A faster FPT algorithm for 3-path vertex cover

Ján Katrenič

doi:10.1016/j.ipl.2015.12.002

Abstract

The k-path vertex cover of a graph G is a subset S of vertices of G such that every path on k vertices in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover set in G. The minimum k-path vertex cover problem (k-PVCP) is to find a k-path vertex cover of size ψk(G). In this paper we present an FPT algorithm to the 3-PVCP with runtime O(1.8172snO(1)) on a graph with n vertices. The algorithm constructs a 3-path vertex cover of size at most s in a given graph G, or reports that no such 3-path vertex cover exists in G. This improves previous O(2snO(1)) upper bound by Tu [5] and O(1.882snO(1)) upper bound by Wu [13].

Highlights

The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G = (V, E) such that G\F does not contain a path on d vertices
We present an algorithm that solves the 5-PVC problem parameterized by the size of the solution k in O∗(4k) time by employing the iterative compression technique
We see the trend of solving 3-PVC, 4-PVC and 5-PVC with the iterative compression technique, so it is natural to ask whether this approach can be further used for 6-PVC or even to d-PVC in general

Summary

Introduction

In this paper we aim on the parameterized analysis of the problem, that is, to confine the exponential part of the running time to a specific parameter of the input, presumably much. When parameterized by the size of the solution k, the d-PVC problem is directly solvable by a trivial FPT algorithm that runs in O∗(dk) time.. For the 2-PVC (Vertex Cover) problem, the algorithm of Chen, Kanj, and Xia [2] has the currently best known running time of O∗(1.2738k). For the 4-PVC problem, Tu and Jin [17] again used iterative compression and achieved a running time O∗(3k) and Tsur [14] claims to have an algorithm with running time O∗(2.619k). We present an algorithm that solves the 5-PVC problem parameterized by the size of the solution k in O∗(4k) time by employing the iterative compression technique. To showcase our technique of proving the correctness of our algorithm, we included the proofs of correctness for Rules (R10) and (R13.1), and proofs of Lemmata 18 and 21

Preliminaries

Algorithm

Preprocessing

3.11 Final remarks

Conclusion