Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

Abstract

Multicut is a fundamental network communication and connectivity problem. It is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth.

Introduction

Motivation and previous results. Multicut in graphs is a fundamental network design problem. It models questions concerning the reliability and robustness of computer and communication networks. Informally speaking, the problem is, given a graph, to determine a minimum size set of either edges or vertices such that the deletion of this set disconnects a prespecified set of pairs of terminal vertices in the graph. In most cases, the problem is NP-complete. There are many results and variants for Multicut and we refer to Costa et al. [8] for a recent survey.

The major part of the literature deals with the “edge deletion variant” of Multicut (Edge Multicut) [8], [10], [16], [20], [21]. Given a graph and m pairs of terminal vertices, this problem is solvable in polynomial time for m = 1 [11] and m = 2 [22], [23]. For m ⩾ 3, the problem is NP-complete [10]. The problem remains NP-hard and MaxSNP-hard even when the input graph is a star [16].

Our main focus here lies on the “vertex deletion variant” (Vertex Multicut). Relatively little seems to be known for Vertex Multicut problems; we are only aware of two recent investigations [9], [26]. Călinescu et al. [9] introduced two variants of Vertex Multicut:

Unrestricted Vertex Multicut (UVMC)

Input: An undirected graph $G=(V\text{,}E)$, a collection H of pairs of vertices $H\subseteq V\times V$, and an integer $k⩾0$.

Task: Find a subset V′ of V with $|V^{\prime }|⩽k$ whose removal separates each pair of vertices in H.

The vertices appearing in the vertex pairs in H are called terminals and, throughout this paper, we use S to denote the set of terminals, i.e., $S≔{\bigcup }_{(u\text{,}v)\in H}\{u\text{,}v\}$. By way of contrast, in the case of Restricted Vertex Multicut the removal of terminal vertices is not allowed.

Restricted Vertex Multicut (RVMC)

Input: An undirected graph $G=(V\text{,}E)$, a collection H of pairs of vertices $H\subseteq V\times V$, and an integer $k⩾0$.

Task: Find a subset V′ of V with $|V^{\prime }|⩽k$ that contains no terminal and whose removal separates each pair of vertices in H.

Since RVMC forbids the removal of terminals, there are instances that have no solution, namely when there is a path between a terminal pair in H consisting only of terminal vertices. As it is easy to determine whether or not an instance has a solution for RVMC by checking whether all terminal pairs are disconnected after removing all nonterminal vertices, we assume that all instances under consideration allow for feasible solutions for RVMC.

RVMC is at least as hard as UVMC in general graphs and many special graph classes: From an instance of UVMC we can obtain an “equivalent” RVMC instance by adding for each terminal s a new degree-1 vertex s′ adjacent only to s. Each terminal pair (s, t) is substituted by $(s^{\prime }\text{,}{t}^{\prime })$. Then, solving RVMC in this new instance is equivalent to solving UVMC in the original instance.

Călinescu et al. [9] showed that RVMC is NP-complete in bounded-degree trees and the “easier” UVMC is polynomially solvable in trees but becomes NP-complete in bounded-degree graphs of treewidth two. Moreover, they gave a polynomial-time approximation scheme (PTAS) for UVMC in graphs of bounded treewidth. Marx [26] extended the results for UVMC (which he called Minimum Node Multicut) by providing an $\mathrm{O}(2^{k\ell }·4^{k^3}·|G{|}^{\mathrm{O}(1)})$ time algorithm for UVMC in general graphs, where k is an upper bound on the vertices to be removed and ℓ is the number of terminal pairs. In other words, UVMC is fixed-parameter tractable (FPT) with respect to the combined parameter $(k\text{,}\ell )$. Because of the huge combinatorial explosion in k (and ℓ), however, this result is of mainly theoretical interest and improvements are highly desirable. Finally, we mention in passing that Garg et al. [17] studied the vertex deletion variant for the closely related Multiway Cut problem.

We continue and complement the work of Călinescu et al. [9] and Marx [26] as follows: We show that the NP-complete RVMC in trees is fixed-parameter tractable with respect to the parameter k (number of vertex deletions) with the modest running time $\mathrm{O}(|S{|}^2·|E|+2^k·\ell )$ (again, ℓ is the number of terminal pairs). Whereas in trees UVMC is polynomial-time solvable but RVMC is NP-complete [9], we have the surprising result that UVMC is NP-complete in interval graphs but RVMC is polynomial-time solvable here; interval graphs are graphs where the vertices correspond to intervals on the real line and there is an edge corresponding to each pair of intersecting intervals [27]. More specifically, the NP-completeness result for UVMC even holds in interval graphs of pathwidth four. We also strengthen the NP-completeness result for RVMC in trees of Călinescu et al. by showing that NP-completeness already holds for maximum-vertex-degree-three trees whereas their result only holds for maximum vertex degree four. Note that RVMC is clearly polynomial-time solvable in paths, that is, trees with maximum vertex degree two. Moreover, we show that RVMC in general graphs is NP-complete even in case of only three terminal pairs, hence excluding fixed-parameter tractability with respect to the parameter “number of terminal pairs”. By way of contrast, we show that RVMC can be solved in $\mathrm{O}(|S{|}^{|S|+\omega +1}·{\omega }^2·|V|)$ time on graphs of treewidth ω; thus, RVMC is fixed-parameter tractable with respect to the combined parameter “treewidth” and “terminal set size”. Observe that there is no hope for fixed-parameter tractability exclusively with respect to either the parameter ∣S∣ or the parameter ω. This fixed-parameter tractability result directly transfers to UVMC as well; indeed, it also works for the Edge Multicut (EMC) variant (note that Bentz [1] independently obtained the EMC fixed-parameter tractability result). On the way to show the NP-completeness of UVMC in interval graphs, we prove that Edge Multicut is NP-complete in caterpillar graphs with maximum vertex degree five.

Table 1 summarizes most of the presented results.

We introduce some additional terminology. By default, we consider only undirected graphs G = (V, E) without self-loops. A tree is called caterpillar if removing its leaves gives a path. For any graph G = (V, E), we can construct its line graph as $(E\text{,}\{\{e_1\text{,}{e}_2\}\in E|e_1\cap e_2\ne \varnothing \})$, that is, the vertices in the line graph one-to-one correspond to the edges in G and two vertices in the line graph are adjacent iff their corresponding edges in G have a common endpoint. For an overview on graph classes, we refer to [6]. We use $G[V^{\prime }]$ to denote the subgraph of G induced by the vertices $V^{\prime }\subseteq V$, and V(G) to refer to the vertex set V. A set of vertices $V^{\prime }\subseteq V$ is called vertex separator if $G[V⧹V^{\prime }]$ has more connected components than G. For a minimization problem, a feasible solution is called minimal if it does not contain another feasible solution as proper subset, and minimum if there is no other feasible solution with better measure.

Many NP-complete graph problems become easy when the input instance is a tree. The notion treewidth, introduced by Robertson and Seymour [31], tries to capture the “tree-likeness” of a graph: “tree-like” graphs have small treewidth, and in particular, trees have treewidth one. Many in general NP-hard graph problems can then be solved in polynomial or even linear time when the underlying graph has a treewidth bounded by a constant [2], [3], [30], [32].

Definition 1

A tree decomposition of G is a pair $〈\{X_i|i\in I\}\text{,}T〉$, where each X_i is a subset of V, called bag, and $T=(I\text{,}F)$ is a tree with node set I and edge set F. The following must hold:

• ${\bigcup }_{i\in I}{X}_i=V$;

• for every edge $\{u\text{,}v\}\in E$, there is an $i\in I$ such that $\{u\text{,}v\}\subseteq X_i$;

• for all $i\text{,}j\text{,}l\in I$, if j lies on the path between i and l in T, then $X_i\cap X_l\subseteq X_j$.

The width of $〈\{X_i|i\in I\}\text{,}T〉$ is $\max \{|X_i\Vert i\in I\}-1$. The treewidth of G is the minimum width over all tree decompositions of G.

A path decomposition is a tree decomposition where T is a path; pathwidth is defined analogous to treewidth. For a more detailed introduction to tree decompositions we refer to [2], [3], [4], [25], [29], [30].

A tree decomposition $〈\{X_i|i\in I\}\text{,}T〉$ can be transformed in linear time and without affecting its width such that it becomes nice [25, Lemma 13.1.3]. Then the following conditions are satisfied:

• T is rooted;

• every node of the tree T has at most two children;

• if a node i has two children j and k, then $X_i=X_j=X_k$ (in this case i is called a join node);

• if a node i has one child j, then either

  • $|X_i|=|X_j|+1$ and $X_j\subset X_i$ (in this case i is called an introduce node), or

  • $|X_i|=|X_j|-1$ and $X_i\subset X_j$ (in this case i is called a forget node).

We investigate Multicut problems in the context of parameterized complexity [12], [14], [29].

A problem is fixed-parameter tractable (FPT) with parameter k if an instance of size n can be solved in $f(k)·n^{\mathrm{O}(1)}$ time, where f is a computable function solely depending on the parameter k. The idea is to restrict the seemingly unavoidable combinatorial explosion that occurs in exact solutions to NP-hard problems to certain, hopefully small problem parameters.