes ([math]-valid and $1$-valid222A $\{0,\infty\}$-valued weighted relation is [math]-valid ($1$-valid) if it assigns value [math] to the all-zero (all-one) tuple.) are s-intractable, thus obtaining a dichotomy classification of Boolean $\{0,\infty\}$-valued languages in the surjective setting.

Contributions

Complexity classification

As our main contribution, we establish a dichotomy classification of all Boolean ($\overline{\mathbb{Q}}$-valued) languages in the surjective setting, which extends the classification from [14]. Let $D=\{0,1\}$. Six of the eight tractable classes of Boolean languages identified by Cohen et al. [12] include constants $\mathcal{C}_{D}$, and thus are also s-tractable. We show that languages in the remaining two classes ([math]-optimal and $1$-optimal333A weighted relation is [math]-optimal ($1$-optimal) if the all-zero (all-one) tuple minimises it.) are s-tractable if, for every weighted relation, the set of feasible tuples and the set of optimal tuples are essentially downsets (in the [math]-optimal case; see Definition 27) or essentially upsets (in the $1$-optimal case); otherwise, they are s-intractable.

Somewhat surprisingly, such languages are s-tractable regardless of the remaining (i.e., finite but non-optimal) values. Those values must, however, bear on the time bound of any polynomial-time algorithm solving surjective VCSPs over such languages (unless P = NP). In particular, we give an example of an infinite language $\Gamma$ that is s-tractable (i.e., $\operatorname{VCSP_{s}}(\Gamma^{\prime})$ can be solved in polynomial time for every finite $\Gamma^{\prime}\subseteq\Gamma$) but $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard (see Example 32). This is quite unusual; all known tractable classes of VCSPs are in fact globally tractable, which means that $\operatorname{VCSP}(\Gamma^{\prime})$ is solvable by the same polynomial-time algorithm for every finite subset $\Gamma^{\prime}$ of a tractable language $\Gamma$, and hence $\operatorname{VCSP}(\Gamma)$ is also polynomial-time solvable [7]. To capture this distinction, our main result (Theorem 19) gives a classification in terms of global s-tractability,444Weighted relations in an instance are assumed to be represented explicitly (by tables of values). We only consider languages of bounded arity; this restriction is vital in some of our proofs. Also, unbounded arity presents new challenges to complexity classification. For example, explicitly representing a weighted relation of an arity that is super-logarithmic in the number of variables requires super-polynomial space. from which a classification for s-tractability easily follows (see Remark 20). We call the condition that describes the borderline of global s-tractability in the [math]-optimal case EDS (see Definition 18), drawing a parallel to the corresponding condition for s-tractability, which involves essentially downsets. The $1$-optimal case is analogous (one only needs to exchange the roles of labels [math] and $1$).

Tractability

While [math]-optimal and $1$-optimal languages are trivially tractable for VCSPs, the algorithm for surjective VCSPs over the newly identified class of languages is nontrivial and constitutes our second main contribution. The global s-tractability part of our result is established by a reduction from $\overline{\mathbb{Q}}$-valued $\operatorname{VCSP_{s}}$ to the generalised Min-Cut problem (defined in Section 5), for which we require to find all $\alpha$-optimal solutions in polynomial time, where $\alpha$ is a constant depending on the valued constraint language. The generalised Min-Cut problem consists in minimising an objective function $f+g$, where $f$ is a superadditive set function given by an oracle and $g$ is a cut function (same as in the Min-Cut problem); see Section 5 for the details. We prove that the running time of our algorithm is roughly $O\left(n^{20\alpha}\right)$, thus improving on the bound of $O\left(n^{3^{3\alpha}}\right)$ established in [43] (one of the two extended conference abstracts of this paper) for the special case of $\{0,1\}$-valued languages.

Hardness

The hardness part of our result is proved by analysing weighted relations that can be obtained from a language using gadgets that preserve (global) s-tractability. Since not all standard gadgets have this property (in particular, minimisation over a variable may affect the surjectivity of a solution), we cannot employ the algebraic approach [11]. Instead, we define a collection of operations that form building blocks of gadgets preserving tractability in the surjective setting (see Definition 9 and Lemma 10). Such gadgets apply to non-Boolean domains as well, and may be useful in future work on non-Boolean surjective VCSPs. Another important ingredient of our proof is the NP-hardness of the Minimum Distance problem [45], which to the best of our knowledge has not previously appeared in the literature on exact solvability of (V)CSPs.

Approximability

By a simple reduction, our main result implies a complexity classification of the approximability of maximising $\mathbb{Q}_{\geq 0}$-valued surjective VCSPs (see Theorem 36).

Enumeration

For the globally s-tractable languages, we also show that all optimal solutions can be enumerated with polynomial delay [44] (see Theorem 22). While this is an easy observation for the already known globally s-tractable languages (since constants $\mathcal{C}_{D}$ allow for a standard self-reduction technique), we prove the same result for the newly discovered classes of languages, which do not include constants $\mathcal{C}_{D}$.

Related work

Recent years have seen some remarkable progress on the computational complexity of CSPs and VCSPs parametrised by the (valued) constraint language. We highlight the resolution of the “bounded width conjecture” [2] and the result that a dichotomy for CSPs, conjectured in [18] and recently established by two independent proofs [6, 48], implies a dichotomy for VCSPs [32, 31]. All this work is for arbitrary (i.e., not necessarily Boolean) finite domains and relies on the algebraic approach initiated in [7] and nicely described in a survey [3].

One of the important aspects of the algebraic approach is the assumption that constants $\mathcal{C}_{D}$ are present in (valued) constraint languages. (This is without loss of generality with respect to polynomial-time solvability.) In the surjective setting, it is the lack of constants that makes it difficult, if not impossible, to employ the algebraic approach. Chen made the first step in this direction [9] but it is not clear how to take his result (for CSPs) further.

For a binary (unweighted) relation $\gamma$, $\operatorname{VCSP_{s}}(\{\gamma\})$ has been studied under the name of surjective $\gamma$-Colouring [4, 25, 24, 35] and vertex-compaction [47]. We remark that our notion of surjectivity is global. For the $\gamma$-Colouring problem, a local version of surjectivity has also been studied [20, 19]. This version corresponds to finding a graph homomorphism such that the neighbourhood of every vertex $v$ is mapped surjectively onto the neighbourhood of the image of $v$.

Under the assumption of the unique games conjecture [30], Raghavendra has shown that the optimal approximation ratio for maximising $\mathbb{Q}_{\geq 0}$-valued VCSPs is achieved by the basic semidefinite programming relaxation [37, 38].

Bach and Zhou have shown that any Max-CSP that is solvable in polynomial time in the non-surjective setting admits a PTAS in the surjective setting, and that any Max-CSP that is APX-hard in the non-surjective setting remains APX-hard in the surjective setting [1].

2 Preliminaries

2.1 Weighted relations and VCSPs

We work in the arithmetic model of computation, i.e., every number is represented in constant space, and basic arithmetic operations take constant time. Let $\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}$ denote the set of extended rationals. For any $c\in\overline{\mathbb{Q}}$, we define $c\leq\infty$ and $\infty+c=c+\infty=\infty$. If $c\geq 0$, we define $c\cdot\infty=\infty\cdot c=\infty$. We leave the result of multiplying $\infty$ undefined for $c<0$.

For any integer $n\geq 1$, let $[n]=\{1,\dots,n\}$.

Definition 1**.**

Let $r\geq 1$ be an integer. An $r$-ary weighted relation over $D$ is a mapping $\gamma:D^{r}\to\overline{\mathbb{Q}}$; the arity of $\gamma$ equals $\operatorname{ar}(\gamma)=r$. We denote by $\operatorname{Feas}(\gamma)$ the underlying feasibility relation of $\gamma$, i.e.

[TABLE]

We denote by $\operatorname{Opt}(\gamma)$ the relation consisting of the minimal-valued tuples, i.e.

[TABLE]

A weighted relation $\gamma$ is called crisp if $\operatorname{Feas}(\gamma)=\operatorname{Opt}(\gamma)$. In other words, there exists a constant $c\in\mathbb{Q}$ such that $\gamma(\mathbf{x})=c$ for all $\mathbf{x}\in\operatorname{Feas}(\gamma)$ and $\gamma(\mathbf{x})=\infty$ for all $\mathbf{x}\in D^{r}\setminus\operatorname{Feas}(\gamma)$.

Weighted relations that differ only by a constant are considered equivalent, as adding a rational constant to a weighted relation changes the value of every solution to the $\operatorname{VCSP}$ by the same amount. Therefore, a crisp weighted relation $\gamma$ can be equated with the relation $\operatorname{Feas}(\gamma)$. Conversely, a relation $\rho$ can be seen as a crisp weighted relation $\gamma_{c}$ with $\operatorname{Feas}(\gamma_{c})=\rho$ and the codomain equal to $\{c,\infty\}$ for some $c\in\mathbb{Q}$. Unless stated otherwise, we choose $c=0$.

Definition 2**.**

We denote by $\rho_{=}$ the binary equality relation $\{(d,d)~{}|~{}d\in D\}$. For any $d\in D$, we denote by $\rho_{d}$ the unary relation $\{(d)\}$, which is called a constant. We denote the set of constants on $D$ by $\mathcal{C}_{D}=\{\rho_{d}~{}|~{}d\in D\}$.

For any relation $\rho$, we denote by $\operatorname{Soft}(\rho)$ the soft variant of $\rho$ defined by $\operatorname{Soft}(\rho)(\mathbf{x})=0$ if $\mathbf{x}\in\rho$ and $\operatorname{Soft}(\rho)(\mathbf{x})=1$ otherwise.

Definition 3**.**

A constraint language (or simply a language) over $D$ is a (possibly infinite) set of weighted relations over $D$.

In this paper, we only consider languages of bounded arity. Note that a crisp language is of a bounded arity if and only if it is finite.

Definition 4**.**

A language $\Gamma$ is called s-tractable if, for every finite $\Gamma^{\prime}\subseteq\Gamma$, $\operatorname{VCSP_{s}}(\Gamma^{\prime})$ can be solved in polynomial time. If $\operatorname{VCSP_{s}}(\Gamma)$ can be solved in polynomial time, language $\Gamma$ is called globally s-tractable.

If there exists a finite $\Gamma^{\prime}\subseteq\Gamma$ such that $\operatorname{VCSP_{s}}(\Gamma^{\prime})$ is NP-hard, language $\Gamma$ is called s-intractable. If $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard, language $\Gamma$ is called globally s-intractable.

Note that a globally s-tractable language is s-tractable, and an s-intractable language is globally s-intractable.

Lemmas 5 and 6 establish a relation between the complexity of the $\operatorname{VCSP}$ and $\operatorname{VCSP_{s}}$. We denote by $\leq_{p}$ the standard polynomial-time Turing reduction.

Lemma 5**.**

For any constraint language $\Gamma$,

[TABLE]

Proof.

Given an instance $I$ of $\operatorname{VCSP}(\Gamma)$, we construct an instance $I^{\prime}$ of $\operatorname{VCSP_{s}}(\Gamma)$ by adding $|D|$ extra variables. Any solution to $I$ can be extended to a surjective solution to $I^{\prime}$ of the same value and, conversely, any (surjective) solution to $I^{\prime}$ induces a solution to $I$ of the same value. ∎

Lemma 6**.**

For any constraint language $\Gamma$,

[TABLE]

Proof.

Given an instance $I=(V,D,\phi_{I})$ of $\operatorname{VCSP_{s}}(\Gamma)$, we iterate through all $O\left(|V|^{|D|}\right)$ injective mappings $f:D\to V$. For each mapping $f$, we construct an instance $I^{\prime}_{f}$ of $\operatorname{VCSP}(\Gamma\cup\mathcal{C}_{D})$ by adding constraints $\rho_{d}(f(d))$ for all $d\in D$. The additional constraints guarantee that only surjective solutions to $I^{\prime}_{f}$ are feasible. Conversely, any surjective solution to $I$ is a feasible solution to $I^{\prime}_{f}$ for some mapping $f$. Therefore, a solution of the smallest value among optimal solutions to $I^{\prime}_{f}$ for all $f$ is an optimal surjective solution to $I$. ∎

Corollary 7**.**

Any (globally) tractable language $\Gamma$ with $\mathcal{C}_{D}\subseteq\Gamma$ is also (globally) s-tractable.

Now we define a few operations on weighted relations that occur throughout the paper.

Definition 8**.**

Let $\gamma$ be an $r$-ary weighted relation.

• •

Addition of a rational constant: For any $c\in\mathbb{Q}$, $\gamma+c=\gamma^{\prime}$ such that $\gamma^{\prime}(\mathbf{x})=\gamma(\mathbf{x})+c$.

• •

Non-negative scaling: For any $c\in\mathbb{Q}_{\geq 0}$, $c\cdot\gamma=\gamma^{\prime}$ such that $\gamma^{\prime}(\mathbf{x})=c\cdot\gamma(\mathbf{x})$. Note that $0\cdot\gamma=\operatorname{Feas}(\gamma)$.

• •

Coordinate mapping: For any arity $r^{\prime}$ and mapping $f:[r]\to[r^{\prime}]$, $f(\gamma)=\gamma^{\prime}$ such that $\gamma^{\prime}(x_{1},\dots,x_{r^{\prime}})=\gamma\left(x_{f(1)},\dots,x_{f(r)}\right)$.

• •

Minimisation: For any $i\in[r]$, the minimisation of $\gamma$ at coordinate $i$ results in $\gamma^{\prime}$ such that $\gamma^{\prime}(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{r})=\min_{x_{i}\in D}\gamma(x_{1},\dots,x_{r})$.

• •

Pinning: For any $d\in D$ and $i\in[r]$, the pinning of $\gamma$ to label $d$ at coordinate $i$ results in $\gamma^{\prime}$ such that $\gamma^{\prime}(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{r})=\gamma(x_{1},\dots,x_{i-1},d,x_{i+1},\dots,x_{r})$. A pinning to label $d$ is called a $d$-pinning.

• •

Addition: For any weighted relations $\gamma_{1},\gamma_{2}$ with $\operatorname{ar}(\gamma_{1})=\operatorname{ar}(\gamma_{2})$, $\gamma_{1}+\gamma_{2}=\gamma^{\prime}$ such that $\gamma^{\prime}(\mathbf{x})=\gamma_{1}(\mathbf{x})+\gamma_{2}(\mathbf{x})$.

We extend operations on weighted relations to languages in the natural way, e.g., $\operatorname{Feas}(\Gamma)=\{\operatorname{Feas}(\gamma)~{}|~{}\gamma\in\Gamma\}$.

A weighted relational clone [11] is a language closed under certain operations (e.g., non-negative scaling and minimisation) that preserve the tractability of languages in the following sense: The VCSP over the smallest weighted relational clone containing a language $\Gamma$ can be reduced in polynomial time to $\operatorname{VCSP}(\Gamma)$. Weighted relational clones are characterised by their weighted polymorphisms (a generalisation of multimorphisms defined in Definition 12), which enables the employment of tools from universal algebra in the effort to obtain a complexity classification of languages.

In the surjective setting, however, minimisation may not preserve the tractability of languages, and thus we need to define a language closure that excludes this operation. Consequently, we are unable to use the algebraic approach in our proofs in Section 4.

Definition 9**.**

A constraint language $\Gamma$ is called closed if it is closed under addition, coordinate mapping, non-negative scaling, addition of a rational constant, operation $\operatorname{Opt}$, and, for all $d\in D$ such that $\rho_{d}\in\Gamma$, $d$-pinning.

We define $\Gamma^{*}$ to be the smallest closed language containing $\Gamma$.

Now we show that these closure operations preserve the complexity of the $\operatorname{VCSP_{s}}$. Note that we require a language to be closed under $d$-pinning only if it contains $\rho_{d}$.

Lemma 10**.**

For any constraint language $\Gamma$,

[TABLE]

Proof.

For most of the closure operations, standard reductions for the $\operatorname{VCSP}$ apply to the surjective setting as well. Let $\gamma_{1},\gamma_{2}\in\Gamma$ be weighted relations with $\operatorname{ar}(\gamma_{1})=\operatorname{ar}(\gamma_{2})$, and let $\gamma^{\prime}=\gamma_{1}+\gamma_{2}$. Then $\operatorname{VCSP_{s}}(\Gamma\cup\{\gamma^{\prime}\})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$, as any constraint of the form $w\cdot\gamma^{\prime}(\mathbf{x})$ can be replaced with a pair of constraints $w\cdot\gamma_{1}(\mathbf{x})$, $w\cdot\gamma_{2}(\mathbf{x})$. Similarly, let $\gamma\in\Gamma$ and $\gamma^{\prime}=f(\gamma)$ where $f:[\operatorname{ar}(\gamma)]\to[\operatorname{ar}(\gamma^{\prime})]$; then any constraint of the form $w\cdot\gamma^{\prime}\left(x_{1},\dots,x_{\operatorname{ar}(\gamma^{\prime})}\right)$ can be replaced with a constraint $w\cdot\gamma\left(x_{f(1)},\dots,x_{f(\operatorname{ar}(\gamma))}\right)$. Non-negative scaling can be achieved by scaling the weight of affected constraints. Addition of a rational constant changes the value of every solution by the same amount, and thus it can be ignored.

Now we show that $\operatorname{VCSP_{s}}(\Gamma\cup\{\operatorname{Opt}(\gamma)\})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$ for any $\gamma\in\Gamma$. Let $I$ be an instance of $\operatorname{VCSP_{s}}(\Gamma\cup\{\operatorname{Opt}(\gamma)\})$. Without loss of generality, assume that the minimum values assigned by $\gamma$ and $\operatorname{Opt}(\gamma)$ equal [math] and all weighted relations in $I$ assign non-negative values (this can be achieved by adding rational constants). We may also assume that $\gamma$ is not crisp (otherwise $\operatorname{Opt}(\gamma)=\gamma$). Let $m$ denote the smallest positive value assigned by $\gamma$, and let $M$ be an upper bound on the value of any feasible solution to $I$ (e.g., the weighted sum of the maximum finite values assigned by the constraints of $I$). We replace every constraint of the form $w\cdot\operatorname{Opt}(\gamma)(\mathbf{x})$ in $I$ with a constraint $(M/m+1)\cdot\gamma(\mathbf{x})$ to obtain an instance $I^{\prime}\in\operatorname{VCSP_{s}}(\Gamma)$. Any feasible solution to instance $I$ gets assigned the same value by $I^{\prime}$. Any infeasible solution to instance $I$ is either infeasible for $I^{\prime}$ as well, or it incurs an infinite value from a constraint of the form $w\cdot\operatorname{Opt}(\gamma)(\mathbf{x})$ in $I$ and thus a value of at least $(M/m+1)\cdot m>M$ in $I^{\prime}$. Therefore, an optimal solution to $I^{\prime}$ is optimal for $I$ as well.

In the case of pinning, we need a different reduction as the standard one relies on minimisation. Suppose that $\rho_{d}\in\Gamma$. Let $\gamma^{\prime}$ be a $d$-pinning of a weighted relation $\gamma\in\Gamma$; without loss of generality, let it be a pinning at the first coordinate. We show that $\operatorname{VCSP_{s}}(\Gamma\cup\{\gamma^{\prime}\})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$. Let $I=(V,D,\phi_{I})$ be an instance of $\operatorname{VCSP_{s}}(\Gamma\cup\{\gamma^{\prime}\})$ with $V=\{x_{1},\dots,x_{n}\}$. In a surjective solution to $I$, at least one variable is assigned label $d$, but we do not a priori know which one. For every $i\in[n]$, we construct an instance $I_{i}=(V,D,\phi_{I_{i}})$ of $\operatorname{VCSP_{s}}(\Gamma)$ by replacing all constraints of the form $\gamma^{\prime}(\mathbf{x})$ with $\gamma(x_{i},\mathbf{x})$ and adding a constraint $\rho_{d}(x_{i})$ to force variable $x_{i}$ to take label $d$. A solution of the smallest value among optimal solutions to $I_{1},\dots,I_{n}$ is an optimal solution to $I$. ∎

2.2 Polymorphisms and multimorphisms

For any $r\geq 1$ and a $k$-ary operation $h:D^{k}\to D$, we extend $h$ to $r$-tuples over $D$ by applying it componentwise. Namely, for $\mathbf{x}_{1},\dots,\mathbf{x}_{k}\in D^{r}$ where $\mathbf{x}_{i}=(x_{i,1},\dots,x_{i,r})$, we define $h(\mathbf{x}_{1},\dots,\mathbf{x}_{k})\in D^{r}$ by

[TABLE]

The following notion is at the heart of the algebraic approach to decision CSPs [7].

Definition 11**.**

Let $\gamma$ be a weighted relation on $D$. A $k$-ary operation $h:D^{k}\to D$ is a polymorphism of $\gamma$ (and $\gamma$ is invariant under or admits $h$) if, for every $\mathbf{x}_{1},\ldots,\mathbf{x}_{k}\in\operatorname{Feas}(\gamma)$, we have $h(\mathbf{x}_{1},\ldots,\mathbf{x}_{k})\in\operatorname{Feas}(\gamma)$. We say that $h$ is a polymorphism of a language $\Gamma$ if it is a polymorphism of every $\gamma\in\Gamma$.

The following notion, which involves a collection of $k$ $k$-ary polymorphisms, plays an important role in the complexity classification of Boolean valued constraint languages [12], as we will see in Theorem 14 in Section 2.3.

Definition 12**.**

Let $\gamma$ be a weighted relation on $D$. A list $\langle h_{1},\ldots,h_{k}\rangle$ of $k$-ary polymorphisms of $\gamma$ is a $k$-ary multimorphism of $\gamma$ (and $\gamma$ admits $\langle h_{1},\ldots,h_{k}\rangle$) if, for every $\mathbf{x}_{1},\ldots,\mathbf{x}_{k}\in\operatorname{Feas}(\gamma)$, we have

[TABLE]

$\langle h_{1},\ldots,h_{k}\rangle$ is a multimorphism of a language $\Gamma$ if it is a multimorphism of every $\gamma\in\Gamma$.

The operations in Definition 9 preserve multimorphisms [12, 22], i.e., any multimorphism of a language $\Gamma$ is also a multimorphism of $\Gamma^{*}$. Consequently, all polymorphisms of a crisp weighted relation are preserved.

2.3 Boolean VCSPs

In the rest of the paper, we consider only Boolean languages (i.e., $D=\{0,1\}$), unless explicitly mentioned otherwise. For any arity $r\geq 1$, we denote by $\mathbf{0}^{r}$ ($\mathbf{1}^{r}$) the zero (one) $r$-tuple. For $r$-tuples $\mathbf{x}=(x_{1},\dots,x_{r})$ and $\mathbf{y}=(y_{1},\dots,y_{r})\in D^{r}$, we define $\mathbf{x}\leq\mathbf{y}$ if and only if $x_{i}\leq y_{i}$ for all $i\in[r]$ (where $0<1$). We also define the following operations on $D$:

• •

For any $a\in D$, $c_{a}$ is the constant unary operation such that $c_{a}(x)=a$ for all $x\in D$.

• •

Operation $\neg$ is the unary negation, i.e. $\neg(0)=1$ and $\neg(1)=0$. For a weighted relation $\gamma$, we define $\neg(\gamma)$ to be the weighted relation $\neg(\gamma)(\mathbf{x})=\gamma(\neg(\mathbf{x}))$. For a language $\Gamma$, we define $\neg(\Gamma)=\{\neg(\gamma)~{}|~{}\gamma\in\Gamma\}$. Note that $\neg(\Gamma)$ can be obtained from $\Gamma$ simply by exchanging the labels $\{0,1\}$, and hence has the same complexity as $\Gamma$.

• •

Binary operation $\oplus$ is the addition modulo $2$ operation. In this case, we use the infix notation, i.e., $0\oplus 0=0=1\oplus 1$ and $0\oplus 1=1=1\oplus 0$.

• •

Binary operation $\min$ ($\max$) returns the smaller (larger) of its two arguments with respect to the order $0<1$.

• •

Binary operation $\operatorname{sub}$ (for subtraction) is defined as $\operatorname{sub}(x,y)=\min(x,\neg y)$.

• •

Ternary operation $\operatorname{Mn}$ (for minority) is the unique ternary operation on $D$ satisfying $\operatorname{Mn}(x,x,y)=\operatorname{Mn}(x,y,x)=\operatorname{Mn}(y,x,x)=y$ for all $x,y\in D$.

• •

Ternary operation $\operatorname{Mj}$ (for majority) is the unique ternary operation on $D$ satisfying $\operatorname{Mj}(x,x,y)=\operatorname{Mj}(x,y,x)=\operatorname{Mj}(y,x,x)=x$ for all $x,y\in D$.

Lemma 13**.**

If a weighted relation admits polymorphism $\operatorname{sub}$, then it also admits polymorphisms $c_{0}$ and $\min$.

Proof.

For every $x,y\in D$, it holds $c_{0}(x)=0=\operatorname{sub}(x,x)$ and $\min(x,y)=\operatorname{sub}(x,\operatorname{sub}(x,y))$. ∎

Cohen et al. [12] established a complexity classification of Boolean constraint languages.

Theorem 14** ([12, Theorem 7.1]).**

Let $\Gamma$ be a Boolean $\overline{\mathbb{Q}}$-valued language. Then $\Gamma$ is tractable if it admits any the following multimorphisms: $\langle c_{0}\rangle$, $\langle c_{1}\rangle$, $\langle\min,\min\rangle$, $\langle\max,\max\rangle$, $\langle\min,\max\rangle$, $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$. Otherwise, $\Gamma$ is intractable.

Note that multimorphism $\langle\min,\max\rangle$ corresponds to submodularity [40]. Constants $\mathcal{C}_{D}=\{\rho_{0},\rho_{1}\}$ admit multimorphisms $\langle\min,\min\rangle$, $\langle\max,\max\rangle$, $\langle\min,\max\rangle$, $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$; hence, these six classes of languages are s-tractable by Lemma 6. However, $\rho_{0}$ does not admit $\langle c_{1}\rangle$ and $\rho_{1}$ does not admit $\langle c_{0}\rangle$.

Remark 15**.**

Although Theorem 14 is stated only for the weaker notion of tractability (i.e., for finite languages) in [12], the proofs there actually establish the same classification for the stronger notion of global tractability as well.

In particular, all the tractable classes (characterised by the eight multimorphisms) are globally tractable. Conversely, any globally intractable language is also intractable: If a language $\Gamma$ does not admit any of the eight multimorphisms, then there exists a finite subset $\Gamma^{\prime}\subseteq\Gamma$ with $|\Gamma^{\prime}|\leq 8$ that does not admit any of the eight multimorphisms (since a single weighted relation suffices to violate a multimorphism).

We note that Theorem 14 is a generalisation of Schaefer’s classification of $\{0,\infty\}$-valued constraint languages [39] and Creignou’s classification of $\{0,1\}$-valued constraint languages [13]. In particular, Theorem 14 implies the following classification of $\mathbb{Q}$-valued languages.

Theorem 16** ([12, Corollary 7.11]).**

Let $\Gamma$ be a Boolean $\mathbb{Q}$-valued language. Then $\Gamma$ is tractable if it admits any of the following multimorphisms: $\langle c_{0}\rangle$, $\langle c_{1}\rangle$, $\langle\min,\max\rangle$. Otherwise, $\Gamma$ is intractable.

Creignou and Hébrard [14] established a complexity classification of Boolean $\{0,\infty\}$-valued languages in the surjective setting.

Theorem 17** ([14]).**

Let $\Gamma$ be a Boolean $\{0,\infty\}$-valued language. Then $\Gamma$ is s-tractable if it is invariant under any of the following operations: $\min$, $\max$, $\operatorname{Mn}$, $\operatorname{Mj}$. Otherwise, $\Gamma$ is s-intractable.

3 Results

We present our results in three parts: Section 3.1 defines the EDS property and states the main classification theorem, Section 3.2 focuses on finite EDS languages, and Section 3.3 gives a classification in terms of approximability for the surjective $\operatorname{Max-VCSP}$.

3.1 Boolean surjective VCSPs

We first define the property EDS (which stands for essentially a downset, see Definition 27) characterising the newly discovered tractable class of weighted relations.

Definition 18**.**

For any $\alpha\geq 1$, an $r$-ary weighted relation $\gamma$ is $\alpha$-EDS if, for every $\mathbf{x},\mathbf{y}\in\operatorname{Feas}(\gamma)$, it holds $\mathbf{0}^{r}\in\operatorname{Feas}(\gamma)$ and

[TABLE]

A weighted relation is EDS if it is $\alpha$-EDS for some $\alpha\geq 1$. A language is EDS if there exists $\alpha\geq 1$ such that every weighted relation in the language is $\alpha$-EDS.

Although this definition does not involve the notion of polymorphisms, it is stated in a similar vein. Let $h$ be a binary operation defined by $h(x,y)=0$; then the requirement “for every $\mathbf{x},\mathbf{y}\in\operatorname{Feas}(\gamma)$, it holds $\mathbf{0}^{r}\in\operatorname{Feas}(\gamma)$” translates to “$\gamma$ is invariant under $h$” (or, equivalently, “$\gamma$ is invariant under $c_{0}$”).555In fact, any EDS weighted relation admits multimorphism $\langle c_{0}\rangle$ (see Lemma 24).666Note that the unary empty relation $\rho_{\emptyset}$ is vacuously $\alpha$-EDS for all $\alpha\geq 1$, as $\operatorname{Feas}(\rho_{\emptyset})=\emptyset$. In the case of $\alpha=1$, inequality (12) translates to that of admitting multimorphism $\langle\operatorname{sub},h\rangle$. For more intuition behind this notion in the general case, see the corresponding definition of EDS for set functions (Definition 62) in Section 5.3. Finite EDS languages admit a simpler equivalent definition, see Corollary 30.

The following classification of $\overline{\mathbb{Q}}$-valued languages is our main result.

Theorem 19**.**

Let $\Gamma$ be a Boolean $\overline{\mathbb{Q}}$-valued language. Then $\Gamma$ is globally s-tractable if it is EDS, or $\neg(\Gamma)$ is EDS, or $\Gamma$ admits any of the following multimorphisms: $\langle\min,\min\rangle$, $\langle\max,\max\rangle$, $\langle\min,\max\rangle$, $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$. Otherwise, $\Gamma$ is globally s-intractable.

Proof.

The global s-tractability of languages admitting any of the six multimorphisms in the statement of the theorem follows from Theorem 14 (see Remark 15) by Lemma 6. The global s-tractability of EDS languages (whether $\Gamma$ or $\neg(\Gamma)$, which is symmetric) follows from Theorem 68, proved in Section 5. Finally, the global s-intractability of the remaining languages follows from Theorem 48, proved in Section 4. ∎

Remark 20**.**

Theorem 19 gives us also a classification in terms of s-tractability. As noted in Section 2.1, any globally s-tractable language is s-tractable. Consider now a globally s-intractable language $\Gamma$. It does not admit any of the six multimorphisms, and hence there exists a finite subset of $\Gamma$ that does not admit them either (see Remark 15). If there exists a finite subset $\Gamma^{\prime}\subseteq\Gamma$ such that neither $\Gamma^{\prime}$ nor $\neg(\Gamma^{\prime})$ is EDS, then $\Gamma$ is s-intractable; otherwise $\Gamma$ is s-tractable. Equivalently (by Corollary 30), $\Gamma$ is s-intractable if neither $\operatorname{Feas}(\Gamma)\cup\operatorname{Opt}(\Gamma)$ nor $\operatorname{Feas}(\neg(\Gamma))\cup\operatorname{Opt}(\neg(\Gamma))$ admit polymorphism $\operatorname{sub}$, and it is s-tractable otherwise.

To see how EDS languages fit into the classification of $\{0,\infty\}$-valued languages established in Theorem 17, note the following. Any $\{0,\infty\}$-valued language of bounded arity is finite. By Corollary 30, any EDS $\{0,\infty\}$-valued language admits polymorphism $\operatorname{sub}$, and hence also polymorphism $\min$ (by Lemma 13).

For $\mathbb{Q}$-valued languages, Theorem 19 gives a tighter classification: the only reasons for global s-tractability are EDS and submodularity.

Theorem 21**.**

Let $\Gamma$ be a Boolean $\mathbb{Q}$-valued language. Then $\Gamma$ is globally s-tractable if it is EDS, or $\neg(\Gamma)$ is EDS, or $\Gamma$ admits the $\langle\min,\max\rangle$ multimorphism. Otherwise, $\Gamma$ is globally s-intractable.

Proof.

We need to show that in the case of $\mathbb{Q}$-valued languages, the remaining globally s-tractable classes from Theorem 19 (which are characterised by polymorphisms $\langle\min,\min\rangle$, $\langle\max,\max\rangle$, $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, and $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$) collapse.

If a $\mathbb{Q}$-valued $r$-ary weighted relation $\gamma$ admits the $\langle\min,\min\rangle$ multimorphism, then it holds $\gamma(\mathbf{x})\geq\gamma(\mathbf{y})$ for all $\mathbf{x}\geq\mathbf{y}$. This implies that, for all $\mathbf{x},\mathbf{y}\in\operatorname{Feas}(\gamma)$, it holds $\gamma(\mathbf{x})\geq\gamma(\operatorname{sub}(\mathbf{x},\mathbf{y}))$ and $\gamma(\mathbf{y})\geq\gamma(\mathbf{0}^{r})$. Hence, $\gamma$ is $1$-EDS. If $\gamma$ admits the $\langle\max,\max\rangle$ multimorphism, then $\neg(\gamma)$ admits the $\langle\min,\min\rangle$ multimorphism. Therefore, if a $\mathbb{Q}$-valued language $\Gamma$ admits $\langle\min,\min\rangle$ or $\langle\max,\max\rangle$ as a multimorphism, then $\Gamma$ or $\neg(\Gamma)$ is EDS.

Multimorphisms $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, and $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$ are covered by the $\langle\min,\max\rangle$ multimorphism: Weighted relations that admit $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$ or $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$ as a multimorphism are crisp [12, Propositions 6.20 and 6.22], and hence, in the $\mathbb{Q}$-valued case, they are constant functions. $\mathbb{Q}$-valued weighted relations that admit the $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$ multimorphism are modular [12, Corollary 6.26], and hence they are submodular. ∎

Enumerating all optimal solutions to an instance with polynomial delay is a fundamental problem [27, 46] studied in the context of CSP [16, 8]. An algorithm outputting a sequence of solutions works with polynomial delay if the time it takes to output the first solution as well as the time it takes between every two consecutive solutions is bounded by a polynomial in the input size.

It is known that, for a tractable constraint language $\Gamma$ that includes constants $\mathcal{C}_{D}$, one can enumerate all optimal solutions with polynomial delay [10]. Our results imply that the newly discovered globally s-tractable EDS languages enjoy the same property (despite not including constants).

Theorem 22**.**

Let $\Gamma$ be a Boolean $\overline{\mathbb{Q}}$-valued language. If $\Gamma$ is globally s-tractable then there is a polynomial-delay algorithm that enumerates all optimal solutions to any instance of $\operatorname{VCSP_{s}}(\Gamma)$.

The theorem is proved in Section 5.3.

3.2 Finite EDS languages

The EDS property can be described in a simpler way for languages of finite size; see the following observation and Corollary 30.

Observation 23**.**

A language of finite size is EDS if and only if it consists of EDS weighted relations.

In the following we prove several useful properties EDS weighted relations.

Lemma 24**.**

Any EDS weighted relation admits multimorphism $\langle c_{0}\rangle$.

Proof.

Let $\gamma$ be an $r$-ary $\alpha$-EDS weighted relation. For any $\mathbf{x}\in\operatorname{Feas}(\gamma)$, it holds $\mathbf{0}^{r}\in\operatorname{Feas}(\gamma)$ and

[TABLE]

as $\operatorname{sub}(\mathbf{x},\mathbf{x})=\mathbf{0}^{r}$, and therefore $\gamma(\mathbf{x})\geq\gamma(\mathbf{0}^{r})$. ∎

Lemma 25**.**

A crisp weighted relation is EDS if and only if it admits polymorphism $\operatorname{sub}$.

Proof.

Any EDS weighted relation admits polymorphism $\operatorname{sub}$. For the converse implication, note that any crisp weighted relation that admits polymorphism $\operatorname{sub}$ (and thus, by Lemma 13, also polymorphism $c_{0}$) satisfies (12) for any $\alpha\geq 1$. ∎

Lemma 26**.**

A weighted relation $\gamma$ is EDS if and only if both $\operatorname{Feas}(\gamma)$ and $\operatorname{Opt}(\gamma)$ are EDS.

Proof.

Let $\gamma$ be an $r$-ary $\alpha$-EDS weighted relation. For any $\mathbf{x},\mathbf{y}\in\operatorname{Feas}(\gamma)$, it holds $\mathbf{0}^{r}\in\operatorname{Feas}(\gamma)$ and

[TABLE]

and hence $\operatorname{sub}(\mathbf{x},\mathbf{y})\in\operatorname{Feas}(\gamma)$. By Lemma 25, $\operatorname{Feas}(\gamma)$ is EDS. Similarly, for any $\mathbf{x},\mathbf{y}\in\operatorname{Opt}(\gamma)$, it holds $\mathbf{0}^{r}\in\operatorname{Opt}(\gamma)$ (by Lemma 24) and

[TABLE]

therefore $\operatorname{sub}(\mathbf{x},\mathbf{y})\in\operatorname{Opt}(\gamma)$ and $\operatorname{Opt}(\gamma)$ is EDS.

To prove the converse implication, let us assume that $\operatorname{Feas}(\gamma)$, $\operatorname{Opt}(\gamma)$ are EDS and consider any $\mathbf{x},\mathbf{y}\in\operatorname{Feas}(\gamma)$. As $\operatorname{Opt}(\gamma)$ admits polymorphism $c_{0}$, it holds $\mathbf{0}^{r}\in\operatorname{Opt}(\gamma)\subseteq\operatorname{Feas}(\gamma)$. Therefore, the left-hand side of (12) is non-negative. Moreover, if it equals [math], then $\mathbf{x},\mathbf{y}\in\operatorname{Opt}(\gamma)$, and hence $\operatorname{sub}(\mathbf{x},\mathbf{y})\in\operatorname{Opt}(\gamma)$ and the right-hand side equals [math] as well. Therefore, (12) holds for large enough $\alpha$, as there are only finitely many choices of $\mathbf{x},\mathbf{y}\in\operatorname{Feas}(\gamma)$. ∎

We show that relations invariant under $\operatorname{sub}$ have a simple structure.

Definition 27**.**

An $r$-ary relation $\rho$ is a downset if, for any $r$-tuples $\mathbf{x},\mathbf{y}$ such that $\mathbf{x}\geq\mathbf{y}$ and $\mathbf{x}\in\rho$, it holds $\mathbf{y}\in\rho$.

An $r$-ary relation $\rho$ is essentially a downset if it can be written as a conjunction of a downset and binary equality relations. Formally, there exists a downset $\rho^{\prime}$ with $\operatorname{ar}(\rho^{\prime})=r^{\prime}\leq r$, a permutation $\pi$ of $[r]$, and indices $a_{r^{\prime}+1},\dots,a_{r}\in\{\pi(1),\dots,\pi(r^{\prime})\}$ such that

[TABLE]

(Note that addition of crisp weighted relations corresponds to conjunction.) In other words, removing duplicate coordinates777A coordinate $i$ is a duplicate of a coordinate $j$ if, for every $(x_{1},\dots,x_{r})\in\rho$, it holds $x_{i}=x_{j}$. of $\rho$ results in a downset.

Example 28**.**

Relation $\rho^{\prime}=\{(0,0),(0,1),(1,0)\}$ is a downset, while $\rho=\{(0,0,0),(0,1,1),(1,0,0)\}$ is only essentially a downset (as $\rho(x,y,z)=\rho^{\prime}(x,y)+\rho_{=}(y,z)$).

Lemma 29**.**

A relation is essentially a downset if and only if it admits polymorphism $\operatorname{sub}$.

Proof.

For any $r$-ary relation $\rho$ that is essentially a downset and $\mathbf{x},\mathbf{y}\in\rho$, we prove that $\mathbf{z}=\operatorname{sub}(\mathbf{x},\mathbf{y})\in\rho$. Let $\mathbf{x}=(x_{1},\dots,x_{r})$, $\mathbf{y}=(y_{1},\dots,y_{r})$, $\mathbf{z}=(z_{1},\dots,z_{r})$. It holds $\mathbf{x}\geq\mathbf{z}$. Moreover, for any coordinates $i,j$ such that $x_{i}=x_{j}$ and $y_{i}=y_{j}$, it holds $z_{i}=z_{j}$. Since $\rho$ can be written as a sum of a downset and equality relations, we have $\mathbf{z}\in\rho$.

We prove the converse implication by contradiction. Suppose that $\rho$ is a smallest-arity relation that admits polymorphism $\operatorname{sub}$ but is not essentially a downset; let us denote its arity by $r$. If there are distinct coordinates $i,j$ such that $z_{i}=z_{j}$ for all $\mathbf{z}=(z_{1},\dots,z_{r})\in\rho$, identifying these coordinates yields an $(r-1)$-ary relation $\rho^{\prime}$ such that $\rho$ can be written as the sum of $\rho^{\prime}$ and a binary equality relation. However, $\rho^{\prime}$ also admits $\operatorname{sub}$, and hence is essentially a downset by the choice of $\rho$, which implies that $\rho$ is essentially a downset as well. Therefore, for any distinct coordinates $i,j$, there exists $\mathbf{z}^{(i,j)}\in\rho$ with $z^{(i,j)}_{i}\neq z^{(i,j)}_{j}$.

As $\rho$ is not a downset, for some $r$-tuples $\mathbf{x},\mathbf{y}$ it holds $\mathbf{x}\geq\mathbf{y}$, $\mathbf{x}\in\rho$, $\mathbf{y}\not\in\rho$. We may assume without loss of generality that, for some $n\in[r]$, the set of coordinates with label $1$ equals $[n]$ for $\mathbf{x}$ and $[n-1]$ for $\mathbf{y}$. Let $\mathbf{e}=(e_{1},\dots,e_{r})\in\rho$ be a tuple with the smallest number of coordinates labelled $1$ such that $e_{n}=1$. We claim that $e_{i}=0$ for all $i\neq n$: Otherwise, either $\operatorname{sub}\left(\mathbf{e},\mathbf{z}^{(i,n)}\right)=\min\left(\mathbf{e},\neg(\mathbf{z}^{(i,n)})\right)$ or $\operatorname{sub}\left(\mathbf{e},\operatorname{sub}\left(\mathbf{e},\mathbf{z}^{(i,n)}\right)\right)=\min\left(\mathbf{e},\mathbf{z}^{(i,n)}\right)$ contradicts the minimality of $\mathbf{e}$. But then $\operatorname{sub}(\mathbf{x},\mathbf{e})=\mathbf{y}\in\rho$, which is a contradiction. ∎

Corollary 30**.**

Let $\Gamma$ be a finite language. The following conditions are equivalent.

Language $\Gamma$ is EDS. 2. 2.

For every $\gamma\in\Gamma$, weighted relation $\gamma$ is EDS. 3. 3.

For every $\gamma\in\Gamma$, both $\operatorname{Feas}(\gamma)$ and $\operatorname{Opt}(\gamma)$ admit polymorphism $\operatorname{sub}$. 4. 4.

For every $\gamma\in\Gamma$, both $\operatorname{Feas}(\gamma)$ and $\operatorname{Opt}(\gamma)$ are essentially downsets.

Remark 31**.**

In [23], a weighted relation $\gamma$ is called PDS if both $\operatorname{Feas}(\gamma)$ and $\operatorname{Opt}(\gamma)$ are essentially downsets. For a $\{0,1\}$-valued weighted relation, this condition is equivalent to that of being almost-min-min [43]. By Corollary 30, PDS and EDS are equivalent concepts for languages of finite size.

As we show in the following example, there exists an infinite non-EDS language $\Gamma$ such that every finite subset $\Gamma^{\prime}\subseteq\Gamma$ is EDS. Hence, $\Gamma$ is s-tractable, although it is globally s-intractable ($\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard).

Example 32**.**

For any $w\in\mathbb{Z}_{\geq 1}$, we define a ternary weighted relation $\mu_{w}$ on $D=\{0,1\}$ by

[TABLE]

Note that $\operatorname{Feas}(\mu_{w})=D^{3}$ and $\operatorname{Opt}(\mu_{w})=\{(0,0,0)\}$ are downsets, and hence $\mu_{w}$ is EDS. However, it is not $\alpha$-EDS for any $\alpha<w/2$: For $\mathbf{x}=(0,1,1)$, $\mathbf{y}=(1,0,1)$, we have $\mu_{w}(\mathbf{x})+\mu_{w}(\mathbf{y})=2$ but $\mu_{w}(\operatorname{sub}(\mathbf{x},\mathbf{y}))=\mu_{w}(0,1,0)=w$. Language $\Gamma=\left\{\mu_{w}~{}|~{}w\in\mathbb{Z}_{\geq 1}\right\}$ is therefore not EDS.

By our classification (Theorem 19), language $\Gamma$ is globally s-intractable; here we show it directly by a reduction from the NP-hard Max-Cut problem. Given an undirected graph $G=(V,E)$ with no isolated vertices, we construct a $\operatorname{VCSP_{s}}(\Gamma)$ instance $I$ as follows. Let $w=2|E|+1$. We introduce a corresponding variable for every vertex in $V$, and add a special variable $z$. For every edge $\{x,y\}\in E$, we impose a constraint $\mu_{w}(x,y,z)$.

Cuts in $G$ are in one-to-one correspondence with assignments to $I$ satisfying $z=1$. In particular, a cut of size $k$ corresponds to an assignment to $I$ with value $k+2(|E|-k)=2|E|-k$. Any surjective assignment with $z=0$ is of value at least $w>2|E|-k$. Thus, solving $I$ amounts to solving Max-Cut in $G$.

3.3 Approximability of maximising surjective VCSP

Although the VCSP is commonly defined with a minimisation objective, it is easy to see that, for exact solvability, its maximisation variant is essentially an identical problem: Minimising a $\mathbb{Q}$-valued function $\phi_{I}$ corresponds to maximising $-\phi_{I}$. When studying approximability, however, the two variants vastly differ (see [34] for a survey).

We focus on maximisation of the $\mathbb{Q}_{\geq 0}$-valued VCSP. This problem generalises the Max-CSP, in which the objective is to maximise the number of satisfied constraints; in particular, the Max-CSP corresponds to maximisation of the $\{0,1\}$-valued VCSP. The complexity of exactly maximising the $\mathbb{Q}_{\geq 0}$-valued VCSP was established by Thapper and Živný [42]. Raghavendra [37] showed that, assuming the unique games conjecture, the basic semidefinite programming relaxation achieves the optimal approximation ratio for the problem. In this section, we consider approximate maximisation of the surjective $\mathbb{Q}_{\geq 0}$-valued VCSP.

Definition 33**.**

An instance $I=(V,D,\phi_{I})$ of the Max-VCSP on domain $D$ is given by a finite set of variables $V=\{x_{1},\ldots,x_{n}\}$ and an objective function $\phi_{I}:D^{n}\to\mathbb{Q}_{\geq 0}$ expressed as a weighted sum of constraints over $V$, i.e.,

[TABLE]

where $\gamma_{i}$ is a $\mathbb{Q}_{\geq 0}$-valued weighted relation, $w_{i}\in\mathbb{Q}_{\geq 0}$ is the weight and $\mathbf{x}_{i}\in V^{\operatorname{ar}(\gamma_{i})}$ the scope of the $i$th constraint.

Given an instance $I$, the goal is to find an assignment $s:V\to D$ of domain labels to the variables that maximises $\phi_{I}$. We denote the maximum value of the objective function by $\operatorname{opt}_{I}$. For any $r\in[0,1]$, an assignment $s$ is an $r$-approximate solution to $I$ if $\phi_{I}(s)\geq r\cdot\operatorname{opt}_{I}$.

An assignment $s$ is surjective if its image equals $D$. We denote the maximum objective value of surjective assignments by $\operatorname{s-opt}_{I}$. For any $r\in(0,1]$, a surjective assignment $s$ is an $r$-approximate surjective solution to $I$ if $\phi_{I}(s)\geq r\cdot\operatorname{s-opt}_{I}$.

We denote by $\operatorname{Max-VCSP_{s}}(\Gamma)$ the surjective $\operatorname{Max-VCSP}$ problem on instances over a language $\Gamma$.

Following the standard definitions, we say that $\operatorname{Max-VCSP_{s}}(\Gamma)$ belongs to APX if, for some $r\in(0,1]$, there exists a polynomial-time algorithm that finds an $r$-approximate surjective solution to every $\operatorname{Max-VCSP_{s}}(\Gamma)$ instance. If such an algorithm exists for every $r<1$, we say that the problem admits a polynomial-time approximation scheme (PTAS). $\operatorname{Max-VCSP_{s}}(\Gamma)$ is APX-hard if there exists a PTAS reduction (an approximation-preserving reduction, see [15]) from every problem in APX to $\operatorname{Max-VCSP_{s}}(\Gamma)$.

First, we prove that a polynomial-time algorithm for exactly maximising the $\mathbb{Q}_{\geq 0}$-valued VCSP over a language $\Gamma$ implies a PTAS for $\operatorname{Max-VCSP_{s}}(\Gamma)$. Second, we establish a complexity classification of Boolean languages in Theorem 36.

Lemma 34**.**

Let $\Gamma$ be a $\mathbb{Q}_{\geq 0}$-valued language and $r,\epsilon\in\mathbb{R}$ such that $0<\epsilon\leq r\leq 1$. There is a polynomial-time algorithm that, given a Max-VCSP instance $I=(V,D,\phi_{I})$ over $\Gamma$ and an $r$-approximate solution $s$ to $I$, outputs an $(r-\epsilon)$-approximate surjective solution $s^{\prime}$ to $I$.

Proof.

Let $a_{\text{max}}$ denote the maximum arity of weighted relations in $\Gamma$, and $n$ the number of variables of $I$. If $n<\frac{r\cdot|D|\cdot a_{\text{max}}}{\epsilon}$, we find an optimal surjective assignment to $I$ by trying all $O(|D|^{n})$ assignments.

Otherwise, we modify the given assignment $s$ in order to obtain a surjective assignment $s^{\prime}$. For any variable $x\in V$, let $B_{x}\subseteq[q]$ be the set of indices of constraints in whose scopes $x$ appears. We define the contribution of $x$ by

[TABLE]

It follows that the total contribution of all variables is at most $a_{\text{max}}\cdot\phi_{I}(s)$.

Let $U$ be a set of $|D|$ variables with the smallest contribution. We assign to them labels $D$ bijectively. The resulting assignment $s^{\prime}$ is surjective, and it holds

[TABLE]

∎

Applying this lemma to an optimal solution to an $\operatorname{Max-VCSP}$ instance (i.e., $r=1$) gives us the following corollary.

Corollary 35**.**

If the $\operatorname{Max-VCSP}$ over a $\mathbb{Q}_{\geq 0}$-valued language $\Gamma$ is solvable in polynomial time, then there is a PTAS for $\operatorname{Max-VCSP_{s}}(\Gamma)$.

Finally, we classify Boolean $\mathbb{Q}_{\geq 0}$-valued languages by the complexity of the corresponding $\operatorname{Max-VCSP_{s}}$. Since multimorphisms and the EDS property are defined in the context of minimisation, the following theorem applies them to language $-\Gamma$ instead of $\Gamma$ (where $-\Gamma=\{-\gamma~{}|~{}\gamma\in\Gamma\}$ and $(-\gamma)(\mathbf{x})=-\gamma(\mathbf{x})$).

Theorem 36**.**

Let $\Gamma$ be a Boolean $\mathbb{Q}_{\geq 0}$-valued language. Then

$\operatorname{Max-VCSP_{s}}(\Gamma)$* is solvable exactly in polynomial time if $-\Gamma$ is EDS, or $-(\neg(\Gamma))$ is EDS, or $-\Gamma$ admits the $\langle\min,\max\rangle$ multimorphism;* 2. 2.

otherwise it is NP-hard to solve exactly, but

1. (a)

it is in PTAS if $-\Gamma$ admits $\langle c_{0}\rangle$ or $\langle c_{1}\rangle$, 2. (b)

and is APX-hard otherwise.

Proof.

Theorem 21 implies the case (1) and NP-hardness in the case (2). Case (2a) follows from Theorem 16 and Corollary 35. By Theorem 16, if $-\Gamma$ does not admit either of $\langle c_{0}\rangle$, $\langle c_{1}\rangle$ and $\langle\min,\max\rangle$, then $\operatorname{Max-VCSP}(\Gamma)$ is NP-hard. The proof of Theorem 16 in [12] actually establishes that $\operatorname{Max-VCSP}(\Gamma)$ is APX-hard. By the approximation-preserving reduction in the proof of Lemma 5, this implies that $\operatorname{Max-VCSP_{s}}(\Gamma)$ is APX-hard as well. ∎

Theorem 36 generalises the result of Bach and Zhou [1, Theorem 16] in two respects. Firstly, we classify all $\mathbb{Q}_{\geq 0}$-valued languages as opposed to $\{0,1\}$-valued languages. Secondly, we classify constraint languages as being in P, in PTAS, or being APX-hard; [1] only distinguishes admitting a PTAS versus being APX-hard. Finally, the main technical component of Theorem 36, Lemma 34, has a slightly simpler proof compared to [1].

4 Hardness proofs

Consider a Boolean language $\Gamma$ over $D=\{0,1\}$ that admits multimorphism $\langle c_{0}\rangle$ (the case of multimorphism $\langle c_{1}\rangle$ is symmetric), but does not admit any of the following multimorphisms: $\langle\min,\min\rangle$, $\langle\max,\max\rangle$, $\langle\min,\max\rangle$, $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$. Suppose that $\Gamma$ is not EDS. We prove that $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard, i.e., $\Gamma$ is globally s-intractable.

We start by showing that there exists a relation such that it is not invariant under $\operatorname{sub}$ and it can be added to $\Gamma$ without changing the complexity of $\operatorname{VCSP_{s}}(\Gamma)$ (see Corollary 40). For finite $\Gamma$, this follows simply from Corollary 30 and Lemma 10, as there exists $\gamma\in\Gamma$ such that $\operatorname{Feas}(\gamma)$ or $\operatorname{Opt}(\gamma)$ is not invariant under $\operatorname{sub}$. In general, however, a different argument is necessary. We prove it by showing that $\Gamma$ contains weighted relations arbitrarily “similar” to a relation which is not invariant under $\operatorname{sub}$, and that this relation may be thus added to $\Gamma$.

Definition 37**.**

For any $\alpha\geq 1$, an $r$-ary weighted relation $\gamma$ is $\alpha$-crisp if its image $\gamma(D^{r})$ lies in ${[0,1]}\cup{(\alpha,\infty]}$. We will denote by $\operatorname{Round}_{\alpha}(\gamma)$ the $r$-ary relation defined as

[TABLE]

Note that an $\alpha$-crisp weighted relation is $\alpha^{\prime}$-crisp for any $\alpha^{\prime}\leq\alpha$. Moreover, a crisp weighted relation $\rho$ is $\alpha$-crisp for any $\alpha\geq 1$, and $\operatorname{Round}_{\alpha}(\rho)=\rho$.

Lemma 38**.**

Let $\Gamma$ be a language and $\rho$ a relation such that, for any $\alpha\geq 1$, there exists an $\alpha$-crisp weighted relation $\gamma\in\Gamma$ with $\operatorname{Round}_{\alpha}(\gamma)=\rho$. Then $\operatorname{VCSP_{s}}(\Gamma\cup\{\rho\})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$.

Proof.

Let $I$ be an instance of $\operatorname{VCSP_{s}}(\Gamma\cup\{\rho\})$ with $k$ constraints that apply relation $\rho$. By scaling and adding rational constants to weighted relations in $I$, we ensure that all the assigned values are non-negative integers. Let $M$ be an upper bound on the maximum value of a feasible solution to $I$ (e.g., the weighted sum of the maximum finite values assigned by the constraints of $I$). Let $\gamma\in\Gamma$ be a $M\cdot(k+1)$-crisp weighted relation such that $\operatorname{Round}_{M\cdot(k+1)}(\gamma)=\rho$. In each constraint applying relation $\rho$, we replace it by $\gamma$ with weight $1/(k+1)$, and thus obtain an instance of $\operatorname{VCSP_{s}}(\Gamma)$. Since $\gamma$ is $M\cdot(k+1)$-crisp, the value of any feasible assignment increases by at most $k/(k+1)<1$, and the value of any infeasible assignment becomes larger than $M$. ∎

Lemma 39**.**

Let $\Gamma$ be a language such that it admits multimorphism $\langle c_{0}\rangle$ but is not EDS. Then there exists a relation $\rho$ that is invariant under $c_{0}$ but not under $\operatorname{sub}$ and, for any $\alpha\geq 1$, there exists an $\alpha$-crisp weighted relation $\gamma\in\Gamma^{*}$ with $\operatorname{Round}_{\alpha}(\gamma)=\rho$.

Proof.

We will show that for any $\alpha\geq 1$, there exists an $\alpha$-crisp weighted relation $\gamma\in\Gamma^{*}$ such that $\operatorname{Round}_{\alpha}(\gamma)$ is a relation of arity at most $4$ that is invariant under $c_{0}$ but not under $\operatorname{sub}$. As there are only finitely many such relations, the claim of the lemma will follow.

Language $\Gamma^{*}$ admits multimorphism $\langle c_{0}\rangle$ as well but is not EDS; in particular, it is not $\alpha^{17}$-EDS. Therefore, there exists an $r$-ary weighted relation $\gamma\in\Gamma^{*}$ and $\mathbf{u},\mathbf{v}\in\operatorname{Feas}(\gamma)$ such that $\gamma(\mathbf{0}^{r})=0$ (as $\Gamma^{*}$ is closed under adding rational constants) and

[TABLE]

We may assume that there are no distinct coordinates $i,j$ where $u_{i}=u_{j}$ and $v_{i}=v_{j}$ (otherwise we identify them), and hence $r\leq 4$. As $\Gamma^{*}$ is closed under scaling, we may also assume that $\gamma(\mathbf{u}),\gamma(\mathbf{v})\leq 1$ and $\gamma(\operatorname{sub}(\mathbf{u},\mathbf{v}))>\alpha^{17}$.

Let us consider, for any $0\leq i\leq 16$, the intersection of the image $\gamma(D^{r})$ with the interval ${\left(\alpha^{i},\alpha^{i+1}\right]}$. Since $|D^{r}|\leq 2^{4}=16$, the intersection is empty for some $i$. Scaling $\gamma$ by $1/\alpha^{i}$ then yields an $\alpha$-crisp weighted relation $\gamma^{\prime}\in\Gamma^{*}$ such that $\operatorname{Round}_{\alpha}(\gamma^{\prime})$ is invariant under $c_{0}$ but not under $\operatorname{sub}$, as $\gamma^{\prime}(\mathbf{0}^{r}),\gamma^{\prime}(\mathbf{u}),\gamma^{\prime}(\mathbf{v})\leq 1$ and $\gamma^{\prime}(\operatorname{sub}(\mathbf{u},\mathbf{v}))>\alpha$. ∎

Corollary 40**.**

Let $\Gamma$ be a language such that it admits multimorphism $\langle c_{0}\rangle$ but is not EDS. Then $\operatorname{VCSP_{s}}(\Gamma\cup\{\rho\})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$ for some relation $\rho$ that is invariant under $c_{0}$ but not under $\operatorname{sub}$.

Proof.

By Lemmas 39 and 38, we have that $\operatorname{VCSP_{s}}(\Gamma^{*}\cup\{\rho\})\leq_{p}\operatorname{VCSP_{s}}(\Gamma^{*})$ for some relation $\rho$ that is invariant under $c_{0}$ but not under $\operatorname{sub}$. By Lemma 10, it holds $\operatorname{VCSP_{s}}(\Gamma^{*})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$. ∎

We define weighted relations $\gamma_{0}=\operatorname{Soft}(\rho_{0})$, $\gamma_{1}=\operatorname{Soft}(\rho_{1})$, and $\gamma_{=}=\operatorname{Soft}(\rho_{=})$; a binary relation $\rho_{\leq}=\{(0,0),(0,1),(1,1)\}$, and, for $r\in\{3,4\}$, an $r$-ary relation

[TABLE]

Assuming that $\Gamma$ does not admit polymorphism $\operatorname{sub}$, we prove that $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard (see Lemma 47). The proof makes use of several sources of hardness. More specifically, we show that at least one of the following cases applies:

• •

$\operatorname{VCSP_{s}}(\operatorname{Feas}(\Gamma)\cup\operatorname{Opt}(\Gamma))$ is NP-hard (by the classification of $\{0,\infty\}$-valued languages, see Theorem 17).

• •

$\operatorname{VCSP}(\Gamma\cup\mathcal{C}_{D})$ reduces to $\operatorname{VCSP_{s}}(\Gamma)$. In particular, it holds $\rho_{\leq}\in\Gamma^{*}$, which can be used to simulate constants (see Lemma 45). The intractability of $\operatorname{VCSP}(\Gamma\cup\mathcal{C}_{D})$ follows from Theorem 14.

• •

The NP-hard Minimum Distance problem [45] reduces to $\operatorname{VCSP_{s}}(\Gamma)$. In particular, it holds $\{A_{3},\gamma_{0}\}\subseteq\Gamma^{*}$ or $\{A_{4},\gamma_{=}\}\subseteq\Gamma^{*}$; the reduction from the Minimum Distance problem to these languages is given in Lemma 46.

Before proving Lemma 47, we need a few auxiliary lemmas to establish the existence of certain weighted relations in $\Gamma^{*}$.

Lemma 41**.**

Let $\rho$ be a relation invariant under $c_{0}$ but not under $\neg$. Then $\rho_{0}\in\{\rho\}^{*}$ or $\rho_{\leq}\in\{\rho\}^{*}$.

Proof.

Let $r$ denote the arity of $\rho$. There exists an $r$-tuple $\mathbf{u}\in\rho$ such that $\neg(\mathbf{u})\not\in\rho$. If $\mathbf{1}^{r}\not\in\rho$, we obtain $\rho_{0}$ by identifying all coordinates of $\rho$. Otherwise, we obtain $\rho_{\leq}$ by identifying all coordinates where $u_{i}=0$ and identifying all coordinates where $u_{i}=1$. ∎

Lemma 42**.**

Let $\gamma$ be a non-crisp weighted relation such that it admits multimorphism $\langle c_{0}\rangle$. Then $\gamma_{0}\in\{\gamma,\rho_{0}\}^{*}$. If in addition $\operatorname{Feas}(\gamma)$ and $\operatorname{Opt}(\gamma)$ are invariant under $\neg$, then $\gamma_{=}\in\{\gamma\}^{*}$.

Proof.

Let $r$ denote the arity of $\gamma$. There exists an $r$-tuple $\mathbf{u}$ such that $\gamma(\mathbf{0}^{r})<\gamma(\mathbf{u})<\infty$. By [math]-pinning at all coordinates where $u_{i}=0$ and identifying all coordinates where $u_{i}=1$, we obtain a unary weighted relation $\gamma^{\prime}\in\{\gamma,\rho_{0}\}^{*}$ such that $\gamma^{\prime}(0)<\gamma^{\prime}(1)<\infty$. From it, we can obtain $\gamma_{0}$ by adding a rational constant and scaling, as $\gamma_{0}=\frac{\gamma^{\prime}-\gamma^{\prime}(0)}{\gamma^{\prime}(1)-\gamma^{\prime}(0)}$.

If $\operatorname{Feas}(\gamma)$ and $\operatorname{Opt}(\gamma)$ are invariant under $\neg$, it holds $\gamma(\mathbf{1}^{r})=\gamma(\mathbf{0}^{r})$ and $\gamma(\mathbf{0}^{r})<\gamma(\neg(\mathbf{u}))<\infty$. By identifying all coordinates where $u_{i}=0$ and identifying all coordinates where $u_{i}=1$, we obtain a binary weighted relation $\gamma^{\prime}\in\{\gamma\}^{*}$. Consider $\gamma^{\prime\prime}\in\{\gamma\}^{*}$ defined as $\gamma^{\prime\prime}(x,y)=\gamma^{\prime}(x,y)+\gamma^{\prime}(y,x)$. It holds $\gamma^{\prime\prime}(0,0)=\gamma^{\prime\prime}(1,1)<\gamma^{\prime\prime}(0,1)=\gamma^{\prime\prime}(1,0)<\infty$. From it, we can obtain $\gamma_{=}$ by adding a rational constant and scaling. ∎

Lemma 43**.**

Let $\rho$ be a relation invariant under $c_{0}$, $\neg$, and $\operatorname{Mn}$, but not under $\operatorname{sub}$. Then $A_{4}\in\{\rho\}^{*}$.

Proof.

Let $\rho^{\prime}$ be a smallest-arity relation in $\{\rho\}^{*}$ that is not invariant under $\operatorname{sub}$, and denote its arity by $r$. As $\mathbf{0}^{r}\in\rho^{\prime}$ and $\operatorname{Mn}(\mathbf{x},\mathbf{y},\mathbf{0}^{r})=\mathbf{x}\oplus\mathbf{y}$, relation $\rho^{\prime}$ is closed under the $\oplus$ operation. Let $\mathbf{u},\mathbf{v}\in\rho^{\prime}$ be $r$-tuples such that $\operatorname{sub}(\mathbf{u},\mathbf{v})\not\in\rho^{\prime}$. There are no distinct coordinates $i,j$ where $u_{i}=u_{j}$ and $v_{i}=v_{j}$, otherwise we could identify them to obtain an $(r-1)$-ary relation not invariant under $\operatorname{sub}$. For any $a,b\in\{0,1\}$, there is a coordinate $i$ where $u_{i}=a$ and $v_{i}=b$, otherwise $\operatorname{sub}(\mathbf{u},\mathbf{v})$ would be equal to $\neg(\mathbf{v})$, $\mathbf{u}\oplus\mathbf{v}$, $\mathbf{0}^{r}$, or $\mathbf{u}$ respectively, which would imply $\operatorname{sub}(\mathbf{u},\mathbf{v})\in\rho^{\prime}$. Therefore, $r=4$, and we may assume without loss of generality that $\mathbf{u}=(0,0,1,1)$, $\mathbf{v}=(0,1,0,1)$. As

[TABLE]

it holds $(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0)\not\in\rho^{\prime}$. Since $\rho^{\prime}$ is closed under $\neg$, we have $\rho^{\prime}=A_{4}$. ∎

Lemma 44**.**

Let $\rho$ be a relation invariant under $c_{0}$ but not under $\operatorname{sub}$. If $\rho$ is invariant under $\operatorname{Mn}$, then $A_{3}\in\{\rho,\rho_{0}\}^{*}$. If $\rho$ is invariant under $\min$ or $\max$, then $\rho_{\leq}\in\{\rho,\rho_{0}\}^{*}$.

Proof.

Let $\rho^{\prime}$ be a smallest-arity relation in $\{\rho,\rho_{0}\}^{*}$ that is not invariant under $\operatorname{sub}$, and denote its arity by $r$. Let $\mathbf{u},\mathbf{v}\in\rho^{\prime}$ be $r$-tuples such that $\operatorname{sub}(\mathbf{u},\mathbf{v})\not\in\rho^{\prime}$. There are no distinct coordinates $i,j$ where $u_{i}=u_{j}$ and $v_{i}=v_{j}$, otherwise we could identify them to obtain an $(r-1)$-ary relation not invariant under $\operatorname{sub}$. For any $b\in\{0,1\}$, there is a coordinate $i$ where $u_{i}=1$ and $v_{i}=b$, otherwise $\operatorname{sub}(\mathbf{u},\mathbf{v})$ would be equal to $\mathbf{0}^{r}$ or $\mathbf{u}$ respectively, which would imply $\operatorname{sub}(\mathbf{u},\mathbf{v})\in\rho^{\prime}$. However, there is no coordinate $i$ where $u_{i}=v_{i}=0$, otherwise we could obtain an $(r-1)$-ary relation not invariant under $\operatorname{sub}$ by [math]-pinning $\rho^{\prime}$ at coordinate $i$. Therefore, $r=2$ or $r=3$. If $r=2$, we have $\rho_{\leq}\in\{\rho,\rho_{0}\}^{*}$, and $\rho$ is not invariant under $\operatorname{Mn}$ (as neither is $\rho_{\leq}$).

If $r=3$, we may assume without loss of generality that $\mathbf{u}=(0,1,1)$ and $\mathbf{v}=(1,0,1)$. Relation $\rho$ is not invariant under $\min$, otherwise it would hold $\min(\mathbf{u},\mathbf{v})=(0,0,1)\in\rho^{\prime}$ and we could obtain a binary relation not invariant under $\operatorname{sub}$ by [math]-pinning $\rho^{\prime}$ at the first coordinate. Similarly, relation $\rho$ is not invariant under $\max$, otherwise it would hold $\max(\mathbf{u},\mathbf{v})=(1,1,1)\in\rho^{\prime}$ and we could obtain a binary relation not invariant under $\operatorname{sub}$ by identifying the first and third coordinate. Finally, assume that relation $\rho$ is invariant under $\operatorname{Mn}$. Then $\rho^{\prime}$ is also closed under the $\oplus$ operation, as $\operatorname{Mn}(\mathbf{x},\mathbf{y},\mathbf{0}^{r})=\mathbf{x}\oplus\mathbf{y}$, and we have $\mathbf{u}\oplus\mathbf{v}=(1,1,0)\in\rho^{\prime}$. Since $\operatorname{sub}(\mathbf{u},\mathbf{v})=(0,1,0)=(0,0,1)\oplus\mathbf{u}=(1,1,1)\oplus\mathbf{v}=(1,0,0)\oplus(\mathbf{u}\oplus\mathbf{v})$, it holds $(0,0,1),(1,1,1),(1,0,0)\not\in\rho^{\prime}$, and therefore $\rho^{\prime}=A_{3}$. ∎

Lemma 45**.**

If $\rho_{\leq}\in\Gamma$, then $\operatorname{VCSP}(\Gamma\cup\mathcal{C}_{D})\leq_{p}\operatorname{VCSP_{s}}(\Gamma)$.

Proof.

For a given instance of $\operatorname{VCSP}(\Gamma\cup\{\rho_{0},\rho_{1}\})$ with variables $V$, we construct an instance of $\operatorname{VCSP_{s}}(\Gamma)$ as follows: We introduce new variables $y_{0}$, $y_{1}$ and impose constraints $\rho_{\leq}(y_{0},x)$, $\rho_{\leq}(x,y_{1})$ for all $x\in V$ to ensure that $y_{0}=0$, $y_{1}=1$ in any feasible surjective assignment. Then we replace each constraint of the form $\rho_{0}(x)$ with $\rho_{\leq}(x,y_{0})$ and each constraint of the form $\rho_{1}(x)$ with $\rho_{\leq}(y_{1},x)$. ∎

Lemma 46**.**

Languages $\{A_{3},\gamma_{0}\}$ and $\{A_{4},\gamma_{=}\}$ are both s-intractable.

Proof.

First we show a reduction from the optimisation variant of the Minimum Distance problem, which is NP-hard [45], to $\operatorname{VCSP_{s}}(\{A_{3},\gamma_{0}\})$. A problem instance is given as an $m\times n$ matrix $H$ over the field $D=\{0,1\}$, and the objective is to find a non-zero vector $\mathbf{x}=(x_{1},\dots,x_{n})\in D^{n}$ satisfying $H\cdot\mathbf{x}=\mathbf{0}^{m}$ with the minimum weight (i.e. $\sum_{i=1}^{n}x_{i}$).

Note that $\rho_{0}=\operatorname{Opt}(\gamma_{0})$, and therefore we may use relation $\rho_{0}$ as well (by Lemma 10). We construct a $\operatorname{VCSP_{s}}$ instance $I$ as follows: Let $x_{1},\dots,x_{n}$ be variables corresponding to the elements of the sought vector $\mathbf{x}$. The requirement $H\cdot\mathbf{x}=\mathbf{0}^{m}$ can be seen as a system of $m$ linear equations, each in the form $\bigoplus_{i=1}^{k}x_{a_{i}}=0$ for a set $\{a_{1},\dots,a_{k}\}\subseteq[n]$ (the set may differ for each equation). We encode such an equation by introducing new variables $y_{0},\dots,y_{k}$ and imposing constraints $\rho_{0}(y_{0})$, $A_{3}(y_{i-1},x_{a_{i}},y_{i})$ for all $i\in[k]$, and $\rho_{0}(y_{k})$. These ensure that each variable $y_{j}$ is assigned the value of the prefix sum $\bigoplus_{i=1}^{j}x_{a_{i}}$, and that the total sum equals [math]. Finally, we encode the objective function of the Minimum Distance problem by imposing constraints $\gamma_{0}(x_{1}),\dots,\gamma_{0}(x_{n})$.

Every vector $\mathbf{x}\in D^{n}$ satisfying $H\cdot\mathbf{x}=\mathbf{0}^{m}$ corresponds to a feasible assignment to $I$. If $\mathbf{x}$ is non-zero, the corresponding assignment is surjective, as at least one of variables $x_{1},\dots,x_{n}$ gets label $1$ and, for every equation, variable $y_{0}$ gets label [math]. Conversely, if a feasible assignment to $I$ is surjective, then it corresponds to a non-zero vector $\mathbf{x}$ (labelling all variables $x_{1},\dots,x_{n}$ with [math] implies that all the prefix sums $y_{j}$ equal [math] as well). The objective value of the assignment corresponding to a vector $\mathbf{x}$ equals the weight of $\mathbf{x}$, and hence finding an optimal surjective solution to $I$ solves the Minimum Distance problem.

Finally, we show that $\{A_{4},\gamma_{=}\}$ is s-intractable by a reduction from $\operatorname{VCSP_{s}}(\{A_{3},\gamma_{0}\})$ to $\operatorname{VCSP_{s}}(\{A_{4},\gamma_{=}\})$. Given an instance $I$, we construct an instance $I^{\prime}$ by introducing a new variable $w$ and replacing each constraint of the form $A_{3}(x,y,z)$ with $A_{4}(x,y,z,w)$ and each constraint of the form $\gamma_{0}(x)$ with $\gamma_{=}(x,w)$. Any surjective assignment to $I$ can be extended to a surjective assignment to $I^{\prime}$ of the same objective value by labelling $w$ with [math]. Conversely, consider a feasible surjective assignment $s^{\prime}$ to $I^{\prime}$; we may assume $s^{\prime}(w)=0$ since language $\{A_{4},\gamma_{=}\}$ admits multimorphism $\langle\neg\rangle$. Restricting $s^{\prime}$ to the variables of $I$ gives us a surjective assignment to $I$ of the same objective value. Note that if $s^{\prime}$ assigns label $1$ to all the variables except $w$, its restriction will not be surjective; however, such $s^{\prime}$ violates constraints $\rho_{0}(y_{0})$ and thus is not feasible.∎

Lemma 47**.**

Let $\Gamma$ be a language such that it admits multimorphism $\langle c_{0}\rangle$ but not polymorphism $\operatorname{sub}$. If $\Gamma\cup\mathcal{C}_{D}$ is intractable, then $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard.

Proof.

Let $\Phi=\operatorname{Feas}(\Gamma)\cup\operatorname{Opt}(\Gamma)\subseteq\Gamma^{*}$. Suppose that $\Phi$ does not admit any of the following polymorphisms: $\min$, $\max$, $\operatorname{Mn}$, and $\operatorname{Mj}$. By the classification of $\{0,\infty\}$-valued languages (see Theorem 17), $\Phi$ is s-intractable. Hence, $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard by Lemma 10. In the rest of the proof, we assume that $\Phi$ admits at least one of polymorphisms $\min$, $\max$, $\operatorname{Mn}$, and $\operatorname{Mj}$. Note that $\Phi$ admits polymorphism $c_{0}$ but not polymorphism $\operatorname{sub}$. Since $\min(x,y)=\operatorname{Mj}(x,y,0)$, we may assume that $\Phi$ admits at least one of polymorphisms $\min$, $\max$, and $\operatorname{Mn}$.

Suppose that $\Phi$ admits polymorphism $\neg$. Then it does not admit $\min$, as $\operatorname{sub}(x,y)=\min(x,\neg y)$, nor it admits $\max$, as $\min(x,y)=\neg\max(\neg x,\neg y)$. Therefore, $\Phi$ admits polymorphism $\operatorname{Mn}$. If $\Gamma$ is crisp, then language $\Gamma\cup\{\rho_{0},\rho_{1}\}$ admits multimorphism $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$ and thus is tractable by Theorem 14, which contradicts an assumption of the lemma. Hence, $\Gamma$ is not crisp. By Lemmas 43 and 42, we have $\{A_{4},\gamma_{=}\}\subseteq\Gamma^{*}$. Therefore, $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard by Lemma 46.

If $\Phi$ does not admit polymorphism $\neg$, then, by Lemma 41, we have $\rho_{0}\in\Gamma^{*}$ or $\rho_{\leq}\in\Gamma^{*}$. If $\rho_{\leq}\in\Gamma^{*}$, $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard by Lemma 45 and we are done; in the rest of the proof we assume that $\rho_{\leq}\not\in\Gamma^{*}$ and hence $\rho_{0}\in\Gamma^{*}$. If $\Phi$ admits polymorphism $\min$ or $\max$, we get $\rho_{\leq}\in\Gamma^{*}$ by Lemma 44, which is a contradiction. Therefore, $\Phi$ admits $\operatorname{Mn}$, and thus $\Gamma$ is not crisp (by the same argument as in the previous paragraph). By Lemmas 44 and 42, we have $\{A_{3},\gamma_{0}\}\subseteq\Gamma^{*}$. Therefore, $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard by Lemma 46. ∎

Theorem 48**.**

Let $\Gamma$ be a language such that it is not EDS, $\neg(\Gamma)$ is not EDS, and $\Gamma$ does not admit any of the following multimorphisms: $\langle\min,\min\rangle$, $\langle\max,\max\rangle$, $\langle\min,\max\rangle$, $\langle\operatorname{Mn},\operatorname{Mn},\operatorname{Mn}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mj}\rangle$, $\langle\operatorname{Mj},\operatorname{Mj},\operatorname{Mn}\rangle$. Then $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard.

Proof.

If $\Gamma$ does not admit at least one of multimorphisms $\langle c_{0}\rangle$ and $\langle c_{1}\rangle$, it is intractable by Theorem 14, and hence $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard by Lemma 5. Language $\Gamma\cup\mathcal{C}_{D}$ is, by the same theorem, intractable. We may assume that $\Gamma$ admits multimorphism $\langle c_{0}\rangle$; if it does not, we consider $\neg(\Gamma)$ instead. By Corollary 40 and Lemma 47, $\operatorname{VCSP_{s}}(\Gamma)$ is NP-hard. ∎

5 Tractability of EDS languages

We prove that EDS languages are globally s-tractable by a reduction to a generalised variant of the Min-Cut problem. The problem is defined in Section 5.1, its tractability is established in Section 5.2, and the reduction is stated in Section 5.3.

5.1 Generalised Min-Cut problem

Let $V$ be a finite set. A set function on $V$ is a function $\gamma:2^{V}\to\mathbb{Q}_{\geq 0}\cup\{\infty\}$ with $\gamma(\emptyset)=0$.

Definition 49**.**

A set function $\gamma:2^{V}\to\mathbb{Q}_{\geq 0}\cup\{\infty\}$ is symmetric if $\gamma(X)=\gamma(V\setminus X)$ for all $X\subseteq V$; it is increasing if $\gamma(X)\leq\gamma(Y)$ for all $X\subseteq Y\subseteq V$; it is superadditive if

[TABLE]

for all disjoint $X,Y\subseteq V$; it is posimodular if

[TABLE]

for all $X,Y\subseteq V$; and it is submodular if

[TABLE]

for all $X,Y\subseteq V$.

Note that any superadditive set function is also increasing, as for all $X\subseteq Y\subseteq V$ it holds $\gamma(X)\leq\gamma(X)+\gamma(Y\setminus X)\leq\gamma(Y)$ by superadditivity. In the case of symmetric set functions, submodularity implies posimodularity, as

[TABLE]

and, similarly, posimodularity implies submodularity.

Example 50**.**

Let $V$ be a finite set and $T\subseteq V$ a non-empty subset. We define a set function $\gamma$ on $V$ by $\gamma(X)=1$ if $T\subseteq X$ and $\gamma(X)=0$ otherwise. Intuitively, this corresponds to a soft NAND constraint imposed on variables $T$. The set function $\gamma$ is superadditive, and hence also increasing.

We now formally define the Min-Cut problem.

Definition 51**.**

An instance of the Min-Cut (MC) problem is given by an undirected graph $G=(V,E)$ with edge weights $w:E\to\mathbb{Q}_{\geq 0}\cup\{\infty\}$. The objective function $g$ of the MC problem is a set function on $V$ defined by

[TABLE]

Function $g$ is a well-known example of a submodular function. Since it is symmetric, it is also posimodular.

A solution to the MC problem is a set $X$ such that $\emptyset\subsetneq X\subsetneq V$. Note that a cut $(X,V\setminus X)$ corresponds to two solutions, namely $X$ and $V\setminus X$. An optimal solution is a solution with the minimum objective value among all solutions. A minimal optimal solution is an optimal solution with no proper subset being an optimal solution.

Note that any two different minimal optimal solutions $X,Y$ must be disjoint, otherwise $X\setminus Y$ or $Y\setminus X$ would be a smaller optimal solution (by the posimodularity of $g$).

Although the definition allows infinite weight edges, those can be easily eliminated by identifying their endpoints, and so we may assume that all edge weights are finite. Edges with weight [math] are conventionally disregarded.

Finally, we define the Generalised Min-Cut problem, which further generalises the problem introduced in [43].

Definition 52**.**

An instance $J$ of the Generalised Min-Cut (GMC) problem is given by an undirected graph $G=(V,E)$ with edge weights $w:E\to\mathbb{Q}_{\geq 0}\cup\{\infty\}$, and an oracle defining a superadditive set function $f$ on $V$. The objective function the GMC problem is a set function on $V$ defined by $J(X)=f(X)+g(X)$, where $g$ is the objective function of the underlying Min-Cut problem on $G$.

A solution to the GMC problem is a set $X$ such that $\emptyset\subsetneq X\subsetneq V$. An optimal solution is a solution with the minimum objective value among all solutions. We denote this minimum objective value by $\lambda$. For any $\alpha\geq 1$, an $\alpha$-optimal solution is a solution $X$ such that $J(X)\leq\alpha\lambda$.

We show in Theorem 59 that, in the case of $0<\lambda<\infty$ and a fixed $\alpha\geq 1$, there are only polynomially many $\alpha$-optimal solutions and they can be found in polynomial time.

5.2 Tractability of the Generalised Min-Cut problem

In this section, we present a polynomial-time algorithm that solves the Generalised Min-Cut problem. We assume that $w(u,v)\in\mathbb{Q}_{>0}$ for all edges $(u,v)$.

Lemma 53**.**

There is a polynomial-time algorithm that, given an instance $J$ of the GMC problem, either finds a solution $X$ with $J(X)=\lambda=0$, or determines that $\lambda=\infty$, or determines that $0<\lambda<\infty$.

Proof.

A solution $X$ with $J(X)=f(X)+g(X)=0$ satisfies $f(X)=g(X)=0$, and hence it does not cut any edge. Since the set function $f$ is increasing, we may assume that $X$ is a single connected component. The algorithm simply tries each connected component as a solution, which takes a linear number of queries to the oracle for $f$.

The case of $\lambda=\infty$ occurs only if $f(X)=\infty$ for all solutions $X$. Since $f$ is increasing, it is sufficient to check all solutions of size $1$. ∎

In view of Lemma 53, we can assume that $0<\lambda<\infty$. Our goal is to show that, for a given $\alpha\geq 1$, all $\alpha$-optimal solutions to a GMC instance can be found in polynomial time. This is proved in Theorem 59; before that we need to prove several auxiliary lemmas on properties of the MC and GMC problems.

Lemma 54**.**

For any instance $J$ of the GMC problem on a graph $G=(V,E)$ and any non-empty set $V^{\prime}\subseteq V$, there is an instance $J^{\prime}$ on the induced subgraph $G[V^{\prime}]$ that preserves the objective value of all solutions $X\subsetneq V^{\prime}$. In particular, any $\alpha$-optimal solution $X$ of $J$ such that $X\subsetneq V^{\prime}$ is $\alpha$-optimal for $J^{\prime}$ as well.

Proof.

Edges with exactly one endpoint in $V^{\prime}$ need to be taken into account separately because they do not appear in the induced subgraph. We accomplish that by defining the new set function $f^{\prime}$ by

[TABLE]

for all $X\subseteq V^{\prime}$. By the construction, $f^{\prime}$ is superadditive, and the objective value $J^{\prime}(X)$ for any $X\subsetneq V^{\prime}$ equals $J(X)$.

Note that the minimum objective value for $J^{\prime}$ is greater than or equal to the minimum objective value for $J$. Therefore, any solution $X\subsetneq V^{\prime}$ that is $\alpha$-optimal for $J$ is also $\alpha$-optimal for $J^{\prime}$. ∎

Lemma 55**.**

Let $X$ be an optimal solution to an instance of the GMC problem over vertices $V$ with $\lambda<\infty$, and $Y$ a minimal optimal solution to the underlying MC problem. Then $X\subseteq Y$, $X\subseteq V\setminus Y$, or $X$ is an optimal solution to the underlying MC problem.

Proof.

Assume that $X\not\subseteq Y$ and $X\not\subseteq V\setminus Y$. If $Y\subseteq X$, we have $f(Y)\leq f(X)$ as $f$ is increasing, and hence $f(Y)+g(Y)\leq f(X)+g(X)<\infty$. Therefore, $Y$ is optimal for the GMC problem and $X$ is optimal for the MC problem. In the rest, we assume that $Y\not\subseteq X$.

By the posimodularity of $g$ we have $g(X)+g(Y)\geq g(X\setminus Y)+g(Y\setminus X)$. Since $Y\setminus X$ is a proper non-empty subset of $Y$, it holds $g(Y\setminus X)>g(Y)$, and hence $g(X)>g(X\setminus Y)$. But then $f(X)+g(X)>f(X\setminus Y)+g(X\setminus Y)$ as $\infty>f(X)\geq f(X\setminus Y)$. Set $X\setminus Y$ is non-empty, and therefore contradicts the optimality of $X$. ∎

The following lemma relates the number of optimal solutions and the number of minimal optimal solutions to the MC problem. Note that this bound is tight for (unweighted) paths and cycles with at most one path attached to each vertex.

Lemma 56**.**

For any instance of the MC problem on a connected graph with $n\geq 2$ vertices and $p$ minimal optimal solutions, there are at most $p(p-1)+2(n-p)$ optimal solutions.

We prove the lemma by induction on $n$, closely following the proof that establishes the cactus representation of minimum cuts in [21]. We note that the cactus representation could be applied directly to obtain a weaker bound of $p(p-1)+O(n)$ but we do not know how to achieve the exact bound using it.

Proof.

For $n=2$, the lemma holds as there are exactly two solutions and both are minimal optimal. Assume $n\geq 3$. We denote the number of optimal solutions by $s$. A solution $X$ is called a star if $|X|=1$ or $|X|=n-1$, otherwise it is called proper.

First we consider the case where every optimal solution is a star. Let us denote the minimum cuts by $(\{v_{1}\},V\setminus\{v_{1}\}),\dots,(\{v_{h}\},V\setminus\{v_{h}\})$. If $h=1$, then we have $s=p=2$ and the bound holds. Otherwise, there are $2h$ optimal solutions but only $h$ of them are minimal (i.e., $\{v_{1}\},\dots,\{v_{h}\}$). Hence,

[TABLE]

as it holds $n\geq h\geq 2$ and $n\geq 3$.

From now on we assume that there is a proper optimal solution, and hence $n\geq 4$. We say that solutions $X,Y$ cross if none of $X\setminus Y$, $Y\setminus X$, $X\cap Y$, $V\setminus(X\cup Y)$ is empty. Note that only proper solutions might cross. If every proper optimal solution is crossed by some optimal solution, then the graph is a cycle with edges of equal weight [21, Lemma 7.1.3]. In that case, there are $n(n-1)$ optimal solutions (all sets of contiguous vertices except for $\emptyset$ and $V$) and $n$ minimal optimal solutions (all singletons), and therefore the bound holds.

Finally, assume that there is a proper optimal solution that is not crossed by any optimal solution, and denote the corresponding minimum cut by $(V_{1},V_{2})$. For any optimal solution $X$, it must hold either $X\subseteq V_{1}$, $V_{1}\subseteq X$, $X\subseteq V_{2}$, or $V_{2}\subseteq X$. For $i\in\{1,2\}$, let $G_{i}$ be the result of shrinking $V_{i}$ into a new vertex $t_{i}$ so that the weight of any edge $(t_{i},v)$ for $v\in V\setminus V_{i}$ equals the sum of weights of edges $(u,v)$ for $u\in V_{i}$. Denote by $n_{i}$, $p_{i}$, and $s_{i}$ the number of vertices, minimal optimal solutions, and optimal solutions to $G_{i}$. It holds $n=n_{1}+n_{2}-2$. Consider any solution $X^{\prime}$ of $G_{i}$: If $t_{i}\not\in X^{\prime}$, it corresponds to a solution $X=X^{\prime}$ of the original graph $G$; otherwise it corresponds to $X=X^{\prime}\setminus\{t_{i}\}\cup V_{i}$. In both cases, the objective values of $X^{\prime}$ and $X$ in their respective problem instances are equal. Therefore, any optimal solution $X$ of $G$ such that $X\subseteq V_{2}$ or $V_{1}\subseteq X$ corresponds to an optimal solution to $G_{1}$, and any optimal solution to $G$ such that $X\subseteq V_{1}$ or $V_{2}\subseteq X$ corresponds to an optimal solution in $G_{2}$. Hence, $p=p_{1}+p_{2}-2$ and $s=s_{1}+s_{2}-2$, as only solutions $V_{1}$ and $V_{2}$ satisfy both conditions simultaneously. By the inductive hypothesis, we get

[TABLE]

as it holds $p_{1},p_{2}\geq 2$. ∎

Lemma 57**.**

For any instance of the GMC problem on $n$ vertices with $0<\lambda<\infty$, the number of optimal solutions is at most $n(n-1)$. There is an algorithm that finds all of them in polynomial time.

Note that the bound of $n(n-1)$ optimal solutions precisely matches the known upper bound of $\binom{n}{2}$ for the number of minimum cuts [29]; the bound is tight for cycles.

Proof.

Let $t(n)$ denote the maximum number of optimal solutions for such instances on $n$ vertices. We prove the bound by induction on $n$. If $n=1$, there are no solutions and hence $t(1)=0$. For $n\geq 2$, let $Y_{1},\dots,Y_{p}$ be the minimal optimal solutions to the underlying MC problem. As there exists at least one minimum cut and the minimal optimal solutions are all disjoint, it holds $2\leq p\leq n$.

First, suppose that $\bigcup Y_{i}=V$. By Lemma 55, any optimal solution to the GMC problem is either a proper subset of some $Y_{i}$ or an optimal solution to the underlying MC problem. Restricting solutions to a proper subset of $Y_{i}$ is, by Lemma 54, equivalent to considering a GMC problem instance on vertices $Y_{i}$, and hence the number of such optimal solutions is bounded by $t(|Y_{i}|)\leq|Y_{i}|\cdot(|Y_{i}|-1)$. Since it holds $\sum|Y_{i}|=n$ and $|Y_{i}|\geq 1$ for all $i$, the sum $\sum|Y_{i}|\cdot(|Y_{i}|-1)$ is maximised when $p-1$ of the sets $Y_{i}$ are singletons and the size of the remaining one equals $n-p+1$. If the graph is connected, then, by Lemma 56, there are at most $p(p-1)+2(n-p)$ optimal solutions to the underlying MC problem. Adding these upper bounds we get

[TABLE]

If the graph is disconnected, the sets $Y_{1},\dots,Y_{p}$ are precisely its connected components. The optimal solutions to the underlying MC problem are precisely unions of connected components (with the exception of $\emptyset$ and $V$), which means that there can be exponentially many of them. However, only the sets $Y_{1},\dots,Y_{p}$ themselves can be optimal solutions to the GMC problem: We have $0<\lambda\leq f(Y_{i})+g(Y_{i})=f(Y_{i})$. Since $f$ is superadditive, it holds

[TABLE]

for any distinct $i_{1},\dots,i_{k}$, and hence no union of two or more connected components can be an optimal solution to the GMC problem. This gives us an upper bound of $p\leq p(p-1)+2(n-p)$, and the rest follows as in the previous case.

Finally, suppose that $\bigcup Y_{i}\neq V$, and hence the graph is connected. Let $Z=V\setminus\bigcup Y_{i}$. By Lemma 55, any optimal solution to the GMC problem is a proper subset of some $Y_{i}$, a proper subset of $Z$, set $Z$ itself, or an optimal solution to the underlying MC problem. Similarly as before, we get an upper bound of

[TABLE]

Using a procedure generating all minimum cuts [46], it is straightforward to turn the above proof into a recursive algorithm that finds all optimal solutions in polynomial time. ∎

Lemma 58**.**

Let $\alpha,\beta\geq 1$. Let $X$ be an $\alpha$-optimal solution to an instance $J$ of the GMC problem over vertices $V$ with $0<\lambda<\infty$, and $Y$ an optimal solution to the underlying MC problem. If $g(Y)<\lambda/\beta$, then

[TABLE]

if $g(Y)\geq\lambda/\beta$, then $X$ is an $\alpha\beta$-optimal solution to the underlying MC problem.

Proof.

If $g(Y)\geq\lambda/\beta$, it holds $g(X)\leq J(X)\leq\alpha\lambda\leq\alpha\beta\cdot g(Y)$, and hence $X$ is an $\alpha\beta$-optimal solution to the underlying MC problem. In the rest we assume that $g(Y)<\lambda/\beta$.

Since $g$ is posimodular, we have

[TABLE]

and hence

[TABLE]

By superadditivity of $f$, it holds $f(X)\geq f(X\setminus Y)+f(X\cap Y)$. The claim then follows from the fact that $f(X)+g(X)+2g(Y)<(\alpha+2/\beta)\lambda$. ∎

Finally, we prove that $\alpha$-optimal solutions to the GMC problem can be found in polynomial time.

Theorem 59**.**

For any instance $J$ of the GMC problem on $n$ vertices with $0<\lambda<\infty$ and $\alpha\in\mathbb{Z}_{\geq 1}$, the number of $\alpha$-optimal solutions is at most $n^{20\alpha-15}$. There is an algorithm that finds all of them in polynomial time.

Note that for a cycle on $n$ vertices, the number of $\alpha$-optimal solutions to the MC problem is $\Theta(n^{2\alpha})$, and thus the exponent in our bound is asymptotically tight in $\alpha$.

Proof.

Let $\beta\in\mathbb{Z}_{\geq 3}$ be a parameter. Throughout the proof, we relax the integrality restriction on $\alpha$ and require only that $\alpha\beta$ is an integer. For $\alpha=1$, the claim follows from Lemma 57, therefore we assume $\alpha\geq 1+1/\beta$ in the rest of the proof.

Define a linear function $\ell$ by

[TABLE]

We prove that the number of $\alpha$-optimal solutions is at most $n^{\ell(\alpha)}$; taking $\beta=4$ then gives the claimed bound. Function $\ell$ was chosen as a slowest-growing function satisfying the following properties required in this proof: It holds $\ell(x)+\ell(y)\leq\ell(x+y-3/\beta)$ for any $x,y$, and $\ell(x)\geq 2\beta x$ for any $x\geq 1+1/\beta$.

We prove the bound by induction on $n+\alpha\beta$. As it trivially holds for $n\leq 2$, we assume $n\geq 3$ in the rest of the proof. Let $Y$ be an optimal solution to the underlying MC problem with $k=|Y|\leq n/2$. If $g(Y)\geq\lambda/\beta$ then, by Lemma 58, any $\alpha$-optimal solution to the GMC problem is an $\alpha\beta$-optimal solution to the underlying MC problem. Since $g(Y)\geq\lambda/\beta>0$, the graph is connected, and hence there are at most

[TABLE]

such solutions by [29]. (In detail, [29, Theorem 6.2] shows that the number of $\alpha\beta$-optimal cuts in an $n$-vertex graph is $2^{2\alpha\beta-1}\binom{n}{2\alpha\beta}$, and every cut corresponds to two solutions.)

From now on we assume that $g(Y)<\lambda/\beta$, and hence inequality (50) holds. Upper bounds in this case may be quite loose; in particular, we use the following inequalities:

[TABLE]

Consider any $\alpha$-optimal solution to the GMC problem $X$.

If $X\subsetneq Y$, then, by Lemma 54, $X$ is an $\alpha$-optimal solution to an instance on vertices $Y$. By the induction hypothesis, there are at most $k^{\ell(\alpha)}\leq(k/128n)\cdot n^{\ell(\alpha)}$ such solutions.

Similarly, if $X\subsetneq V\setminus Y$, then $X$ is an $\alpha$-optimal solution to an instance on vertices $V\setminus Y$, and there are at most

[TABLE]

such solutions.

If $Y\subsetneq X$, then $X\setminus Y$ is an $(\alpha-1+2/\beta)$-optimal solution on vertices $V\setminus Y$ by (50) and the fact that $J(X\cap Y)\geq\lambda$. Similarly, if $V\setminus Y\subsetneq X$, then $X\cap Y$ is an $(\alpha-1+2/\beta)$-optimal solution on vertices $Y$. In either case, we bound the number of such solutions depending on the value of $\alpha$: For $\alpha<2-2/\beta$, there are trivially none; for $\alpha=2-2/\beta$, Lemma 57 gives a bound of $n(n-1)\leq n^{\ell(\alpha)-2\beta}$; and for $\alpha>2-2/\beta$ we get an upper bound of $n^{\ell(\alpha-1+2/\beta)}\leq n^{\ell(\alpha)-2\beta}$ by the induction hypothesis. The number of solutions is thus at most $n^{\ell(\alpha)-2\beta}\leq(1/128n)\cdot n^{\ell(\alpha)}$ for any $\alpha$.

Finally, we consider $X$ such that $\emptyset\subsetneq X\setminus Y\subsetneq V\setminus Y$ and $\emptyset\subsetneq X\cap Y\subsetneq Y$, i.e., $X\setminus Y$ and $X\cap Y$ are solutions on vertices $V\setminus Y$ and $Y$ respectively. Let $i$ be the integer for which

[TABLE]

Then, by (50), it holds $J(X\setminus Y)<(\alpha-1-(i-2)/\beta)\lambda$. Therefore, $X\cap Y$ is a $(1+(i+1)/\beta)$-optimal solution on vertices $Y$ and $X\setminus Y$ is an $(\alpha-1-(i-2)/\beta)$-optimal solution on vertices $V\setminus Y$. Since $0\leq i\leq(\alpha-2)\beta+1$, we can bound the number of such solutions by the induction hypothesis as at most

[TABLE]

which is at most $2\cdot(k/128n)\cdot n^{\ell(\alpha)}$ in total for all $i$.

By adding up the bounds we get that the number of $\alpha$-optimal solutions is at most $n^{\ell(\alpha)}$. A polynomial-time algorithm that finds the $\alpha$-optimal solutions follows from the above proof using a procedure generating all $\alpha\beta$-optimal cuts [46]. ∎

Remark 60**.**

For our reduction from the $\operatorname{VCSP_{s}}$ over EDS languages, we need to find all $\alpha$-optimal solutions to the GMC problem. However, if one is only interested in a single optimal solution, the presented algorithm can be easily adapted to an even more general problem.

Let $f,g$ be set functions on $V$ given by an oracle such that $f:2^{V}\to\mathbb{Q}_{\geq 0}\cup\{\infty\}$ is increasing and $g:2^{V}\to\mathbb{Q}_{\geq 0}$ satisfies the posimodularity and submodularity inequalities for intersecting pairs of sets (i.e. sets $X,Y$ such that neither of $X\cap Y,X\setminus Y,Y\setminus X$ is empty). The objective is to minimise the sum of $f$ and $g$.

The case when the optimum value $\lambda=\infty$ can be recognised by checking all solutions of size $1$. Assuming $\lambda<\infty$, note that the proof of Lemma 55 works even for this more general problem. Let $Y$ be a minimal optimal solution to $g$. It follows that there is an optimal solution $X$ to $f+g$ such that $X\subseteq Y$, $X\subseteq V\setminus Y$, or $X$ is itself a minimal optimal solution to $g$ (as $f$ is increasing). We can find all minimal optimal solutions to $g$ in polynomial time [36, Theorem 10.11]. Restricting $f,g$ to a subset of $V$ preserves the required properties, and hence we can recursively solve the problem on $Y$ and $V\setminus Y$. Therefore, an optimal solution to $f+g$ can be found in polynomial time.

5.3 Reduction to the Generalised Min-Cut problem

At the heart of our reduction is an observation that EDS weighted relations can be approximated by instances of the Generalised Min-Cut problem. We define this notion of approximability in Definition 64. In Theorem 65, we show how to approximate any EDS weighted relation with a constant factor. However, that construction does not yield a sufficient bound on the approximation factor; we present it only in order to provide some intuition for the more opaque construction in Theorem 67. Using that, we establish the global s-tractability of EDS languages in Theorem 68.

In this section, we equate weighted relations admitting multimorphism $\langle c_{0}\rangle$ with set functions; the correspondence is formally stated in the following definition. Note that we may without loss of generality assume that the minimum assigned value equals [math], as adding a rational constant to a weighted relation preserves tractability.

Definition 61**.**

Let $\gamma$ be an $r$-ary weighted relation such that, for any $r$-tuple $\mathbf{x}$, $\gamma(\mathbf{x})\geq\gamma(\mathbf{0}^{r})=0$. The corresponding set function $\gamma^{\prime}$ on $[r]$ is defined by $\gamma^{\prime}(X)=\gamma(\mathbf{x})$ where $x_{i}=1\iff i\in X$.

The definition of $\alpha$-EDS weighted relations then translates into the following:

Definition 62**.**

For any $\alpha\geq 1$, a set function $\gamma$ on $V$ is $\alpha$-EDS if, for every $X,Y\subseteq V$, it holds

[TABLE]

Remark 63**.**

Inequality (12) could be modified so that (62) becomes symmetric, say

[TABLE]

It is easy to see that, although the set of $\alpha$-EDS weighted relations for a fixed $\alpha$ would be different, this change would not affect the set of EDS weighted relations. We opt for the shorter, albeit asymmetric, definition.

Definition 64**.**

Let $J$ be an instance of the GMC problem on vertices $V$ and $\gamma$ a set function on $V$. For any $\alpha\geq 1$, we say that $J$ $\alpha$-approximates $\gamma$ if, for all $X\subseteq V$,

[TABLE]

A set function is $\alpha$-approximable if there exists a GMC instance that $\alpha$-approximates it, and it is approximable if it is $\alpha$-approximable for some $\alpha\geq 1$.

Theorem 65**.**

Any $\alpha$-EDS set function is approximable.

Proof.

Let $\gamma$ be an $\alpha$-EDS set function on $[n]$ and $\gamma^{\prime}$ the corresponding $n$-ary weighted relation. By Corollary 30, both $\operatorname{Feas}(\gamma^{\prime})$ and $\operatorname{Opt}(\gamma^{\prime})$ are essentially downsets. The rest of the proof relies only on this property and does not depend on the value of $\alpha$. The intuition behind our construction is that a downset can be represented by a superadditive function on $[n]$, and binary equality relations can be represented by edges.

There exist $A_{\operatorname{Feas}},A_{\operatorname{Opt}}\subseteq[n]$, downsets $\mathcal{S}_{\operatorname{Feas}}\subseteq 2^{A_{\operatorname{Feas}}}$, $\mathcal{S}_{\operatorname{Opt}}\subseteq 2^{A_{\operatorname{Opt}}}$, and sets of pairs of distinct coordinates $E_{\operatorname{Feas}},E_{\operatorname{Opt}}$ such that $|A_{\operatorname{Feas}}|+|E_{\operatorname{Feas}}|=|A_{\operatorname{Opt}}|+|E_{\operatorname{Opt}}|=n$ and

[TABLE]

We construct an instance $J$ of the GMC problem on vertices $[n]$ as follows. Let $w_{\operatorname{Feas}}(i,j)=\infty$ if $\{i,j\}\in E_{\operatorname{Feas}}$ and $w_{\operatorname{Feas}}(i,j)=0$ otherwise. Let $w_{\operatorname{Opt}}(i,j)=1$ if $\{i,j\}\in E_{\operatorname{Opt}}$ and $w_{\operatorname{Opt}}(i,j)=0$ otherwise. Then the weight of edge $(i,j)$ is $w(i,j)=w_{\operatorname{Feas}}(i,j)+w_{\operatorname{Opt}}(i,j)$. Let $f_{\operatorname{Feas}}$ be a set function on $[n]$ defined by $f_{\operatorname{Feas}}(X)=0$ if $X\cap A_{\operatorname{Feas}}\in\mathcal{S}_{\operatorname{Feas}}$ and $f_{\operatorname{Feas}}(X)=\infty$ otherwise; $f_{\operatorname{Feas}}$ is superadditive because $\mathcal{S}_{\operatorname{Feas}}$ is a downset. Let $f_{\operatorname{Opt}}$ be a set function on $[n]$ defined by $f_{\operatorname{Opt}}(X)=0$ if $X\cap A_{\operatorname{Opt}}\in\mathcal{S}_{\operatorname{Opt}}$ and $f_{\operatorname{Opt}}(X)=|X\cap A_{\operatorname{Opt}}|$ otherwise; $f_{\operatorname{Opt}}$ is superadditive because $\mathcal{S}_{\operatorname{Opt}}$ is a downset. Then the superadditive function defining instance $J$ is $f=f_{\operatorname{Feas}}+f_{\operatorname{Opt}}$.

By the construction, it holds $\gamma(X)<\infty\iff J(X)<\infty$ and $\gamma(X)=0\iff J(X)=0$. Moreover, for any $X$ such that $0<J(X)<\infty$, it holds $1\leq J(X)\leq n$. If the set

[TABLE]

is empty, then instance $J$ $1$-approximates $\gamma$; otherwise let $b_{\text{min}},b_{\text{max}}$ denote the minimum and maximum of $B$. We scale the weights of the edges $w$ and the superadditive function $f$ by a factor of $b_{\text{min}}/n$ to obtain an instance $J^{\prime}$ such that $J^{\prime}(X)\leq\gamma(X)$ for all $X$. Instance $J^{\prime}$ then $(n\cdot b_{\text{max}}/b_{\text{min}})$-approximates $\gamma$. ∎

To establish the tractability of infinite EDS languages, we need a better bound on the approximability of $\alpha$-EDS set functions than the one given in Theorem 65. This is achieved in Theorem 67, which we prove using the following technical lemma. We refer the reader to [17, Theorem 1.1] for an example of the application of this proof technique in a simpler setting.

Lemma 66**.**

Let $\gamma$ be an $\alpha$-EDS set function on $V$ for some $\alpha\geq 1$. For any distinct $u,v\in V$, let $T_{\{u,v\}}$ be a subset of $V$ such that $\left|T_{\{u,v\}}\cap\{u,v\}\right|=1$. Then, for any $R\subseteq S\subseteq V$, it holds

[TABLE]

Proof.

First, we show by induction that, for any $X,Y_{1},\dots,Y_{n}\subseteq V$, it holds

[TABLE]

For $n=1$, this is equivalent to (62). As for the inductive step, assume that (69) holds for $n\geq 1$. By the inductive hypothesis and (62), we get

[TABLE]

If $\gamma(S)=\infty$, the inequality claimed by this lemma trivially holds. In the rest of the proof, we assume $\gamma(S)<\infty$. For any $u\in R$, $v\in S\setminus R$, we define a set $T_{uv}^{\prime}$ such that $T_{uv}^{\prime}\cap\{u,v\}=\{v\}$: If $v\in T_{\{u,v\}}$, let $T_{uv}^{\prime}=T_{\{u,v\}}$; otherwise let $T_{uv}^{\prime}=S\setminus T_{\{u,v\}}$. We claim that

[TABLE]

This is trivially true in the case of $T_{uv}^{\prime}=T_{\{u,v\}}$, and it follows from (62) in the case of $T_{uv}^{\prime}=S\setminus T_{\{u,v\}}$. By (72), it holds

[TABLE]

For any $u\in R$, let

[TABLE]

By properties of $T_{uv}^{\prime}$, it holds $u\in W_{u}\subseteq R$. Moreover, we have

[TABLE]

by (69), which together with (74) gives us

[TABLE]

As it holds $\bigcup_{u\in R}W_{u}=R$, we have

[TABLE]

and hence

[TABLE]

As it holds $\alpha\cdot(\gamma(S)+\gamma(S\setminus R))\geq\gamma(R)$, this proves the claimed inequality. ∎

Theorem 67**.**

Any $\alpha$-EDS set function on $V$ is $\alpha^{n+2}\left(n^{3}+2n\right)$-approximable, where $n=|V|$.

Proof.

Let $\gamma$ be an $\alpha$-EDS set function on $V$ for some $\alpha\geq 1$. We construct an instance $J$ of the GMC problem on vertices $V$ such that it $\alpha^{n+2}\left(n^{3}+2n\right)$-approximates $\gamma$. The weight of edge $(u,v)$ is

[TABLE]

Let $f$ be a set function on $V$ defined as

[TABLE]

We claim that $f$ is a superadditive set function. As $\gamma(\emptyset)=0$, it holds $f(\emptyset)=0$. Consider any disjoint $X,Y\subseteq V$ and let $Z\supseteq X\cup Y$ be a minimiser in (82) for $f(X\cup Y)$. It holds $f(X)\leq|X|\cdot\left(|Z|^{2}+2\right)\cdot\gamma(Z)/\left(n^{3}+2n\right)$ and $f(Y)\leq|Y|\cdot\left(|Z|^{2}+2\right)\cdot\gamma(Z)/\left(n^{3}+2n\right)$, and hence

[TABLE]

The edge weights $w$ and superadditive set function $f$ define the GMC instance $J$. Now we prove that it $\alpha^{n+2}\left(n^{3}+2n\right)$-approximates $\gamma$.

First, we show that $J(R)\leq\gamma(R)$ for all $R\subseteq V$. By (82), we have $f(R)\leq|R|\cdot\left(|R|^{2}+2\right)\cdot\gamma(R)/\left(n^{3}+2n\right)$. For any edge $(u,v)$ cut by $R$ (i.e. $|R\cap\{u,v\}|=1$), it holds $w(u,v)\leq\gamma(R)/\left(n^{3}+2n\right)$ by (81), and hence $g(R)\leq|R|\cdot|V\setminus R|\cdot\gamma(R)/\left(n^{3}+2n\right)$. Together, this gives

[TABLE]

Second, we show that $\alpha^{n+2}\left(n^{3}+2n\right)\cdot J(R)\geq\gamma(R)$ for all $R\subseteq V$. For $R=\emptyset$, the inequality holds, as $J(\emptyset)=\gamma(\emptyset)=0$. Otherwise, let $S\supseteq R$ be a minimiser in (82) for $f(R)$, and $T_{\{u,v\}}$ a minimiser in (81) for any edge $(u,v)$. It holds

[TABLE]

and therefore, by Lemma 66, $\alpha^{n+2}\left(n^{3}+2n\right)\cdot J(R)\geq\alpha^{|S|+2}\left(n^{3}+2n\right)\cdot J(R)\geq\gamma(R)$. ∎

Theorem 68**.**

Any EDS language is globally s-tractable.

Proof.

Let $\Gamma$ be an EDS language and $\alpha^{\prime}\geq 1$ such that every weighted relation in $\Gamma$ is $\alpha^{\prime}$-EDS. Without loss of generality, we may assume that $\gamma\left(\mathbf{0}^{\operatorname{ar}(\gamma)}\right)=0$ for every $\gamma\in\Gamma$, and hence identify weighted relations with their corresponding set functions. Weighted relations in $\Gamma$ are of bounded arity and therefore, by Theorem 67, there exists $\alpha$ such that every $\gamma\in\Gamma$ is $\alpha$-approximable. We will denote by $J_{\gamma}$ a GMC instance that $\alpha$-approximates $\gamma$.

Given a $\operatorname{VCSP_{s}}(\Gamma)$ instance $I$ with an objective function

[TABLE]

we denote by $\phi_{I}$ the corresponding set function and construct a GMC instance $J$ that $\alpha$-approximates $\phi_{I}$. For $i\in[q]$, we relabel the vertices of $J_{\gamma_{i}}$ to match the variables in the scope $\mathbf{x}^{i}$ of the $i$th constraint (i.e., vertex $j$ is relabelled to $x^{i}_{j}$) and identify vertices in case of repeated variables. As the constraint is weighted by a non-negative factor $w_{i}$, we also scale the weights of the edges of $J_{\gamma_{i}}$ and the superadditive function by $w_{i}$. (Note that non-negative scaling preserves superadditivity.) Instance $J$ is then obtained by adding up GMC instances $J_{\gamma_{i}}$ for all $i\in[q]$.

Let $\mathbf{x}\in D^{n}$ denote a surjective assignment minimising $\phi_{I}^{\prime}$, $X\subseteq[n]$ the corresponding set $\{i\in[n]~{}|~{}x_{i}=1\}$, $Y\subseteq[n]$ an optimal solution to $J$, and $\lambda=J(Y)$. Since $J$ $\alpha$-approximates $\phi_{I}$, it holds

[TABLE]

and hence $X$ is an $\alpha$-optimal solution to $J$. By Lemma 53, we can determine whether $\lambda=0$, in which case any optimal solution to $J$ is also optimal for $\phi_{I}$; and whether $\lambda=\infty$. If $0<\lambda<\infty$, we find all $\alpha$-optimal solutions by Theorem 59. ∎

We now prove Theorem 22.

Proof.

We only need to prove the theorem in the case of an EDS language (whether $\Gamma$ or $\neg(\Gamma)$, which is symmetric), as the remaining classes of globally s-tractable languages include constants $\mathcal{C}_{D}$ and thus admit a polynomial-delay algorithm using standard self-reduction techniques [14, 10].

Let $\Gamma$ be an EDS language. As in the proof of Theorem 68, we may assume that every weighted relation in $\Gamma$ assigns [math] as the minimum value. Given an instance of $\operatorname{VCSP_{s}}(\Gamma)$, we can determine in polynomial time, by Lemma 53, whether $\lambda=0$, $0<\lambda<\infty$, or $\lambda=\infty$. If $\lambda=0$, then optimal solutions incur the minimum value from every constraint. By applying $\operatorname{Opt}$ to all constraints, we obtain a CSP instance invariant under $\min$ (by Lemma 13), and hence are able to enumerate all optimal solutions with a polynomial delay by the results in [14]. If $0<\lambda<\infty$, then the claim follows from the proof of Theorem 68; moreover, the number of optimal solutions is polynomially bounded (see Theorem 59). Finally, the case $\lambda=\infty$ is trivial. ∎

6 Conclusions

We have established the complexity classification of surjective VCSPs on two-element domains. An obvious open problem is to consider surjective VCSPs on three-element domains. A complexity classification is known for $\{0,\infty\}$-valued languages [5] and $\mathbb{Q}$-valued languages [26] (the latter generalises the $\{0,1\}$-valued case obtained in [28]). In fact, [31] implies a dichotomy for $\overline{\mathbb{Q}}$-valued languages on a three-element domain. However, all these results depend on the notion of core and the presence of constants $\mathcal{C}_{D}$ in the language, and thus it is unclear how to use them to obtain a complexity classification in the surjective setting. Moreover, one special case of the CSP on a three-element domain is the 3-No-Rainbow-Colouring problem [4], whose complexity status is open.

Acknowledgements

We would like to thank Yuni Iwamasa, who prompted us to extend the complexity classification to languages of infinite size (and bounded arity). We also thank the anonymous reviewers of the two extended abstracts [43, 23] of this paper.