On knot-free vertex deletion: Fine-grained parameterized complexity analysis of 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)∖V(Q). Knots are important graph structures in Computer Science because they characterize the existence of deadlocks in a classical distributed computation model, the so-called OR-model. Given a directed graph G, and a positive integer k, the Knot-Free Vertex Deletion (KFVD) problem consists of determining whether G has a subset S⊆V(G) with size at most k such that G[V∖S] contains no knot. KFVD is an NP-complete graph problem with natural applications in deadlock resolution. In this paper, we present a parameterized complexity analysis of KFVD. We prove that: KFVD is W[1]-hard when parameterized by the size of the solution; it can be solved in 2klog⁡φ⋅nO(1) time, but assuming SETH, it cannot be solved in (2−ϵ)klog⁡φ⋅nO(1) time, where φ is the size of the largest strongly connected subgraph of G; it can be solved in 2ϕ⋅nO(1) time, but assuming ETH, it cannot be solved in 2o(ϕ)⋅nO(1) time, where ϕ is the number of vertices with out-degree at most k; it can be solved in 2O(twlog⁡tw)⋅nO(1) time, but assuming ETH it cannot be solved in 2o(tw)⋅nO(1) time, where tw is the treewidth of the underlying undirected graph; and, it can be solved in FPT time when parameterized by the clique-width of the directed graph. In addition, lower bounds for kernelization are also provided.