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

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) \ 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 subset of V (G) of size at most k such that G[V \ 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) (phi)n (O(1)) time, but assuming SETH it cannot be solved in (2 - epsilon)(k) (log) (phi)n(O(1)) time, where phi 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 = Sigma(3)(p), KFVD does not admit polynomial kernel even when phi = 2 and k is the parameter.