Covering problems in edge- and node-weighted graphs

Choosing a set of edges in a graph that optimizes some objective function under constraints on the chosen edges constitutes a typical combinatorial optimization problem and has been investigated in many varieties. For example, the spanning tree problem seeks an acyclic edge set that spans all nodes in a graph, the edge cover problem finds an edge set such that each node is incident to at least one edge in the set, and the shortest path problem selects an edge set that connects two specified nodes. All these problems seek to minimize the sum of the weights assigned to edges.

This paper discusses several graph covering problems. Formally, the graph covering problem is defined as follows in this paper. Given a graph $G=(V,E)$ and family $\mathcal{E}\subseteq 2^{E}$, find a subset $F$ of $E$ that satisfies $F\cap C\neq\emptyset$ for each $C\in\mathcal{E}$, while optimizing some function depending on $F$. As indicated above, the popular approaches assume an edge weight function $w\colon E\rightarrow\mathbb{R}_{+}$ is given, where $\mathbb{R}_{+}$ denotes the set of non-negative real numbers, and seeks to minimize $\sum_{e\in F}w(e)$. On the other hand, we aspire to simultaneously minimize edge and node weights. Formally, we let $V(F)$ denote the set of end nodes of edges in $F$. Given a graph $G=(V,E)$ and weight function $w\colon E\cup V\rightarrow\mathbb{R}_{+}$, we seek a subset $F$ of $E$ that minimizes $\sum_{e\in F}w(e)+\sum_{v\in V(F)}w(v)$ under the constraints on $F$. Hereafter, we denote $\sum_{e\in F}w(e)$ and $\sum_{v\in V(F)}w(v)$ by $w(F)$ and $w(V(F))$, respectively.

Most previous investigations of the graph covering problem have focused on edge weights. By contrast, node weights have been largely neglected, except in the problems of choosing node sets, such as the vertex cover and dominating set problems. To our knowledge, when node weights have been considered in graph covering problems for choosing edge sets, they have been restricted to the Steiner tree problem or its generalizations, possibly because the inclusion of node weights greatly complicates the problem. For example, the Steiner tree problem in edge-weighted graphs can be approximated within a constant factor (the best currently known approximation factor is 1.39 [5, 15]). Conversely, the Steiner tree problem in node-weighted graphs is known to extend the set cover problem (see [20]), indicating that achieving an approximation factor of $o(\log|V|)$ is NP-hard. The literature is reviewed in Section 2. As revealed later, the inclusion of node weights generalizes the set cover problem in numerous fundamental problems.

However, from another perspective, node weights can introduce rich structure into the above problems. In fact, node weights provide useful optimization problems. The objective function counts the weight of a node only once, even if the node is shared by multiple edges. Hence, the objective function defined from node weights includes a certain subadditivity, which cannot be captured by edge weights.

The aim of the present paper is to give algorithms for fundamental graph covering problems in edge- and node-weighted graphs. In solving the problems, we adopt a basic linear programming (LP) technique. Many algorithms for combinatorial optimization problems are typically designed using LP relaxations. However, in problems with node-weighted graphs, the integrality gap of natural relaxations may be excessively large. Therefore, we propose tighter LP relaxations that preclude unnecessary integrality gaps. We then discuss upper bounds on the integrality gap of these relaxations in two fundamental graph covering problems: the edge dominating set (EDS) problem and multicut problem in trees. We prove upper bounds by designing primal-dual algorithms for both problems. The approximation factors of our proposed algorithms match the current best approximations in purely edge-weighted graphs.

The EDS problem covers edges by choosing adjacent edges in undirected graphs. For any edge $e$, let $\delta(e)$ denote the set of edges that share end nodes with $e$, including $e$ itself. We say that an edge $e$ dominates another edge $f$ if $f\in\delta(e)$, and a set $F$ of edges dominates an edge $f$ if $F$ contains an edge that dominates $f$. Given an undirected graph $G=(V,E)$, a set of edges is called an EDS if it dominates each edge in $E$. The EDS problem seeks to minimize the weight of the EDS. In other words, the EDS problem is the graph covering problem with $\mathcal{E}=\{\delta(e)\colon e\in E\}$.

In the multicut problem, an instance specifies an undirected graph $G=(V,E)$ and demand pairs $(s_{1},t_{1}),\ldots,(s_{k},t_{k})\in V\times V$. A multicut is an edge set $C$ whose removal from $G$ disconnects the nodes in each demand pair. This problem seeks a multicut of minimum weight. Let $\mathcal{P}_{i}$ denote the set of paths connecting $s_{i}$ and $t_{i}$. The multicut problem is equivalent to the graph covering problem with $\mathcal{E}=\bigcup_{i=1}^{k}\mathcal{P}_{i}$.

Our proposed algorithms for solving these problems assume that the given graph $G$ is a tree. In fact, our algorithms are applicable to the prize-collecting versions of these problems, which additionally specifies a penalty function $\pi\colon\mathcal{E}\rightarrow\mathbb{R}_{+}$. In this scenario, an edge set $F$ is a feasible solution even if $F\cap C=\emptyset$ for some $C\in\mathcal{E}$, but imposes a penalty $\pi(C)$. The objective is to minimize the sum of $w(F)$, $w(V(F))$, and the penalty $\sum_{C\in\mathcal{E}:F\cap C=\emptyset}\pi(C)$. The prize-collecting versions of the EDS and multicut problems are referred to as the prize-collecting EDS problem and the prize-collecting multicut problem, respectively.

Thus far, the EDS problem has been applied only to edge-weighted graphs. The vertex cover problem can be reduced to the EDS problem while preserving the approximation factors [6]. The vertex cover problem is solvable by a 2-approximation algorithm, which is widely regarded as the best possible approximation. Indeed, assuming the unique game conjecture, Khot and Regev [19] proved that the vertex cover problem cannot be approximated within a factor better than $2$. Fujito and Nagamochi [11] showed that a 2-approximation algorithm is admitted by the EDS problem, which matches the approximation hardness known for the vertex cover problem. In the Appendix, we show that the EDS problem in bipartite graphs generalizes the set cover problem if assigned node weights and generalizes the non-metric facility location problem if assigned edge and node weights. This implies that including node weights increases difficulty of the problem even in bipartite graphs.

On the other hand, Kamiyama [18] proved that the prize-collecting EDS problem in an edge-weighted graph admits an exact polynomial-time algorithm if the graph is a tree. As one of our main results, we show that this idea is extendible to problems in edge- and node-weighted trees.

Theorem 1 will be proven in Section 4. As demonstrated in the Appendix, the prize-collecting EDS problem in general edge- and node-weighted graphs admits an $O(\log|V|)$-approximation, which matches the approximation hardness on the set cover problem and the non-metric facility location problem.

The multicut problem is hard even in edge-weighted graphs; the best reported approximation factor is $O(\log k)$ [13]. The multicut problem is known to be both NP-hard and MAX SNP-hard [10], and admits no constant factor approximation algorithm under the unique game conjecture [7]. However, Garg, Vazirani, and Yannakakis [14] developed a 2-approximation algorithm for the multicut problem with edge-weighted trees. They also mentioned that, although the graphs are restricted to trees, the structure of the problem is sufficiently rich. They showed that the tree multicut problem includes the set cover problem with tree-representable set systems. They also showed that the vertex cover problem in general graphs is simply reducible to the multicut problem in star graphs, while preserving the approximation factor. This implies that the 2-approximation seems to be tight for the multicut problem in trees. As a second main result, we extended this 2-approximation to edge- and node-weighted trees, as stated in the following theorem.

Both algorithms claimed in Theorems 1 and 2 are primal-dual algorithms, that use the LP relaxations we propose. These algorithms fall in the same frameworks as those proposed in [14, 18] for edge-weighted graphs. However, they need several new ideas to achieve the claimed performance because our LP relaxations are much more complicated than those used in [14, 18].

The remainder of this paper is organized as follows. After surveying related work in Section 2, we define our LP relaxation for the prize-collecting graph covering problem in Section 3. Using this relaxation, we prove Theorems 1 and 2 in Sections 4 and 5, respectively. The paper concludes with Section 6. In the Appendix, we show that the prize-collecting EDS problem in edge- and node-weighted graphs generalizes the set cover problem and facility location problem, and admits an $O(\log|V|)$-approximation algorithm.

In this case, a leaf edge $e$ of maximum depth satisfies $\pi(e)>0$. Since Case A is not the base case, the depth of $l_{e}$ exceeds one. Let $u$ denote the upper end node of $e$. Also let $v_{0}$ denote the parent of $u$, and let $v_{1},\ldots,v_{k}$ be the children of $u$. Throughout this paper, the sets $\{1,\ldots,k\}$ and $\{0,\ldots,k\}$ are denoted by $[k]$, and $[k]^{*}$, respectively. The edge joining $v_{i}$ and $u$ is called $e_{i}$, with $i\in[k]^{*}$ ($e$ is included in $\{e_{1},\ldots,e_{k}\}$). The relationships between these nodes and edges are illustrated in Figure 1. We define $\beta_{1}=\min_{i=0}^{k}(w(e_{i})+w(u)+w(v_{i}))$, $\beta_{2}=\sum_{i=1}^{k}\pi(e_{i})$, and $\beta=\min\{\beta_{1},\beta_{2}\}$. Let $i^{*}\in[k]^{*}$ be the index of an edge $e_{i^{*}}$ that attains $\beta_{1}=w(e_{i^{*}})+w(u)+w(v_{i^{*}})$.

The algorithm constructs an instance $I^{\prime}=(G^{\prime},w^{\prime},\pi^{\prime})$ as follows. Suppose that $\beta_{1}>\beta_{2}$. In this case, $G^{\prime}$ is defined as $G$, and $\pi^{\prime}\colon E^{\prime}\rightarrow\mathbb{R}_{+}$ is defined by

$$\pi^{\prime}(e)=\begin{cases}0&\mbox{if }e\in\{e_{1},\ldots,e_{k}\},\\ \pi(e)&\mbox{otherwise.}\end{cases}$$

For $\psi\in\mathbb{R}$, we denote $\max\{0,\psi\}$ by $(\psi)_{+}$. The weight function $w^{\prime}\colon V^{\prime}\cup E^{\prime}\rightarrow\mathbb{R}_{+}$ is defined by

$$w^{\prime}(e)=\begin{cases}(w(e_{i})-(\beta-w(u))_{+})_{+}&\mbox{if }e=e_{i},i\in[k]^{*},\\ w(e)&\mbox{otherwise}\end{cases}$$

for each $e\in E^{\prime}$, and

$$w^{\prime}(v)=\begin{cases}(w(u)-\beta)_{+}&\mbox{if }v=u,\\ w(v_{i})-(\beta-w(u)-w(e_{i}))_{+}&\mbox{if }v=v_{i},i\in[k]^{*},\\ w(v)&\mbox{otherwise}\end{cases}$$

for each $v\in V^{\prime}$. If $\beta_{1}\leq\beta_{2}$, then $G^{\prime}=(V^{\prime},E^{\prime})$ is defined as the tree obtained by removing nodes $v_{1},\ldots,v_{k}$ and edges $e_{1},\ldots,e_{k}$ from $G$, and $\pi^{\prime}\colon E^{\prime}\rightarrow\mathbb{R}_{+}$ is defined by

$$\pi^{\prime}(e)=\begin{cases}0&\mbox{if }e=e_{0},\\ \pi(e)&\mbox{otherwise.}\end{cases}$$

$w^{\prime}$ is defined identically to the case $\beta_{1}>\beta_{2}$, ignoring $w^{\prime}(v_{i})$ and $w^{\prime}(e_{i})$ for $i\in[k]$.

If $\beta_{1}\leq\beta_{2}$, the number of nodes with depth exceeding one is lower in $G^{\prime}$ than in $G$. Hence, the algorithm inductively finds a solution $F^{\prime}$ to $I^{\prime}$ and a feasible dual solution $(\xi^{\prime},\nu^{\prime},\mu^{\prime})$ to $D(I^{\prime})$ that satisfy (5). Otherwise, the number of leaf edges $e$ of maximum depth with $\pi^{\prime}(e)>0$ is lower in $G^{\prime}$ than in $G$. If $G^{\prime}$ lacks edges of this type, then instance $I^{\prime}$ is categorized into Case B, and $F^{\prime}$ and $(\xi^{\prime},\nu^{\prime},\mu^{\prime})$ are found as demonstrated below. If such edges do exist in $G^{\prime}$, the algorithm finds $F^{\prime}$ and $(\xi^{\prime},\nu^{\prime},\mu^{\prime})$ by induction on the number of such edges. Therefore, it suffices to show that the required $F$ and $(\xi,\nu,\mu)$ can be constructed from $F^{\prime}$ and $(\xi^{\prime},\nu^{\prime},\mu^{\prime})$, provided that $F^{\prime}$ and $(\xi^{\prime},\nu^{\prime},\mu^{\prime})$ exist.

We now define $F$ and $(\xi,\nu,\mu)$. $F$ is defined by

$$F=\begin{cases}F^{\prime}\cup\{e_{0}\}&\mbox{if }\delta_{F^{\prime}}(v_{0})\neq\emptyset,\beta>w(u)+w(e_{0}),\\ F^{\prime}&\mbox{if }\delta_{F^{\prime}}(v_{0})=\emptyset\mbox{ or }\beta\leq w(u)+w(e_{0}),\beta_{1}>\beta_{2},\\ F^{\prime}\cup\{e_{i^{*}}\}&\mbox{if }\delta_{F^{\prime}}(v_{0})=\emptyset\mbox{ or }\beta\leq w(u)+w(e_{0}),\beta_{1}\leq\beta_{2}.\\ \end{cases}$$

In this case, $\pi(e)=0$ holds for all leaf edges $e$ of maximum depth. Let $s$ be the grandparent of a leaf node of maximum depth. Also, let $u_{1},\ldots,u_{k}$ be the children of $s$, and $e_{i}$ be the edge joining $s$ and $u_{i}$ for $i\in[k]$. In the following discussion, we assume that $s$ has a parent, and that each node $u_{i}$ has at least one child. This discussion is easily modified to cases in which $s$ has no parent or some node $u_{i}$ has no child. We denote the parent of $s$ by $u_{0}$, and the edge between $u_{0}$ and $s$ by $e_{0}$. For each $i\in[k]$, let $V_{i}$ be the set of children of $u_{i}$, and $H_{i}$ be the set of edges joining $u_{i}$ to its child nodes in $V_{i}$. Also define $h_{i}=u_{i}v_{i}$ as an edge that attains $\min_{u_{i}v\in H_{i}}(w(u_{i}v)+w(v))$. The relationships between these nodes and edges are illustrated in Figure 2.

Now define $\theta_{1}=\min_{i=0}^{k}(w(e_{i})+w(u_{i})+w(s))$, $\theta_{2}=\sum_{i=1}^{k}\min\{w(u_{i})+w(v_{i})+w(h_{i}),\pi(e_{i})\}$, and let $\theta=\min\{\theta_{1},\theta_{2}\}$. We denote the index $i\in[k]$ of an edge $e_{i}$ that attains $\theta_{1}=w(e_{i})+w(u_{i})+w(s)$ by $i^{*}$, and specify $K=\{i\in[k]\colon w(u_{i})+w(v_{i})+w(h_{i})\leq\pi(e_{i})\}$.