Elsevier

Operations Research Letters

Volume 47, Issue 6, November 2019, Pages 587-593
Operations Research Letters

A simple algorithm for the multiway cut problem

https://doi.org/10.1016/j.orl.2019.09.012Get rights and content

Abstract

Multiway Cut is a classic graph partitioning problem in which the goal is to disconnect a given set of special vertices called terminals. This problem admits a rich sequence of works. Unfortunately, most of these works resort to the mix of multiple algorithmic components, resulting in both complicated algorithms and analysis. We present a simple algorithm for the Multiway Cut problem that is comprised of a single algorithmic component and achieves an approximation of 118=1.375.

Introduction

The Multiway Cut problem in undirected graphs is a prime example for the success of the metric approach, a prevalent paradigm in the design and analysis of approximation algorithms for many NP-hard graph cut problems, e.g., [3], [4], [9], [13], [20], [24], [25]. In this problem we are given an undirected graph G=(V,E) equipped with non-negative edge weights w:ER+, and a set T=t1,t2,,tkV of k terminals. The goal is to find a minimum weight subset of edges XE such that all terminals are disconnected in (V,EX). When k=2 the Multiway Cut problem is simply the minimum {s,t}-cut problem in undirected graphs, whereas for k=3 it is already known to be NP-hard [12].

The first approximation algorithm for the Multiway Cut problem was given by Dalhaus et al. [12]. They suggested a simple combinatorial heuristic that obtains a 2(11k)-approximation. Călinescu et al. [9] suggested a geometric relaxation for the Multiway Cut problem. In this relaxation each vertex uV is embedded into the k-dimensional simplex Δk={xRk:x0,i=1kxi=1}, while the terminals are mapped bijectively to the vertices of Δk. Călinescu et al. [9] observed that any rounding of this relaxation is in fact a partitioning of the simplex into k parts, one for each terminal, i.e., the th part must contain t. They presented a simple and elegant simplex partitioning algorithm achieving an approximation guarantee of 1.51k. The main idea of the algorithm of Călinescu et al. [9] is to iterate over the terminals in a random order and cut an 1 sphere of random radius around each terminal, thus partitioning the simplex. This idea was later used in the design and analysis of algorithms for numerous other problems. Notable examples include the tight probabilistic approximation of metrics by tree metrics [15] and the 0 -Extension problem [10], [14], [22]. Therefore, it is no surprise that the Călinescu et al. algorithm is an important component of many introductory courses to approximation algorithms and books on this topic (see, e.g., Vazirani [29], Chapter 19, and Williamson and Shmoys [30], Chapter 8.2).

It is important to note that since the introduction of the above geometric relaxation all subsequent approximation algorithms for the Multiway Cut problem are based on rounding it. In fact, Manokaran et al. [26] proved that assuming the unique games conjecture the integrality gap of this geometric relaxation can be translated to a hardness result for the Multiway Cut problem with the exact same value. Thus, assuming the correctness of the unique games conjecture the best possible rounding of the geometric relaxation provides a tight approximation for Multiway Cut.

In a sequence of works following Călinescu et al. [9] improved methods for partitioning the simplex, and thus approximating Multiway Cut, were given. A main component in all these subsequent works, whether described explicitly or implicitly, is the application of the algorithm of Călinescu et al. [9] on a transformed simplex. Several types of transformations were proposed, e.g.: a local transformation f:[0,1][0,1] that is applied independently to each of the k coordinates of each point in the simplex (in some works, e.g., [21], the local transformation of the simplex is implicitly given by the equivalent description of choosing a radius r for the 1 sphere from a non-uniform distribution); and a global transformation f:Δk[0,1]k that is applied to each point in the simplex. It is worth noting that the latter type of transformation is more general than the former as it might introduce dependencies between the different coordinates.

Unfortunately, all works subsequent to that of Călinescu et al. [9] do not settle only for the above described approach and resort to the mixing of multiple algorithmic components. In many cases this results in both complicated algorithms and analysis. For example, the currently best known approximation of 1.2965 (given by Sharma and Vondrák [28]) requires the mixing of four different algorithmic components and its analysis is computer assisted. We note that using computer assisted proofs to numerically optimize the mixing of different algorithmic components for partitioning the simplex is not unique to the work of [28]. Table 1 summarizes most of the suggested algorithms for the Multiway Cut problem along with the mix of the algorithmic components that they use for partitioning the simplex.

We present a simple algorithm for the Multiway Cut problem that is comprised of a single algorithmic component. Our single component algorithm achieves an approximation factor of 118=1.375 and is based on the global transformation approach of [8]. We note that our algorithm is the first and currently only known algorithm that is not comprised of the mixing of several different algorithmic components for partitioning the simplex and achieves an approximation guarantee better than the 1.5 guarantee of Călinescu et al. [9]. For simplicity of presentation we assume that k is large and thus ignore the factor that depends on k that is shaved off the approximation guarantee.

Theorem 1.1

There exists a transformation f:Δk[0,1]k such that applying it prior to the algorithm of [9] results in an approximation of118=1.375 for the Multiway Cut problem.

We complement the above result in two ways. First, we note that the algorithm of Theorem 1.1 can be easily optimized for small values of k. We demonstrate this for k=3 and obtain a simple approximation guarantee of 2825=1.12, which is very close to the known tight guarantee of 12111.09 [11], [21]. Second, we focus on algorithms that first transform the simplex and then partition it using the algorithm of [9]. For such algorithms we show that the use of a global transformation f:Δk[0,1]k, as opposed to a local transformation, is necessary if one wishes to obtain an approximation better than 1.5 for all values of k. Specifically, we show that using a local transformation f:[0,1][0,1] (that is applied independently to each coordinate of each point uΔk prior to executing the algorithm of Călinescu et al. [9]) it is not possible to obtain an approximation better than 1.5. The exact statement appears in Section 4.

Dahlhaus et al. [12] showed that Multiway Cut is APX-hard. Thus, there exists a constant c>1 such that no polynomial-time algorithm can find a solution within a factor of c of the optimum, unless P=NP. For the geometric relaxation of Călinescu et al. [9], Freund and Karloff [16] exhibited an integrality gap of at least 8(7+1(k1)). This was improved by Angelidakis et al. [1] who proved that the integrality gap is at least 6(5+1(k1)). Additionally, the latter was recently improved by Bérczi et al. [6] who proved that for every ϵ>0 there exists an instance of Multiway Cut whose integrality gap (with respect to the relaxation of [9]) is at least 1.20016ϵ. When k=3, Karger et al. [21] and Cunningham and Teng [11] present an approximation factor of 1211 that matches the integrality gap for this value of k [21]. Thus, Multiway Cut is resolved when k=3. For k=4 and 5, Karger et al. [21] provide improved approximation factors of 1.1539 and 1.2161, respectively.

Other variants and special cases of Multiway Cut were also studied. When considering dense unweighted graphs, Arora et al. [2] and Frieze and Kannan [17] provide a polynomial time approximation scheme. The Node Multiway Cut problem asks for the least weight subset of vertices whose removal from the graph disconnects all terminals. This variant was studied by Garg et al. [19] who present a 2(11k)-approximation algorithm for the problem. They also prove that any improvement to the latter factor would also lead to an improvement of the approximation guarantee for Vertex Cover, for which it is known that no approximation better than 2 can be achieved assuming the unique games conjecture (Khot and Regev [23]). The node version when k is fixed was studied in [5]. The Directed Multiway Cut problem asks for the least weight subset of edges whose removal from the graph disconnects all directed paths connecting terminals. Clearly, Directed Multiway Cut generalizes Node Multiway Cut. For Directed Multiway Cut, Naor and Zosin [27] give an approximation of 2, improving upon the O(logk)-approximation of Garg et al. [19]. The Multicut problem resembles Multiway Cut, however its goal is to separate k pairs of given terminals {si,ti}. The best known approximation for Multicut is O(logk) and is given by Garg et al. [18].

Section snippets

Preliminaries

In the Multiway Cut problem we are given an undirected graph G=(V,E) equipped with non-negative edge weights w:ER+ and a set T=t1,t2,,tkV of k terminals. The goal is to find a minimum weight subset of edges XE such that all terminals are disconnected in (V,EX). Equivalently, the output is a partition of V into k clusters, denoted by g:V1,,k, such that the ith cluster contains ti, i.e., g(ti)=i for every i=1,,k. Obviously, any edge e=(u,v)E satisfies that eX if and only if g(u)g(v).

A Simple 118=1.375 Approximation Algorithm

In this section we present a simple global simplex transformation f:Δk[0,1]k which, when applied prior to executing Algorithm 1, achieves an approximation of 118=1.375. Following [8], our transformation f:Δk[0,1]k, given uΔk, first sorts the coordinates of u in a non-increasing order and then applies a transformation to each of the coordinates that depends on its position in the sorting. As already previously mentioned in Section 2, and without loss of generality, rename the coordinates of u

Extesnsions

Small Values of k: For small values of k it is possible to further optimize the transformation f yielding improved approximation guarantees. Additionally, we claim that our approximation guarantees for small values of k are simpler than the current best ones with respect to both algorithm and analysis.

A special case of particular interest is when k=3, for which the algorithm of Călinescu et al. [9] achieves an approximation of 761.166 and a tight guarantee of 12111.09 is known [11], [21]. It

Acknowledgments

The research of Niv Buchbinder is supported by ISF, Israel grant 2233/19 and by BSF, Israel grant 2018352. The research of Roy Schwartz is supported by ISF, Israel grant 1336/16.

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