Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an ${\rm M}^{\natural}$-Concave Set Function

TL;DR

This paper proves that every finite distributive lattice can be represented as the set of minimizers of an ${ m M}^{ atural}$-concave set function, extending the known relationship between lattices and submodular functions.

Contribution

It establishes that any finite distributive lattice is isomorphic to the minimizer set of an ${ m M}^{ atural}$-concave function, broadening the understanding of discrete convex analysis.

Findings

Any finite distributive lattice can be realized as a minimizer set of an ${ m M}^{ atural}$-concave function.

The result extends the classical correspondence between distributive lattices and submodular functions.

Provides a new connection between lattice theory and discrete convex analysis.

Abstract

Submodularity is an important concept in combinatorial optimization, and it is often regarded as a discrete analog of convexity. It is a fundamental fact that the set of minimizers of any submodular function forms a distributive lattice. Conversely, it is also known that any finite distributive lattice is isomorphic to the minimizer set of a submodular function, through the celebrated Birkhoff's representation theorem. ${\rm M}^{\natural}$-concavity is a key concept in discrete convex analysis. It is known for set functions that the class of ${\rm M}^{\natural}$-concavity is a proper subclass of submodularity. Thus, the minimizer set of an ${\rm M}^{\natural}$-concave function forms a distributive lattice. It is natural to ask if any finite distributive lattice appears as the minimizer set of an ${\rm M}^{\natural}$-concave function. This paper affirmatively answers the question.

Equations

$$f(X)+f(Y)\ge f(X\cup Y)+f(X\cap Y)$$

$$f(X)$$

$$f(X)\{\begin{array}{ll}=0& \text{if X∈I(P),}\\ >0& \text{otherwise,}\end{array}$$

$$f(X)+f(Y)\le f(X-i)+f(Y+i)$$

$$f(X)+f(Y)\le f(X-i+j)+f(Y+i-j)$$

$$f(X)\{\begin{array}{ll}=0& \text{if X∈I(P),}\\ >0& \text{otherwise,}\end{array}$$

$$f(X)=\max\{w(M):\mbox{$M\subseteq E$ is a matching satisfying $\partial M\cap U=X$}\}$$

$$\displaystyle w_{0}(\{u_{i},v_{j}\})=\begin{cases}0&\mbox{if $i=j$,}\\ 1&\mbox{otherwise,}\end{cases}$$

$$f_{0}(X)=\max{\{w_{0}(M):\mbox{$M\subseteq E_{0}$ is a matching satisfying $\partial M\cap U=U_{X}$}\}}$$

$$w_1(\{u_i,v_j\})=\max \{|S|-1:\text{S⊆N is a chain such that j≼s≼i for any s∈S}\},$$

$$f_{1}(X)=\max{\{w_{1}(M):\mbox{ $M\subseteq E_{1}$ is a matching satisfying $\partial M\subseteq U_{X}\cup V_{\overline{X}}$}\}}$$

$$\displaystyle w_{2}(\{u_{i},v_{j}\})=\begin{cases}0&\mbox{if $i=j$,}\\ w_{1}(\{u_{i},v_{j}\})&\mbox{otherwise,}\end{cases}$$

$$f_{2}(X)=\max{\{w_{2}(M):\mbox{$M\subseteq E_{2}$ is a matching satisfying $\partial M\cap U=X$}\}}$$

$$g_r(X)=\exp (-rf(X))$$

$$g_{\infty }(X)=\{\begin{array}{ll}1& \text{mbox}ifX\in \mathcal{I}(\mathcal{P}),\\ 0& \text{mbox}otherwise,\end{array}$$

Full text

**Keywords: ** ${\rm M}^{\natural}$-concave, valuated matroid, submodular, distributive lattice, #BIS

1 Introduction

1.1 The minimizer set of a submodular function forms a distributive lattice

A set function $f\colon 2^{N}\to\mathbb{R}$ for a finite set $N$ is submodular if

[TABLE]

holds for any $X,Y\in 2^{N}$. Submodularity is often regarded as a discrete analog of convexity [19, 9, 21, 26]. A submodular function is efficiently minimized [28, 12, 13, 18], and it has many applications in economics, machine learning, etc. On the other hand, maximization of a submodular function, e.g., max cut, is a celebrated NP-hard problem, and approximation is recently investigated with applications in machine learning, see e.g., [27, 8].

It is a fundamental fact on submodular functions that the set of minimizers of a submodular function forms a distributive lattice. Conversely, any finite distributive lattice “appears” as the minimizers of a submodular function. For a finite partially ordered set (poset) $\mathcal{P}=(N,\preccurlyeq)$, $I\subseteq N$ is an ideal of $\mathcal{P}$ if $x\preccurlyeq y\in I\Rightarrow x\in I$ holds for any $x,y\in N$. Let ${\cal I}({\cal P})$ denote the set of whole ideals of the poset ${\cal P}$. Then, ${\cal I}({\cal P})$ forms a distributive lattice. The following celebrated theorem is due to Birkhoff [1].

Theorem 1.1** (Birkhoff’s representation theorem [1, 9]).**

For any finite distributive lattice ${\cal D}$, there exists a poset ${\cal P}$ such that ${\cal I}({\cal P})$ is isomorphic to ${\cal D}$.

Using Theorem 1.1, it is known that any finite distributive lattice is isomorphic to the minimizer set of a submodular function, as follows.

Proposition 1.2** (see e.g., [9]).**

As given a finite poset ${\cal P}=(N,\preccurlyeq)$, let $f\colon 2^{N}\to\mathbb{R}$ be defined by

[TABLE]

for any $X\in 2^{N}$, where $j\prec i$ denotes $j\preccurlyeq i$ and $j\neq i$. Then $f$ is submodular, and it satisfies

[TABLE]

for any $X\in 2^{N}$.

Proposition 1.2 provides a representation of a finite distributive lattice with a submodular function; any finite distributive lattice is represented as the minimizer set of a submodular function. Another interesting representation theorem for finite distributive lattices is described by stable matchings; John Conway showed that the set of stable matchings forms a distributive lattice under the preferences of Men (or Women, similarly) [17]. Blair [2] showed that any finite distributive lattice is isomorphic to the distributive lattice of the stable matchings for a stable marriage instance.

1.2 ${\rm M}^{\natural}$-concavity is a proper subclass of submodularity

A set function $f\colon 2^{N}\rightarrow\mathbb{R}$ is ${\rm M}^{\natural}$-concave (cf. [24]) if, for any $X,Y\in 2^{N}$ and $i\in X\setminus Y$, we have

[TABLE]

or else

[TABLE]

holds for some $j\in Y\setminus X$, where $X-i,Y+i,X-i+j,Y+i-j$ are abbreviations of $X\setminus\{i\},Y\cup\{i\},(X\setminus\{i\})\cup\{j\},(Y\cup\{i\})\setminus\{j\}$, respectively. ${\rm M}^{\natural}$-concavity is introduced by Murota [21, 25] as a quantitative version of matroid extending the exchange property to set functions [24], and it is a closely related to valuated matroid introduced by Dress and Wenzel [5, 6]. Fujishige and Yang [11] showed that ${\rm M}^{\natural}$-concavity is equivalent to gross substitutes property of Kelso and Crawford [15] in economics. ${\rm M}^{\natural}$-concavity is also found in many areas such as systems analysis, inventory theory in operations research, and mathematical economics and game theory including stable matching [24, 20, 30, 10]. ${\rm M}^{\natural}$-concavity is extensionally defined on multidimensional integer lattice, and it is a key concept in the theory of “discrete convex analysis” [21, 22, 23, 24, 25, 29, 30].

Interestingly, any ${\rm M}^{\natural}$-concave set function is submodular [21]. Thus, an ${\rm M}^{\natural}$-concave set function is minimized efficiently, using an algorithm for submodular minimization. In contrast, not every submodular function is ${\rm M}^{\natural}$-concave; meaning that ${\rm M}^{\natural}$-concavity is a proper subclass of submodularity for set functions. In fact, any ${\rm M}^{\natural}$-concave set function is efficiently maximized by a greedy algorithm, so is a matroid rank function [6, 29, 22]. It is known that the set of maximizers of an ${\rm M}^{\natural}$-concave set function forms a generalized matroid (a.k.a. ${\rm M}^{\natural}$-convex family), and conversely any generalized matroid appears as the maximizer set of an ${\rm M}^{\natural}$-concave set function [21, 23].

Since an ${\rm M}^{\natural}$-concave set function is submodular, the minimizer set of an ${\rm M}^{\natural}$-concave set function forms a distributive lattice. It is a natural question if any finite distributive lattice appears as the minimizer set of an ${\rm M}^{\natural}$-concave function. A naive candidate may be the function given by (2). We briefly remark that this is not the case.

Proposition 1.3**.**

The set function $f$ given by (2) is NOT ${\rm M}^{\natural}$-concave, in general.

Proof.

Let ${\cal P}=(\{1,2,3\},\preccurlyeq)$ be given by $1\prec 3$ and $2\prec 3$. Then, ${\cal I}({\cal P})=\{\emptyset,\{1\},\{2\},\{1,2\},\{1,2,3\}\}$. We will check that the set function $f$ given by (2) is not M*♮*-concave for ${\cal P}$ (see Figure 1). Let $X=\{3\}$, $Y=\{1,2\}$ and $i=3$. Then, it is easy to observe that $f$ does not satisfy (3), because $f(X)=2$, $f(Y)=0$, $f(X-i)=0$ and $f(Y+i)=0$. It is also easily confirmed that $f$ does not satisfy (4) for any $j\in\{1,2\}$. For $j=1$, $f(X-i+j)=f(\{1\})=0$ and $f(Y+i-j)=f(\{2,3\})=1$, then (4) does not hold. For $j=2$, $f(X-i+j)=f(\{2\})=0$ and $f(Y+i-j)=f(\{1,3\})=1$, then (4) does not hold. ∎

This paper presents a representation theorem for finite distributive lattices, and affirmatively answers the question. Our result supports the fact that ${\rm M}^{\natural}$-concavity covers a large part of submodularity for set functions; the minimizer sets of ${\rm M}^{\natural}$-concave set functions represent all finite distributive lattices, so do submodular functions.

2 $\mathrm{M}^{\natural}$-concave functions with the minimizer set ${\cal I}({\cal P})$

This section establishes the following theorem.

Theorem 2.1**.**

For any finite poset ${\cal P}=(N,\preccurlyeq)$, the exists an $\mathrm{M}^{\natural}$-concave function $f\colon 2^{N}\to\mathbb{R}$ satisfying

[TABLE]

for any $X\in 2^{N}$.

In proofs of Theorem 2.1, we will use the following known fact (see Appendix for a proof).

Proposition 2.2** (cf. [23] (Section 3.6 “Example” 6, p. 173)).**

Let $G=(U,V;E)$ be a bipartite graph with vertex bipartition $(U,V)$ and edge set $E$, and let $w\colon E\to\mathbb{R}$ be an edge weight function. For $M\subseteq E$, we denote by $\partial M$ the set of the vertices incident to some edge in $M$, and call $M$ a matching if $|U\cap\partial M|=|M|=|V\cap\partial M|$. For $X\subseteq U$ denote by $f(X)$ the maximum weight of a matching that precisely matched $X$ in $U$, i.e.,

[TABLE]

with $w(M)=\sum_{e\in M}w(e)$, where $f(X)=-\infty$111 Here we omit the argument on “effective domain” concerning $f(X)=-\infty$ (see e.g., [23]). The ${\rm M}^{\natural}$-concave functions appearing in this paper satisfy $f(X)>-\infty$ for any $X\in 2^{N}$. if no such $M$ exists for $X$. Then $f\colon 2^{U}\to\mathbb{R}\cup\{-\infty\}$ is an $M^{\natural}$-concave function.

2.1 A simple ${\rm M}^{\natural}$-concave set function

Now, we present an ${\rm M}^{\natural}$-concave set function for Theorem 2.1. As given an arbitrary finite poset ${\cal P}=(N,\preccurlyeq)$ of order $n$, let $G_{0}=(U,V;E_{0})$ be a bipartite graph with vertex set consisting of $U=\{u_{1},u_{2},\dots,u_{n}\}$ and $V=\{v_{1},v_{2},\dots,v_{n}\}$, and edge set given by E_{0}=\{\{u_{i},v_{j}\}:\mbox{u_{i}\in U$,$v_{j}\in V$and$j\preccurlyeq i$on$\mathcal{P}}\}. Let $w_{0}\colon E_{0}\to\mathbb{Z}_{\geq 0}$ be the edge weight function given by

[TABLE]

for any edge $\{u_{i},v_{j}\}\in E_{0}$. Let $f_{0}\colon 2^{N}\rightarrow\mathbb{R}$ be defined by

[TABLE]

for any $X\in 2^{N}$, where $w_{0}(M)=\sum_{e\in M}w_{0}(e)$ and $U_{X}=\{u_{i}\in U:i\in X\}$.

Lemma 2.3**.**

For any finite poset, $f_{0}$ is $\mathrm{M}^{\natural}$-concave, and it satisfies (5).

Proof.

The ${\rm M}^{\natural}$-concavity of $f_{0}$ follows from Proposition 2.2. (5) is not difficult, as follows. If $X\not\in{\cal I}(P)$, there exist $i\in X$ and $j\not\in X$ such that $j\prec i$. Then, there is a matching $M$ such that $\{u_{i},v_{j}\}\in M$ and $\partial M\cap U=U_{X}$. Since $w(\{u_{i},v_{j}\})=1$, $f(X)\geq w(M)>0$, and we obtain (5) in this case. Suppose $X\in{\cal I}(P)$. Then, $\{\{u_{i},v_{i}\}:i\in X\}$ is a trivial matching, and its weight is zero. We claim that the trivial matching is the unique matching satisfying $\partial M\cap U=U_{X}$ in the case. Let $\emptyset=Y_{0}\subset Y_{1}\subset\cdots\subset Y_{k}=X$ be a maximal chain from $\emptyset$ to $X$ on ${\cal I}({\cal P})$ where $k=|X|$. Notice that $|\Gamma(U_{Y_{l}}\})|=l$ holds for $l=0,1,\ldots,k$ since $Y_{l}\in{\cal I}({\cal P})$, where $\Gamma(U^{\prime})$ for $U^{\prime}\subseteq U$ denotes the adjacent vertices of $U^{\prime}$, i.e., $\Gamma(U^{\prime})=\{v\in V:\exists u\in U^{\prime},\ \{u,v\}\in E_{0}\}$. By an induction on $l$, we can see that only the trivial matching for $Y_{l}$ satisfies the condition $\partial M\cap U=U_{Y_{l}}$. Thus, $f_{0}(X)=0$ in the case. ∎

Theorem 2.1 follows Lemma 2.3.

2.2 Another $\mathrm{M}^{\natural}$-concave set function

This subsection presents another $\mathrm{M}^{\natural}$-concave function for Theorem 2.1, in fact it is presented in a preliminary version of this manuscript. Here, we give a simpler proof. As given an arbitrary finite poset ${\cal P}=(N,\preccurlyeq)$ of order $n$, let $G_{1}=(U,V;E_{1})$ be a bipartite graph with vertex set consisting of $U=\{u_{1},u_{2},\dots,u_{n}\}$ and $V=\{v_{1},v_{2},\dots,v_{n}\}$, and edge set given by E_{1}=\{\{u_{i},v_{j}\}:\mbox{u_{i}\in U$,$v_{j}\in V$and$j\prec i$on$\mathcal{P}}\}. Note that $E_{1}=E_{0}\setminus\{\{u_{i},v_{i}\}:i\in N\}$. Let $w_{1}\colon E_{1}\to\mathbb{Z}_{\geq 0}$ be the edge weight function given by

[TABLE]

for any edge $\{u_{i},v_{j}\}\in E_{1}$, where $S\subseteq N$ is a chain of $S$ if $(S,\preccurlyeq)$ is a totally ordered set, i.e., $w_{1}(\{u_{i},v_{j}\})$ denotes the “length” of a maximum chain between $j$ and $i$ for $j\prec i$. Let $f_{1}\colon 2^{N}\rightarrow\mathbb{R}$ be defined by

[TABLE]

for any $X\in 2^{N}$, where $U_{X}=\{u_{i}\in U:i\in X\}$ and $V_{\overline{X}}=\{v_{i}\in V:i\not\in X\}$.

Lemma 2.4**.**

For any finite poset, $f_{1}$ is $\mathrm{M}^{\natural}$-concave, and it satisfies (5).

It is not trivial from Proposition 2.2 that $f_{1}$ is ${\rm M}^{\natural}$-concave222 We gave a naive proof in the preliminary version. . To prove Lemma 2.4, we introduce another set function $f_{2}$, as follows. As given an arbitrary finite poset ${\cal P}=(N,\preccurlyeq)$ of order $n$, let $G_{2}=(U,V;E_{2})$ be given by $G_{2}=G_{0}$. Let $w_{2}\colon E_{2}\to\mathbb{Z}_{\geq 0}$ be the edge weight function given by

[TABLE]

for any edge $\{u_{i},v_{j}\}\in E_{2}$. Let $f_{2}\colon 2^{N}\rightarrow\mathbb{R}$ be defined by

[TABLE]

for any $X\in 2^{N}$, where $U_{X}=\{u_{i}\in U:i\in X\}$.

Lemma 2.5**.**

For any finite poset, $f_{2}\equiv f_{1}$. Furthermore, $f_{2}$ is $\mathrm{M}^{\natural}$-concave, and it satisfies (5).

Proof.

Firstly, we prove $f_{1}(X)\leq f_{2}(X)$ for any $X\in 2^{N}$. Suppose that $M\subseteq E_{1}$ is a matching attaining $f_{1}(X)=w_{1}(M)$, i.e., $\partial M\subseteq U_{X}\cup V_{\overline{X}}$. Let $M^{\prime}=M\cup\{\{u_{i},v_{i}\}:u_{i}\in U_{X}\setminus\partial M\}$. Then, $\partial M^{\prime}\cap U=U_{X}$ holds, and hence $f_{2}(X)\geq w_{2}(M^{\prime})=w_{1}(M)=f_{1}(X)$.

Next, we prove $f_{1}(X)\geq f_{2}(X)$ for any $X\in 2^{N}$. Suppose that $M\subseteq E_{2}$ is a matching attaining $f_{2}(X)=w_{2}(M)$, i.e., $\partial M\cap U=U_{X}$. We iteratively construct a matching $M^{\prime}\subseteq E_{1}$ satisfying $\partial M^{\prime}\subseteq U_{X}\cup V_{\overline{X}}$, as follows. To begin with, set $M^{\prime}_{1}=M\setminus\{\{u_{i},v_{i}\}:i\in X\}$. If $\partial M^{\prime}_{1}\cap V_{X}=\emptyset$ then $M^{\prime}_{1}$ is a desired matching, and we obtain $f_{1}(X)\geq f_{2}(X)$ in the case, since $w_{1}(M)=w_{2}(M^{\prime}_{1})$, clearly. Suppose $v_{i}\in\partial M^{\prime}_{1}\cap V_{X}$. Here, we remark that $M^{\prime}_{1}$ satisfies the condition () “if $v_{i}\in\partial M^{\prime}_{1}\cap V_{X}$ then $u_{i}\in\partial M^{\prime}_{1}$,” since $\partial M\cap U=U_{X}$. Without loss of generality, we may assume that $\{u_{k},v_{i}\}\in M^{\prime}_{1}$ and $\{u_{i},v_{j}\}\in M^{\prime}_{1}$. This implies that $j\prec i\prec k$, and hence $\{u_{k},v_{j}\}\in E_{2}$. Set $M^{\prime}_{2}:=M^{\prime}_{1}\cup\{\{u_{k},v_{j}\}\}\setminus\{\{u_{k},v_{i}\},\{u_{i},v_{j}\}\}$. Then, we obtain a matching $M^{\prime}_{2}$, such that $\partial M^{\prime}_{2}\cap V_{X}=(\partial M^{\prime}_{1}\cap V_{X})\setminus\{v_{i}\}$ holds and $M^{\prime}_{2}$ inherits the condition (). We also remark that $w_{2}(M^{\prime}_{2})\geq w_{2}(M^{\prime}_{1})$ holds, since $w_{2}(\{u_{k},v_{j}\})\geq w_{2}(\{u_{k},v_{i}\})+w_{2}(\{u_{i},v_{j}\})$ by (9). Recursively applying the above arguments, we eventually obtain a matching $M^{\prime}$ such that $\partial M^{\prime}\cap V_{X}=\emptyset$, meaning that $\partial M^{\prime}\subseteq U_{X}\cup V_{\overline{X}}$. It is not difficult to see from the above argument, $f_{1}(X)\geq w_{1}(M^{\prime})\geq w_{2}(M)=f_{2}(X)$.

The ${\rm M}^{\natural}$-concavity of $f_{2}$ follows from Proposition 2.2. It is easy to see that $f_{2}$ satisfies (5), in a similar way as Lemma 2.3. ∎

Lemma 2.4 is immediate from Lemma 2.5. Each of Lemmas 2.4 and 2.5 implies Theorem 2.1.

3 The indicator function of a distributive lattice

Let $r\in\mathbb{R}_{>0}$ be an arbitrary. As given a finite poset ${\cal P}=(N,\preccurlyeq)$, we define $g_{r}\colon 2^{N}\to\mathbb{R}$ by

[TABLE]

for $X\in 2^{N}$, where $f$ is an ${\rm M}^{\natural}$-concave set function satisfying (5). For convenience, let $g_{\infty}(X)=\lim_{r\to\infty}g_{r}(X)$, then

[TABLE]

holds for any $X\in 2^{N}$, meaning that $g_{\infty}$ is the indicator function of the distributed lattice ${\cal I}({\cal P})$.

Specifically, if we set $r=(n+2)\ln 2$, then we obtain $g_{r}(X)=2^{-(n+2)f(X)}$, and then $|{\cal I}({\cal P})|-\frac{1}{4}\leq\sum_{X\in 2^{N}}g_{r}(X)\leq|{\cal I}({\cal P})|+\frac{1}{4}$ holds. This implies that if we have an approximation algorithm for the partition function of a log-${\rm M}^{\natural}$-convex function, then we can approximate the number of ideals of a poset. By standard arguments (cf. [14, 7]) about a fully polynomial-time randomized approximation scheme (FPRAS), we can conclude as follows.

Theorem 3.1**.**

If there is a polynomial time approximate sampler for a log-${\rm M}^{\natural}$-convex distribution, then counting bipartite independent set (#BIS) has an FPRAS.

4 Concluding Remarks

We have shown that any finite distributive lattice is isomorphic to the minimizer set of an ${\rm M}^{\natural}$-concave set function. The result implies that sampling from log-${\rm M}^{\natural}$-convex set function is #BIS-hard under the polynomial-time randomized approximate reduction. It is a major open problem if an FPRAS exists for #BIS (cf. [3]), with some applications such as stable matching [4, 16].

Acknowledgments

The authors are grateful to Kazuo Murota for his suggestion about the problem, and for his kind advice on our preliminary manuscript. The authors also grateful to Satoru Fujishige, Akihisa Tamura and Naoyuki Kamiyama for their valuable comments. This work is partly supported by JST PRESTO Grant Number JPMJPR16E4, Japan.

Appendix A Proof of Proposition 2.2

Proof of Proposition 2.2 (cf. [20] (Example 5.2.4, p. 282)).

Suppose $f(X)>-\infty$ and $f(Y)>-\infty$ for $X,Y\subseteq U$. Let $M_{X}$ and $M_{Y}$ be respectively matchings attaining $f(X)$ and $f(Y)$, i.e., $f(X)=w(M_{X})$ and $f(Y)=w(M_{Y})$. Let $M=M_{X}\cup(P\cap M_{Y})\setminus(P\cap M_{X})$ and let $M^{\prime}=M_{Y}\cup(P\cap M_{X})\setminus(P\cap M_{Y})$, where $P\subseteq M_{X}\cup M_{Y}$ is the alternating path from $u\in X\setminus Y$. Let $v\in U\cup V$ denote the other end of $P$. In case of $v\in V$, $\partial M\cap U=X-u$ and $\partial M^{\prime}\cap U=Y+u$ hold. This implies that $f(X-u)+f(Y+u)\geq w(M)+w(M^{\prime})=w(M_{X})+w(M_{Y})=f(X)+f(Y)$, and we obtain (3). In the other case, i.e., $v\in U$, $\partial M\cap U=X-u+v$ and $\partial M^{\prime}\cap U=Y+u-v$ hold. This implies that $f(X-u+v)+f(Y+u-v)\geq w(M)+w(M^{\prime})=w(M_{X})+w(M_{Y})=f(X)+f(Y)$, and we obtain (4). ∎