A $(4+\epsilon)$-approximation for $k$-connected subgraphs

TL;DR

This paper presents a new approximation algorithm for the $k$-Connected Subgraph problem, achieving ratios that improve upon previous bounds for large graphs and constant $k$, with implications for related covering problems.

Contribution

It introduces a $(4+rac{1}{ ext{ell}})$-approximation ratio for the problem, improving existing bounds for large $n$ and constant $k$, and extends results to crossing supermodular biset functions.

Findings

Improves approximation ratio to $2(2+1/ell)$ for large $n$

Achieves ratio $4+\epsilon$ for constant $k$

Matches best known ratios for augmentation and specific $k$ values

Abstract

We obtain approximation ratio $2(2+\frac{1}{\ell})$ for the (undirected) $k$-Connected Subgraph problem, where $\ell \approx \frac{1}{2} (\log_k n-1)$ is the largest integer such that $2^{\ell-1} k^{2\ell+1} \leq n$. For large values of $n$ this improves the $6$-approximation of Cheriyan and V\'egh when $n =\Omega(k^3)$, which is the case $\ell=1$. For $k$ bounded by a constant we obtain ratio $4+\epsilon$. For large values of $n$ our ratio matches the best known ratio $4$ for the augmentation version of the problem, as well as the best known ratios for $k=6,7$. Similar results are shown for the problem of covering an arbitrary crossing supermodular biset function.

Tables

Table 1. Table 1: Known approximation ratios for k 𝑘 k -Connected Subgraph and k 𝑘 k -Connectivity Augmentation problems. Here μ = ⌊ n ⌊ ( n − k ) / 2 ⌋ + 1 ⌋ 𝜇 𝑛 𝑛 𝑘 2 1 \mu=\left\lfloor\frac{n}{\lfloor(n-k)/2\rfloor+1}\right\rfloor , and note that if μ = 1 𝜇 1 \mu=1 then k ∈ { 1 , 2 } 𝑘 1 2 k\in\{1,2\} , and that if n ≥ 3 ​ k − 2 𝑛 3 𝑘 2 n\geq 3k-2 then μ = 2 𝜇 2 \mu=2 and H ​ ( μ ) = 3 / 2 𝐻 𝜇 3 2 H(\mu)=3/2 .

General		Augmentation
Undirected	Directed	Undirected	Directed
$O\left(\ln \frac{n}{n-k}\cdot \ln k\right)$ [28]	$O\left(\ln \frac{n}{n-k}\cdot \ln k\right)$ [28]	$2H\left(\mu \right)+1$ [29]	$H\left(\mu \right)+3/2$ [29]
$6$ if $n\ge k^3$ [4] (see also [16])		$O\left(\ln \left(n-k\right)\right)$ [28]	$O\left(\ln \left(n-k\right)\right)$ [28]
$\lceil \left(k+1\right)/2\rceil$ if $k\le 7$ [2, 6, 22]	$k+1$ if $k\le 6$ [22]

Table 2. Table 2: Known approximation ratios of k 𝑘 k -Connected Subgraph problems.

Costs	Undirected	Directed
$\{0,1\}$	$\min \{2,1+\frac{k^2}{2\mathrm{𝗈𝗉𝗍}}\}$ [12, 18]	in P [12]
$\{1,\infty \}$	$1-\frac{1}{k}+\frac{n}{\mathrm{𝗈𝗉𝗍}}\le 1+\frac{1}{k}$ [3] (see also [27])	$1-\frac{1}{k}+\frac{2n}{\mathrm{𝗈𝗉𝗍}}\le 1+\frac{1}{k}$ [3] (see also [27])
metric	$2+\left(k-1\right)/n$ [22]	$2+k/n$ [22]

Equations

$$f_{k{\sf\mbox{-}CS}}(\mathbb{A})=\left\{\begin{array}[]{ll}k-|\partial\mathbb{A}|&\mbox{if }\mathbb{A}\mbox{ is proper}\\ 0&\mbox{otherwise}\end{array}\right.$$

$$\begin{array}[]{llllllll}&&\tau(f)=&\min&c\cdot x&&\\ &\mbox{\bf(Biset-LP)}&&\mbox{s.t.}&x(\delta_{\hat{E}}(\mathbb{A}))\geq f(\mathbb{A})&\forall\mathbb{A}&\\ &&&&0\leq x_{e}\leq 1&\forall e&\end{array}$$

$$f(\mathbb{A}\cap \mathbb{B})+f(\mathbb{A}\cup \mathbb{B})\ge f(\mathbb{A})+f(\mathbb{B}).$$

$$f_R(\mathbb{A})=f(\mathbb{A})-\underset{\mathbb{A}}{\max }f(\mathbb{A})\cdot |A\cap R|.$$

$$k_f=1+\max \{|\partial \mathbb{A}|:f(\mathbb{A})>0\}.$$

$$f_R(\mathbb{A})+f_R(\mathbb{B})=f(\mathbb{A})+f(\mathbb{B})\le f(\mathbb{A}\cap \mathbb{B})+f(\mathbb{A}\cup \mathbb{B})=f_R(\mathbb{A}\cap \mathbb{B})+f_R(\mathbb{A}\cup \mathbb{B}).$$

$$c(I)\le c(I^{\prime })\le \sum _{e\in E^{\prime }}{c}_e{x}_e^{\prime }=\sum _{e\in \delta (R)}{c}_e{x}_e+2\sum _{e\in \gamma (V\setminus R)}{c}_e{x}_e$$

$$|U(f^I,k_f)\cup R|\le r^{\prime }[(2(k-1{)}^2+k]+(r-r^{\prime })\le r[(2(k-1{)}^2+k]=r(2k^2-3k+2)$$

$$\tau ={\gamma }_i+{\delta }_i+{\bar{\gamma }}_i$$

$$c(I)$$

$$c(I^{\prime })$$

$$c(J_i)$$

$$\sum _{i=1}^{\ell }c(J_i)$$

$$c(J_i)\le 2\tau (1+1/\ell )$$

Full text

1 Introduction

A graph is $k$-connected if it contains $k$ internally disjoint paths from every node to any other node. We consider the following problem on undirected graphs:

$k$-Connected Subgraph

Input:   A graph $\hat{G}=(V,\hat{E})$ with edge costs $\{c_{e}:e\in\hat{E}\}$ and an integer $k$.

Output: A minimum cost $k$-connected spanning subgraph of $\hat{G}$.

For undirected graphs, the problem is NP-hard for $k=2$ (the case $k=1$ is the MST problem), even when all edges in $\hat{G}$ have unit costs. This is since any feasible solution with $n=|V|$ edges is a Hamiltonian cycle. For directed graphs the problem is NP-hard already for $k=1$, by a similar reduction. Let $\vec{\rho}(k,n)$ and $\rho(k,n)$ denote the best possible approximation ratios for the directed and undirected variants, respectively. A standard bi-direction reduction gives $\rho(k,n)\leq 2\vec{\rho}(k,n)$, while by [24] $\rho(k,n)\geq\vec{\rho}(k-n/2,n/2)$ for $k\geq n/2+1$. All in all we get that for $k\geq n/2+1$ the approximability of directed and undirected cases is the same, up to a $2$ factor. This however does not exclude that the undirected case is easier when $n$ is much larger than $k$.

In the $k$-Connectivity Augmentation problem $\hat{G}$ contains a spanning $(k-1)$-connected subgraph of cost [math]. A feasible solution for the $k$-Connected Subgraph problem can be obtained by solving sequentially $\ell$-Connectivity Augmentation problems for $\ell=1,\ldots k$, but this reduction usually invokes a factor of $H(k)$ in the ratio, where $H(k)$ denotes the $k$-th harmonic number. Several ratios for $k$-Connected Subgraph were obtained in this way, c.f. [5, 23, 8, 28]. The currently best known ratios for the general and the augmentation version, for both directed and undirected graphs, are summarized in Table 1.

Note that we consider the node-connectivity version of the problem, for which classic techniques like the primal dual method [17] and the iterative rounding method [20, 9] do not seem to be applicable directly. Ravi and Williamson [32] gave an example of a $k$-Connectivity Augmentation instance when the primal dual method has ratio $\Omega(k)$. A related example of Aazami, Cheriyan and Laekhanukit [1] rules out the iterative rounding method. On the other hand, several works showed that the problem can be decomposed into a small number of “good” sub-problems. However, attempts to achieve a constant ratio for $k=n-o(n)$ (e.g., for $k=n-\Theta(\sqrt{n})$) failed even in the easier augmentation case, thus Cheriyan and Végh [4] suggested the following question:

What ratio can we achieve when $n$ is lower bounded by a function of $k$?

This essentially addresses the issue of ”asymptotic approximability” – as a function of the single parameter $k$, for all sufficiently large $n$. For undirected graphs Cheriyan and Végh [4] gave an elegant $6$ approximation when $n\geq k^{4}$, and this bound was slightly improved to $n\geq k^{3}$ in [16].

Note that the “asymptotic approximability” question seems almost settled for the augmentation version: by [28, 29] for both directed and undirected graphs we have a constant ratio unless $k=n-o(n)$; furthermore, for undirected graphs we have ratio $4$ for $n\geq 3k-2$ (ratio $3$ for directed graphs) [29], and this is also the best known ratios when $k=6,7$ for the general version [22].

From now and on we consider undirected graphs, unless stated otherwise. Note that $4$ is a current “lower bound” on the “asymptotic approximability” of the problem, in the sense that no better ratio is known for much easier sub-problems. Our main result shows that this “lower bound” is (almost) achievable.

Theorem 1.1

$k$-Connected Subgraph* admits approximation ratio $2(2+1/\ell)$ where $\ell$ is the largest integer such that $n\geq k[(k^{2}-1)(2k^{2}-3k+2)^{\ell-1}+1]$.*

Note that $\ell\approx\frac{1}{2}(\log_{k}n-1)$ and that Theorem 1.1 implies ratio $5$ for $n\geq 2k^{5}$ and ratio $4+\epsilon$ if $k$ is bounded by a constant. In fact, we prove a generalization of Theorem 1.1, stated in biset function terms, see the next section.

We note that our result can be used to improve approximation ratios for the Min-Cost Degree Bounded $k$-Connected Subgraph problem, see [16, 7].

We refer the reader to [31, 30] for surveys on approximation algorithms for node-connectivity problems, and to [11, 13] for a survey on polynomially solvable cases. Here we briefly mention the status of some restricted $k$-Connected Subgraph problems.

We may assume that the input graph $\hat{G}$ is complete, by assigning infinite costs to “forbidden” edges. Under this assumption, except for general edge costs, three main types of costs are considered in the literature:

• •

$\{0,1\}$-costs: Here we are given a graph $G$, and the goal is to find a minimum size set $J$ of new edges (any edge is allowed) such that $G\cup J$ is $k$-connected.

• •

$\{1,\infty\}$-costs: Here we seek a $k$-connected spanning subgraph of $\hat{G}$ with minimum number of edges.

• •

metric costs: The costs satisfy the triangle inequality $c_{uv}\leq c_{uw}+c_{wv}$ for all $u,w,v\in V$.

The currently best known approximation ratios for these costs types are summarized in Table 2, and we mention some additional results. For $\{0,1\}$-costs the complexity status of the problem is not known for undirected graphs, but for any constant $k$ an optimal solution can be computed in polynomial time [19]. When $\hat{G}$ contains a spanning $(k-1)$-connected subgraph of cost [math] the $\{0,1\}$-costs case can be solved in polynomial time for any $k$ [33]. In the case of $\{1,\infty\}$-costs, directed $1$-Connected Subgraph admits ratio $3/2$ [34]. In the case of metric costs $2$-Connected Subgraph admit ratio $3/2$ [15].

2 Biset functions and $k$-connectivity problems

While edge-cuts of a graph correspond to node subsets, a natural way to represent a node-cut of a graph is by a pair of sets called a “biset”.

Definition 1

An ordered pair $\mathbb{A}=(A,A^{+})$ of subsets of $V$ with $A\subseteq A^{+}$ is called a biset; $A$ is the inner part and $A^{+}$ is the outer part of $\mathbb{A}$, and $\partial\mathbb{A}=A^{+}\setminus A$ is the boundary of $\mathbb{A}$. The co-set of $\mathbb{A}$ is $A^{*}=V\setminus A^{+}$; the co-biset of $\mathbb{A}$ is $\mathbb{A}^{*}=(A^{*},V\setminus A)$. We say that $\mathbb{A}$ is void if $A=\emptyset$, co-void if $A^{+}=V$ (namely, if $A^{*}=\emptyset$), and $\mathbb{A}$ is proper otherwise.

A biset function assigns to every biset $\mathbb{A}$ a real number; in our context, it will always be an integer (possibly negative).

Definition 2

An edge covers a biset $\mathbb{A}$ if it goes from $A$ to $A^{*}$. For an edge-set/graph $J$ let $\delta_{J}(\mathbb{A})$ denote the set of edges in $J$ covering $\mathbb{A}$. The residual function of a biset function $f$ w.r.t. $J$ is defined by $f^{J}(\mathbb{A})=f(\mathbb{A})-|\delta_{J}(\mathbb{A})|$. We say that $J$ $f$-covers $\mathbb{A}$ if $|\delta_{J}(\mathbb{A})|\geq f(\mathbb{A})$, and we say that $J$** covers $f$** or that $J$ is an $f$-cover if $|\delta_{J}(\mathbb{A})|\geq f(\mathbb{A})$ for all $\mathbb{A}$.

In biset terms, Menger’s Theorem says that the maximum number of internally disjoint $st$-paths in $G$ equals to $\min\{|\partial\mathbb{A}|+|\delta_{G}(\mathbb{A})|:s\in A,t\in A^{*}\}$. Consequently, $G$ is $k$-connected if and only if $|\delta_{G}(\mathbb{A})|\geq k-|\partial\mathbb{A}|$ for every proper biset $\mathbb{A}$; note that non-proper bisets cannot and are not required to be covered. Thus $G$ is $k$-connected if and only if $G$ covers the $k$-connectivity biset function $f_{k{\sf\mbox{-}CS}}$ defined by

[TABLE]

We thus will consider the following generic problem:

Biset-Function Edge-Cover

Input:  A graph $\hat{G}=(V,\hat{E})$ with edge costs and a biset function $f$ on $V$.

Output: A minimum cost edge-set $E\subseteq\hat{E}$ that covers $f$.

Here $f$ may not be given explicitly, and an efficient implementation of algorithms requires that certain queries related to $f$ can be answered in time polynomial in $n$. We will consider later implementation details. In the application discussed here, relevant polynomial time oracles are available via min-cut computations. In particular, we have a polynomial time separation oracle for the LP-relaxation due to Frank and Jordán [12]:

[TABLE]

This LP is particularly useful if the biset function $f$ has good uncrossing/supermodularity properties. To state these properties, we need to define the intersection and the union of bisets.

Definition 3

The intersection and the union of two bisets $\mathbb{A},\mathbb{B}$ are defined by $\mathbb{A}\cap\mathbb{B}=(A\cap B,A^{+}\cap B^{+})$ and $\mathbb{A}\cup\mathbb{B}=(A\cup B,A^{+}\cup B^{+})$. The biset $\mathbb{A}\setminus\mathbb{B}$ is defined by $\mathbb{A}\setminus\mathbb{B}=(A\setminus B^{+},A^{+}\setminus B)$. We say that $\mathbb{A},\mathbb{B}$ intersect if $A\cap B\neq\emptyset$, and cross if $A\cap B\neq\emptyset$ and $A^{+}\cup B^{+}\neq V$.

The following properties of bisets are easy to verify.

Fact 1

For any bisets $\mathbb{A},\mathbb{B}$ the following holds. If a directed/undirected edge $e$ covers one of $\mathbb{A}\cap\mathbb{B},\mathbb{A}\cup\mathbb{B}$ then $e$ covers one of $\mathbb{A},\mathbb{B}$; if $e$ is an undirected edge, then if $e$ covers one of $\mathbb{A}\setminus\mathbb{B},\mathbb{B}\setminus\mathbb{A}$, then $e$ covers one of $\mathbb{A},\mathbb{B}$. Furthermore $|\partial\mathbb{A}|+|\partial\mathbb{B}|=|\partial(\mathbb{A}\cap\mathbb{B})|+|\partial(\mathbb{A}\cup\mathbb{B})|=|\partial(\mathbb{A}\setminus B)|+|\partial(\mathbb{B}\setminus\mathbb{A})|$.

For a biset function $f$ and bisets $\mathbb{A},\mathbb{B}$ the supermodular inequality is

[TABLE]

We say that a biset function $f$ is supermodular if the supermodular inequality holds for all $\mathbb{A},\mathbb{B}$, and modular if the supermodular inequality holds as equality for all $\mathbb{A},\mathbb{B}$; $f$ is symmetric if $f(\mathbb{A})=f(\mathbb{A}^{*})$ for all $\mathbb{A}$. Using among others Fact 1, one can see the following.

• •

For any directed/undirected graph $G$ the function $-d_{G}(\cdot)$ is supermodular.

• •

The function $|\partial(\cdot)|$ is modular.

• •

For any $R\subseteq V$ the function $|A\cap R|$ is modular.

We say that a biset $\mathbb{A}$ is $f$-positive if $f(\mathbb{A})>0$. Some important types of biset functions are given in the following definition.

Definition 4

A biset function $f$ is intersecting/crossing supermodular if the supermodular inequality holds whenever $\mathbb{A},\mathbb{B}$ intersect/cross; $f$ is positively intersecting supermodular if the supermodular inequality holds for any pair of intersecting $f$-positive bisets.

Biset-Function Edge-Cover with positively intersecting supermodular $f$ admits a polynomial time algorithm due to Frank [10] that for directed graphs computes an $f$-cover of cost $\tau(f)$ (this also can be deduced using the iterative rounding method); for undirected graphs the cost is at most $2\tau(f)$, by a standard bi-direction reduction. Note however that the function $f_{k{\sf\mbox{-}CS}}$ that we want to cover is obtained by zeroing an intersecting supermodular function on co-void bisets, but $f_{k{\sf\mbox{-}CS}}$ itself is not positively intersecting supermodular.

In general, changing a supermodular function on void bisets gives an intersecting supermodular function, while changing an intersecting supermodular function on co-void bisets gives a crossing supermodular function (not all crossing supermodular functions arise in this way – see [13]). In particular, zeroing a supermodular function on non-proper bisets gives a crossing supermodular function. For example, the $k$-connectivity function $f_{k{\sf\mbox{-}CS}}$ is obtained in this way from the modular function $k-|\partial\mathbb{A}|$, thus $f_{k{\sf\mbox{-}CS}}$ is crossing supermodular.

A common way to find a “cheap” partial cover of $f_{k{\sf\mbox{-}CS}}$ is to find a $2$-approximate cover of the fan function $g_{R}$ obtained by zeroing the function $k-|\partial\mathbb{A}|-|A\cap R|$ on void bisets, where $R\subseteq V$ with $|R|=k$. Note that $g_{R}$ is intersecting supermodular and that $g_{R}$ is non-positive on co-void bisets (e.g., $g_{R}((V,\emptyset))=k-0-k=0$). Fan functions were used in many previous works on $k$-Connected Subgraph problems starting from Khuller and Raghavachari [21], and also by Cheriyan and Végh [4]. In fact, covering $g_{R}$ is equivalent to the following connectivity problem. Let us say that a graph is $k$-in-connected to $r$ if it has $k$ internally disjoint $vr$ paths for every $v\in V$. Construct a graph $G_{r}$ by adding to $\hat{G}$ a new node $r$ and a set $F_{r}$ of zero cost edge from each $v\in R$ to $r$; then $H=(V\cup\{r\},J_{r})$ is a $k$-in-connected to $r$ spanning subgraph of $G_{r}$ if and only if $J=J_{r}\setminus F_{r}$ covers $g_{R}$. The problem of finding an optimal $k$-in-connected spanning subgraph can be solved in strongly polynomial time for directed graphs [14] (see also [10]), and this implies a $2$-approximation for undirected graphs.

Fan functions are considered as the “strongest” intersecting supermodular functions for the purpose of finding a partial cover of $f_{k{\sf\mbox{-}CS}}$. However, an inclusion minimal directed cover $J$ of a fan function may be difficult to decompose, since $J$ may have directed edges with tail in $R$; this is so since a fan function requires to cover to some extent bisets $\mathbb{A}$ with $A\cap R\neq\emptyset$. We therefore use a different type of functions defined below, that are “weaker” but have “better” decomposition properties.

For $R\subseteq V$ the area function of $f$ is defined by

[TABLE]

Note that $f_{R}(\mathbb{A})=f(\mathbb{A})$ if $A\subseteq V\setminus R$ and $f_{R}(\mathbb{A})\leq 0$ otherwise, so $f_{R}$ requires to $f$-cover only those bisets whose inner part is contained in the “area” $V\setminus R$. In the next two lemmas we give some properties of area functions. Let us denote

[TABLE]

Lemma 2

If $|R|\geq k_{f}$ then: $f_{R}$ is non-positive on co-void bisets, $f_{R}$ is intersecting supermodular if $f$ is, and $f_{R}$ is positively intersecting supermodular if $f$ is crossing supermodular and $|R|\geq 2k_{f}-1$.

Proof

The first two statements are easy, so we prove only the last statement. Let $\mathbb{A},\mathbb{B}$ be intersecting $f_{R}$-positive bisets. Then $A\cap R=B\cap R=\emptyset$, and thus $(A\cap B)\cap R=(A\cup B)\cap R=\emptyset$. Consequently, $f_{R}=f$ on the bisets $\mathbb{A},\mathbb{B},\mathbb{A}\cap\mathbb{B},\mathbb{A}\cup\mathbb{B}$. Moreover, $A^{*}\cap B^{*}\cap R\neq\emptyset$, since $|\partial\mathbb{A}\cup\partial\mathbb{B}|<2(k_{f}-1)+1\leq|R|$. Thus $\mathbb{A},\mathbb{B}$ must cross, and since $f$ is crossing supermodular

[TABLE]

Consequently, the supermodular inequality holds for $\mathbb{A},\mathbb{B}$ and $f_{R}$. ∎

For $S\subseteq V$ let $\gamma(S)$ denote the set of edges in $\hat{E}$ with both end in $S$. Consider the following algorithm for covering $f_{R}$.

If $f_{R}$ is positively intersecting supermodular, then step 2 in the algorithm can be implemented in polynomial time if the Biset-LP for $f_{R}$ can be solved in polynomial time. In many specific cases strongly polynomial algorithms are available. E.g., if $f$ is obtained by zeroing the function $k-|\partial\mathbb{A}|$ on void bisets then we can use the Frank-Tardos algorithm [14] or the algorithm of Frank [10] for finding a directed min-cost $k$-in-connected subgraph – in the above reduction described for covering a fan function $g_{R}$, the edge set $F_{r}$ will have $k=\max_{\mathbb{A}}f(\mathbb{A})$ parallel directed edges from each $v\in R$ to the root $r$.

The following lemma relates the cost of the solution computed by Algorithm 1 to the Biset-LP value.

Lemma 3

Let $f_{R}$ be positively intersecting supermodular and let $x$ be a feasible Biset-LP solution for covering $f_{R}$. Then Algorithm 1 returns an $f_{R}$-cover $I$ of cost $c(I)\leq\sum_{e\in\delta(R)}c_{e}x_{e}+2\sum_{e\in\gamma(V\setminus R)}c_{e}x_{e}$.

Proof

Edges in $\gamma(R)$ do not cover $f_{R}$-positive bisets, hence they can be removed. Let $E^{\prime}$ be the bi-direction of $\hat{E}$, where each undirected edge $e$ with ends $u,v$ is replaced by two arcs $uv,vu$ of cost $c_{e}$ and value $x_{e}$ each. Let $x^{\prime}$ be be obtained by zeroing the value of arcs leaving $R$; these arcs do not cover $f$-positive bisets. We claim that:

[TABLE]

The first inequality is obvious. The second inequality is since $f_{R}$ is positively intersecting supermodular and since $x^{\prime}$ is a directed feasible Biset-LP solution for $f_{R}$ while $I^{\prime}$ is an optimal one. The equality is by the construction. ∎

Assuming that for any residual function of $f^{I}$ of $f$, Algorithm 1 can be implemented in polynomial time whenever $f^{I}_{R}$ is positively intersecting supermodular, and that the Biset-LP for covering $f^{I}$ can be solved in polynomial time, we prove the following theorem that implies Theorem 1.1.

Theorem 2.1

Undirected Biset-Function Edge-Cover with symmetric crossing supermodular $f$ admits approximation ratio $2(2+1/\ell)$, where $\ell$ is the largest integer such that:

• •

$n\geq(2k_{f}-1)[(2k_{f}^{2}-3k_{f}+2)^{\ell}+1]$* for symmetric crossing supermodular $f$.*

• •

$n\geq k_{g}[(2k_{g}^{2}-3k_{g}+2)^{\ell}+1]$* if $f$ is obtained by zeroing an intersecting supermodular function $g$ on co-void bisets,*

• •

$n\geq k[(k^{2}-1)(2k^{2}-3k+2)^{\ell-1}+1]$* if $f=f_{k{\sf\mbox{-}CS}}$.*

3 Covering crossing supermodular functions (Theorem 2.1)

A biset function $f$ is positively skew-supermodular if the supermodular inequality or the co-supermodular inequality $f(\mathbb{A}\setminus\mathbb{B})+f(\mathbb{B}\setminus\mathbb{A})\geq f(\mathbb{A})+f(\mathbb{B})$ holds for $f$-positive bisets.

The corresponding Biset-Function Edge-Cover problem, when $f$ is positively skew-supermodular, admits ratio $2$ (assuming the Biset-LP can be solved in polynomial time) [9]; see also [4, 16] for a simpler proof along the proof line of [25] for the set functions case.

We say that $\mathbb{A},\mathbb{B}$ co-cross if $\mathbb{A}\setminus\mathbb{B}$ and $\mathbb{B}\setminus\mathbb{A}$ are both non-void, and that $\mathbb{A},\mathbb{B}$ independent if they do not cross nor co-cross. One can verify that $\mathbb{A},\mathbb{B}$ are independent if and only if at least one of the following holds: $A\subseteq\partial\mathbb{B}$, or $A^{*}\subseteq\partial\mathbb{B}$, or $B\subseteq\partial\mathbb{A}$, or $B^{*}\subseteq\partial\mathbb{A}$. A biset function $f$ is independence-free if no pair of $f$-positive bisets are independent. It is easy to see that if $f$ is symmetric and if $|A|\geq k_{f}$ holds for every $f$-positive biset $\mathbb{A}$ then $f$ is independence-free.

Lemma 4 ([19])

Let $f$ be a symmetric crossing supermodular biset function. If $\mathbb{A},\mathbb{B}$ are not independent then the supermodular or the co-supermodular inequality holds for $\mathbb{A},\mathbb{B}$ and $f$. Thus if $f$ is independence-free then $f$ is positively skew-supermodular.

Proof

If $\mathbb{A},\mathbb{B}$ cross then the supermodular inequality holds for $\mathbb{A},\mathbb{B}$. Assume that $\mathbb{A},\mathbb{B}$ co-cross. Then $\mathbb{A}$ and $\mathbb{B}^{*}$ cross, and thus the supermodular inequality holds for $\mathbb{A},\mathbb{B}^{*}$ and $f$. Note that (i) $\mathbb{A}\setminus\mathbb{B}=\mathbb{A}\cap\mathbb{B}^{*}$; (ii) $\mathbb{A}\cup\mathbb{B}^{*}$ is the co-biset of $\mathbb{B}\setminus\mathbb{A}$, hence $f(\mathbb{A}\cup\mathbb{B}^{*})=f(\mathbb{B}\setminus\mathbb{A})$, by the symmetry of $f$. Thus we get $f(\mathbb{A}\setminus\mathbb{B})+f(\mathbb{B}\setminus\mathbb{A})=f(\mathbb{A}\cap\mathbb{B}^{*})+f(\mathbb{A}\cup\mathbb{B}^{*})\geq f(\mathbb{A})+f(\mathbb{B}^{*})=f(\mathbb{A})+f(\mathbb{B})$. ∎

This suggests a two phase strategy for covering an “almost” supermodular function $f$. First, find a “cheap” edge set $J$ such that the residual function $f^{J}$ will be independence-free so $f^{J}$ will have “good uncrossing properties”. Second, use some “known” algorithms to cover $f^{J}$. The idea is due to Jackson and Jordán [19], and it is also the basis of the algorithm of Cheriyan and Végh [4] (see also [26] where the same idea was used for a related problem). Specifically, if $f$ is crossing supermodular, we will seek a cheap $J$ that $f$-covers all bisets $\mathbb{A}$ with $|A|\leq k_{f}-1$; by Lemma 4 the residual function $f^{J}$ will be positively skew-supermodular so we can use the $2$-approximation algorithms of [9] to cover $f^{J}$.

The algorithm of Cheriyan and Végh [4] finds an edge $J$ as above of cost $\leq 4\tau(f)$, by covering two fan functions. Our algorithm covers a pair of area functions. In fact, we will cover a sequence of $\ell\geq\ell$ pairs of area functions, and with the help of Lemma 3 will show that the sum of their costs is at most $2\tau(\ell^{\prime}+1)$; we choose the cheapest pair cover that will have cost $\leq 2\tau(1+1/\ell^{\prime})$.

For an integer $p$ let $U(f,p)=\bigcup\{A:f(\mathbb{A})>0,|A|\leq p\}$ be the union of inner parts of size $\leq p$ of the $f$-positive bisets. Note that if $R\subseteq V\setminus U(f,p)$ and if $I$ covers $f_{R}$ then $f^{I}(\mathbb{A})\leq 0$ whenever $|A|\leq p$. Thus from Lemma 4 we get:

Corollary 1

If $R\subseteq V\setminus U(f,k_{f})$ and if $I^{\prime}$ is an $f_{R}$-cover, then the residual function $f^{I^{\prime}}$ of $f$ w.r.t. $I^{\prime}$ is independence-free and thus is positively skew-supermodular.

Thus we just need to find $R\subseteq V\setminus U(f,k_{f})$ with $|R|\geq k_{f}$ and compute a $2$-approximate cover of $f_{R}$ – the residual function will be independence-free and thus positively skew-supermodular. However, such $R$ may not exist, e.g., for $f=f_{k{\sf\mbox{-}CS}}$ we have $U(f,k)=V$. The idea of Cheriyan and Végh [4] resolves this difficulty as follows: first find a “cheap” edge set $I$ such that $|U(f^{I},k_{f})|\leq n-k_{f}$ will hold for the residual function $f^{I}$, and only then compute for $f^{I}$ an edge set $I^{\prime}$ as in Corollary 1. Then the function $f^{I\cup I^{\prime}}$ is independence-free and thus is positively skew-supermodular.

Variants of the next lemma were proved in [4, 16] (our bound is just slightly better), and we use it to show that $I$ as above can be a cover of an area function, provided that $n$ is large enough. Let us say that a biset family ${\cal F}$ is weakly posi-uncrossable if for any $\mathbb{A},\mathbb{B}\in{\cal F}$ such that both bisets $\mathbb{A}\setminus\mathbb{B},\mathbb{B}\setminus\mathbb{A}$ are non-void, one of them is in ${\cal F}$. If $f$ is crossing supermodular and symmetric then the family ${\cal F}$ of $f$-positive bisets is weakly posi-uncrossable, see [4, 16].

Lemma 5 ([4, 16])

Let ${\cal F}$ be a weakly posi-uncrossable biset family, let $p=\max_{\mathbb{A}\in{\cal F}}|A|$, $q=\max_{\mathbb{A}\in{\cal F}}|\partial\mathbb{A}|$, $U=\bigcup_{\mathbb{A}\in{\cal F}}A$, and let $\nu$ be the maximum number of pairwise inner part disjoint bisets in ${\cal F}$. Then $|U|\leq\nu[(2q(p-1)+p]$.

Proof

Let ${\cal F}^{\prime}$ be an inclusion minimal subfamily of ${\cal F}$ such that $\bigcup_{\mathbb{A}\in{\cal F}^{\prime}}A=U$. By the minimality of $|{\cal F}^{\prime}|$, for every $\mathbb{A}_{i}\in{\cal F}^{\prime}$ there is $v_{i}\in A_{i}$ such that $v_{i}\notin A_{j}$ for every $j\neq i$. For every $i$ let $\mathbb{C}_{i}$ be an inclusion minimal member of the family $\{\mathbb{C}\in{\cal F}:\mathbb{C}\subseteq\mathbb{A}_{i},v_{i}\in C\}$, where here $\mathbb{A}\subseteq\mathbb{B}$ means that $A\subseteq B$ and $A^{+}\subseteq B^{+}$. Since ${\cal F}$ is weakly posi-uncrossable, the minimality of $\mathbb{C}_{i}$ implies that exactly one of the following holds for any $i\neq j$:

• •

$v_{i}\in\partial\mathbb{C}_{j}$ or $v_{j}\in\partial\mathbb{C}_{i}$;

• •

$\mathbb{C}_{i}=\mathbb{C}_{i}\setminus\mathbb{C}_{j}$ or $\mathbb{C}_{j}=\mathbb{C}_{j}\setminus\mathbb{C}_{i}$.

Construct an auxiliary directed graph ${\cal J}$ on node set ${\cal C}=\{\mathbb{C}_{i}:\mathbb{A}_{i}\in{\cal F}^{\prime}\}$. Add an arc $\mathbb{C}_{i}\mathbb{C}_{j}$ if $v_{i}\in\partial\mathbb{C}_{j}$. The in-degree in ${\cal J}$ of a node $\mathbb{C}_{i}$ is at most $|\partial\mathbb{C}_{i}|\leq q$. Thus every subgraph of the underlying graph of ${\cal J}$ has a node of degree $\leq 2q$. A graph is $d$-degenerate if every subgraph of it has a node of degree $\leq d$. It is known that any $d$-degenerate graph is $(d+1)$-colorable. Hence ${\cal J}$ is $(2q+1)$-colorable, so its node set can be partitioned into at most $2q+1$ independent sets, say ${\cal C}_{1},{\cal C}_{2},\ldots$, where the bisets in each independent set are pairwise inner part disjoint. W.l.o.g. we may assume that ${\cal C}_{1}$ is a maximal subfamily in ${\cal C}$ of pairwise inner part disjoint bisets, so any $\mathbb{C}\in{\cal C}\setminus{\cal C}_{1}$ intersects some biset in ${\cal C}_{1}$. Let ${\cal F}^{\prime}_{i}$ be the subfamily of ${\cal F}^{\prime}$ that corresponds to ${\cal C}_{i}$, so $|{\cal F}^{\prime}_{i}|=|{\cal C}_{i}|\leq\nu$. Let $U_{i}=\cup_{\mathbb{A}\in{\cal F}_{i}}A$. An easy argument shows that $|U_{1}|\leq\nu p$ and that $|U_{i}\setminus U_{1}|\leq\nu(p-1)$ for $i\geq 2$. Consequently, $|U|\leq\nu p+2q\nu(p-1)$, as claimed. ∎

Corollary 2

If $f$ is symmetric crossing supermodular and if $I$ is a cover of $f_{R}$ then $|U(f^{I},k_{f})\cup R|\leq|R|(2k_{f}^{2}-3k_{f}+2)$.

Proof

Denote $r=|R|$, $r^{\prime}=|U(f^{I},k_{f})\cap R|$, and $k=k_{f}$. Substituting $q+1=p=k_{f}$ and observing that $\nu\leq r^{\prime}$ in Lemma 5 we get

[TABLE]

as required. ∎

Let us skip for a moment implementation details, and focus on bounding the cost of an edge set $J$ computed by the following algorithm.

Let us fix some optimal Biset-LP solution $x$. For an edge set $F$ the $x$-cost of $F$ is defined as $\sum_{e\in F}c_{e}x_{e}$. Let us use the following notation:

• •

$\tau=\sum_{e\in\hat{E}}c_{e}x_{e}$ is the optimal solution value.

• •

$\gamma_{i}$ is the $x$-cost of the edges with both ends in $R_{i}$.

• •

$\delta_{i}$ is the $x$-cost of the edges with one end in $R_{i}$ and the other in $V\setminus R_{i}$.

• •

$\bar{\gamma}_{i}$ is the $x$-cost of the edges with both ends in $V\setminus R_{i}$.

Clearly, for any $i$ we have

[TABLE]

By Lemma 3, the cost of the covers $I,I^{\prime}$ computed at iteration $i$ is bounded by

[TABLE]

Thus we get

[TABLE]

Summing this over $\ell$ iterations and observing that the sum is telescopic we get

[TABLE]

Thus there exists an index $i$ such that

[TABLE]

Note that if $R_{i+1}=R_{i}$ for some $i$ then $c(J_{i})\leq c(I)+c(I^{\prime})\leq 2\delta_{i}+2\gamma_{i}+2\bar{\gamma}_{i}=2\tau$, hence in this case the algorithm can terminate with $J=J_{i}$ and $c(J)\leq 2\tau$.

Next we use Corollary 2 to lower bound $n$ to ensure that the algorithm will have $\ell$ iterations. Let $r=|R_{1}|$, and $r$ is also a lower bound on $n-|R_{\ell}|$. To see the bounds on $n$ in Theorem 2.1 note the following.

• •

In the case of intersecting supermodular $f$ we choose $r=2k_{f}-1$ and need $r(2k_{f}^{2}-3k_{f}+2)^{\ell}\leq n-r$, namely, $n\geq(2k_{f}-1)[(2k_{f}^{2}-3k_{f}+2)^{\ell}+1]$.

• •

If $f$ is obtained by zeroing an intersecting supermodular function $g$ on co-void bisets we choose $r=k_{g}$ and need $r(2k_{g}^{2}-3k_{g}+2)^{\ell}\leq n-r$, namely, $n\geq k[(2k_{g}^{2}-3k_{g}+2)^{\ell}+1]$.

• •

When $f=f_{k{\sf\mbox{-}CS}}$, [16] shows a choice of $R_{1}$ such that $|R_{2}|\leq k^{3}-k$. We need $(k^{3}-k)(2k^{2}-3k+2)^{\ell-1}\leq n-k$, namely $n\geq k[(k^{2}-1)(2k^{2}-3k+2)^{\ell-1}+1]$.

To get a polynomial time implementation we need to find in step 4 the set $R_{i+1}=R_{i}\cup U(f^{I},k_{f})$ in polynomial time. We modify the algorithm by relaxing the step 4 condition $R_{i+1}=R_{i}\cup U(f^{I},k_{f})$ to $R_{i}\subseteq R_{i+1}\subseteq R_{i}\cup U(f^{I},k_{f})$ (so $R_{1}\subseteq R_{2}\subseteq\ldots$ will be a nested family), but require that for each $J_{i}=I\cup I^{\prime}$ the algorithm will compute a cover $F_{i}$ of $f^{J_{i}}$ of cost $c(F_{i})\leq 2\tau(f^{J_{i}})$. This can be done in the same way as in [4], as follows.

The iterative rounding $2$-approximation algorithm of [9] for covering a positively skew supermodular biset function, when applied on an arbitrary biset function $h$, either returns a $2$-approximate cover $J$ of $h$, or a failure certificate: a pair $\mathbb{A},\mathbb{B}$ of bisets with $h(\mathbb{A})>0$ and $h(\mathbb{B})>0$ for which both the supermodular and the co-supermodular inequality does not hold. In our case this can happen only if $\mathbb{A},\mathbb{B}$ are independent, by Lemma 4.

Now consider some iteration $i$ of the algorithm. Since $f^{I\cup I^{\prime}}$ is symmetric, then by interchanging the roles of $\mathbb{A},\mathbb{A}^{*},\mathbb{B},\mathbb{B}^{*}$, we can assume w.l.o.g. that our failure certificate $\mathbb{A},\mathbb{B}$ satisfies $A\subseteq\partial\mathbb{B}$. We thus apply the following procedure. Start with $R_{i+1}=R_{i}$. Then iteratively, find $I^{\prime}$ as in step 5 and apply the $2$-approximation algorithm of [9] for covering $h=f^{I\cup I^{\prime}}$; if the algorithm returns an edge set $F$ of cost $c(F)\leq 2\tau(h)$, we keep the current $R_{i+1}$, set $J_{i}\leftarrow I\cup I^{\prime}$ and $F_{i}\leftarrow F$, and continue to the next iteration. Else, we have a failure certificate pair $\mathbb{A},\mathbb{B}$ of $h$-positive bisets with $A\subseteq\partial\mathbb{B}$. Then $A\subseteq U(f^{I},k_{f})$ and $A\setminus R_{i+1}\neq\emptyset$ (since $I^{\prime}$ $f$-covers bisets whose inner part is contained in $R_{i+1}$), and we can apply the same procedure with a larger candidate set $R_{i+1}\leftarrow R_{i+1}\cup A$.

This concludes the proof of Theorem 2.1.