On the Complexity of Connected $(s,t)$-Vertex Separator

On the Complexity of Connected (s, t)-Vertex Separator N.S.Narayanaswamy 1 and N.Sadagopan 2 1 Department of Computer Science and Engineering, Indian Institute of Technology Madras, India 2 Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, India. {swamy@cse.iitm.ac.in, sadagopan@iiitdm.ac.in} Abstract. We investigate the complexity of minimum connected (s, t)-vertex separator ((s, t)-CVS) and show that (s, t)-CVS is Ω(log2−ϵn)-hard for any ϵ > 0 unless NP has quasi-polynomial Las-Vegas algorithms. i.e., for any ϵ > 0 and for some δ > 0, (s, t)-CVS is unlikely to have δ.log2−ϵn-approximation algorithm. We then present an interesting chordality dichotomy: we show that (s, t)-CVS is NP-complete on graphs of chordality at least 5 and present a polynomial-time algorithm for (s, t)-CVS on chordality 4 graphs. We also present a ⌈c 2⌉-approximation algorithm for (s, t)-CVS on graphs with chordality c. Finally, from the parameterized setting, we show that (s, t)-CVS parameterized above the (s, t)-vertex connectivity is W[2]-hard. 1 Introduction The vertex or edge connectivity of a graph and the corresponding separators are of fundamental interest in Computer Science and Graph Theory. Many kinds of vertex separators, stable vertex separators [1], clique vertex separators [2], constrained vertex separators [3], and α-balanced separators [3] are of interest to the research community. As far as complexity results are concerned, finding a minimum vertex separator and a clique vertex separator are polynomial-time solvable, whereas, stable vertex separator and other constrained separators reported in [3] are NP-hard. This shows that imposing an appropriate constraint on the well- studied vertex separator problem makes the problem NP-hard. Interestingly, constrained vertex separators have received much attention in parameterized complexity as well [3,4]. In particular, Marx in [3] considered the parameterized complexity of constrained separators satisfying some hereditary properties. For example, clique separators and stable separators. It is shown in [3] that the above problems have an algorithm whose running time is f(k).nO(1), where k is the size of a constrained separator. Algorithms of this nature are pop- ularly known as fixed-parameter tractable algorithms with parameter as the solution size [5]. While many constrained vertex separators have attracted researchers from both classical and parameterized complexity, the related problem of finding a minimum connected (s, t)-vertex separator is open. In light of [3], this question can also be looked at as finding a (s, t)-vertex separator satisfying some non-hereditary property, like connectedness. Moreover, the results in [3] do not carry over to connected (s, t)-vertex separator and the complexity of it remains open. With these motivations, in this paper, we focus our attention on the computational complexity of minimum connected (s, t)-vertex separator ((s, t)-CVS). Remark: The (s, t)-CVS can also be motivated from the theory of graph minors. We observe that there is an equivalence between the computational problems of finding a minimum connected (s, t)-vertex separator and a minimum set of edges whose contraction reduces the (s, t)-vertex connectivity to one. It is important to note that the analogous computational problem of reducing the (s, t)-edge connectivity to zero by a min- imum number of edge deletions is polynomial-time solvable, because this is computationally equivalent to finding a minimum (s, t)-cut and deleting all edges in it. Our Results: In this paper, we consider connected undirected unweighted simple graphs. For a graph G, let (s, t) denote a fixed non-adjacent pair of vertices in G. Throughout this paper, when we refer to edge contraction, we do not contract edges incident on s and edges incident on t. 1. We establish a polynomial-time reduction from the Group Steiner Tree [ND12, [6]] to (s, t)-CVS. Con- sequently, it follows that there is no polynomial-time approximation algorithm with approximation factor δ.log2−ϵn for some δ > 0 and for any ϵ > 0, unless NP has quasi-polynomial Las-Vegas algorithms. 2. We then observe that on chordal graphs finding a minimum (s, t)-CVS is polynomial-time solvable as every arXiv:1111.1814v2 [cs.DM] 2 Mar 2022 minimal vertex separator is a clique. We show that deciding (s, t)-CVS is NP-complete on chordality 5 graphs and on chordality 4 graphs (s, t)-CVS is polynomial-time solvable. We also present a ⌈c 2⌉-approximation al- gorithm for (s, t)-CVS on graphs with chordality c. 3. We then consider designing algorithms for (s, t)-CVS whose running time is f(k).nO(1) where k is the parameter of interest and f(k) is a function independent of n. If the parameter of interest is the chordality c of the graph, then it follows from the above result that (s, t)-CVS is unlikely to have an algorithm whose running time is f(c).nO(1), c ≥5, unless P=NP. Whereas, on graphs of treewidth ω, we show the exi

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