Lossy Reduction Rules for the Directed Feedback Vertex Set Problem

Abstract

Given a directed graph G(V, A), the DIRECTED FEEDBACK VERTEX SET (DFVS) problem asks for the smallest sized subset of V whose removal makes G acyclic. The problem is NP-complete and efficient constant-factor approximation algorithms are ruled out under UGC. Attempting to get an exact DFVS in practice usually involves the application of reduction rules that decrease the instance size without compromising the optimal solution. If the reduced graph gets sufficiently small, the respective instance can then be solved to optimality e.g. by a branching algorithm. However, one might need to resort to heuristics in the end in case the reduced instance is still huge. In this paper, we propose novel reduction rules for DFVS with a special focus on lossy rules. Here, the idea is that an optimal solution on the reduced graph combined with the information gained in the reduction process provides an α-approximation for the original instance. We present several rules that ensure small α, and discuss how to combine and engineer them. We also propose a taxonomy to study general types of lossy rules. In an extensive experimental analysis, we evaluate the impact of exact and lossy rules on the running time, the size of the reduced instance, and the solution quality. It turns out that the lossy rules are indeed very effective and that it is often possible to solve instances by using reduction rules only.