FPT Algorithms for Diverse Collections of Hitting Sets

TL;DR

This paper develops fixed-parameter tractable algorithms for finding diverse collections of solutions to the $d$-Hitting Set and Feedback Vertex Set problems, using network flow techniques to maximize diversity based on Hamming distances.

Contribution

It introduces a novel FPT approach for generating diverse solution collections in parameterized problems, employing a problem-independent network flow formulation.

Findings

Both problems are FPT in parameters k + r for the diversity measures.

A network flow formulation efficiently computes maximally diverse solution collections.

The approach addresses information loss in problem abstraction processes.

Abstract

In this work, we study the $d$-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of $r$ solutions of size at most $k$ each, which has recently been introduced to the field of parameterized complexity [Baste et al., 2019]. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process which models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are FPT in $k + r$ for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of `base' solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest.