Fine-Grained Parameterized Complexity Analysis of Knot-Free Vertex Deletion – A Deadlock Resolution Graph Problem

Abstract

A knot in a directed graph G is a strongly connected subgraph Q of G with size at least two, such that no vertex in V(Q) is an in-neighbor of a vertex in \(V(G)\setminus V(Q)\). Knots are a very important graph structure in the networked computation field, because they characterize deadlock occurrences into a classical distributed computation model, the so-called OR-model. Given a directed graph G and a positive integer k, in this paper we present a parameterized complexity analysis of the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether G has a subset \(S \subseteq V(G)\) of size at most k such that \(G[V\setminus S]\) contains no knot. KFVD is a graph problem with natural applications in deadlock resolution, and it is closely related to Directed Feedback Vertex Set. It is known that KFVD is NP-complete on planar graphs with bounded degree, but it is polynomial time solvable on subcubic graphs. In this paper we prove that: KFVD is W[1]-hard when parameterized by the size of the solution; it can be solved in \(2^{k\log \varphi }n^{O(1)}\) time, but assuming SETH it cannot be solved in \((2-\epsilon )^{k\log \varphi }n^{O(1)}\) time, where \(\varphi \) is the size of the largest strongly connected subgraph of G; it can be solved in \(2^{\phi }n^{O(1)}\) time, but assuming ETH it cannot be solved in \(2^{o(\phi )}n^{O(1)}\) time, where \(\phi \) is the number of vertices with out-degree at most k; unless \(PH = \varSigma _p^3\), KFVD does not admit polynomial kernel even when \(\varphi =2\) and k is the parameter.