Parameterized Algorithms for Diverse Multistage Problems

TL;DR

This paper introduces a framework for diverse multistage problems, demonstrating fixed-parameter tractability for several classic problems by leveraging diversity, with potential applications in fairness and wear minimization.

Contribution

The paper presents a novel framework for diverse multistage problems and proves fixed-parameter tractability for multiple classical problems based on diversity.

Findings

Fixed-parameter tractability established for diverse multistage problems

Development of colored variants of classical problems

Applicability to fairness and wear minimization scenarios

Abstract

The world is rarely static -- many problems need not only be solved once but repeatedly, under changing conditions. This setting is addressed by the "multistage" view on computational problems. We study the "diverse multistage" variant, where consecutive solutions of large variety are preferable to similar ones, e.g. for reasons of fairness or wear minimization. While some aspects of this model have been tackled before, we introduce a framework allowing us to prove that a number of diverse multistage problems are fixed-parameter tractable by diversity, namely Perfect Matching, s-t Path, Matroid Independent Set, and Plurality Voting. This is achieved by first solving special, colored variants of these problems, which might also be of independent interest.