Complexity and approximability of the k‐way vertex cut

Abstract

In this article, we consider k -way vertex cut: the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of components. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs. Also, we show that k -way vertex cut is polynomially solvable on graphs of bounded treewidth and fixed-parameter tractable on planar graphs with the size of the separator as the parameter.

The firebreak problem

“…We now proceed to show that the Firebreak and Key Player decision problems are both NP‐complete, even when restricted to the class of split graphs. To do so we will refer to the t ‐ Way Vertex Cut problem studied by Berger et al [7], expressed as a decision problem as follows:…”

“…Theorem When restricted to split graphs , the Firebreak and Key Player decision problems are both NP ‐complete . Proof Relying on a construction of Marx [26] that is restricted to split graphs, Berger et al show that the t ‐ Way Vertex Cut problem is NP‐complete when restricted to split graphs [7]. By using an oracle for the Key Player problem, it is straightforward to answer any given instance ( G , k , t ) of the t ‐ Way Vertex Cut problem.…”

“…Some of the early papers about this problem in the literature attest to its applicability to network tolerance and robustness [5,28,32]. More recent results have considered it from a computational complexity perspective, establishing that it is NP-hard for various classes of graphs yet being solvable in polynomial time for graphs having bounded treewidth [1,7,26,33]. A recent survey by Lalou et al on the topic of detecting critical nodes in networks also touches on this problem [24].…”

Models and algorithms for decomposition problems

“…Literature review. The k-Vertex Cut problem is polynomially solvable for k = 2 [12], and it is NP-hard for k ≥ 3, when k is part of the input [14]. Only very recently, in [11] the authors show that even for a fixed value of k, the problem remains NP-hard for k ≥ 3.…”

“…For our experiments we considered only the six classes (11,12,13,14,15,16) that are denoted as hard by Pisinger [94] for KP algorithms, and used five different values of R (namely, R = 10 3 , 10 4 , 10 5 , 10 6 and 10 7 ) and five different values of n (n = 20, 50, 100, 200, and 500). It turned out that the generator returned integer overflow when generating instances of class 16 with R ≥ 10 5 ; thus, we disregarded the corresponding instances.…”