Degreewidth: a New Parameter for Solving Problems on Tournaments

TL;DR

This paper introduces the degreewidth parameter for tournaments, explores its computational complexity, and analyzes classical problems like Dominating Set and Feedback Vertex Set in relation to this new measure.

Contribution

It defines degreewidth, proves NP-hardness of computing it, provides approximation algorithms, and studies related classical problems in the context of degreewidth.

Findings

Degreewidth is NP-hard to compute.

A 3-approximation algorithm for degreewidth exists.

Dominating Set is fixed parameter tractable with respect to degreewidth.

Abstract

In the paper, we define a new parameter for tournaments called degreewidth which can be seen as a measure of how far is the tournament from being acyclic. The degreewidth of a tournament $T$ denoted by $\Delta(T)$ is the minimum value $k$ for which we can find an ordering $\langle v_1, \dots, v_n \rangle$ of the vertices of $T$ such that every vertex is incident to at most $k$ backward arcs (\textit{i.e.} an arc $(v_i,v_j)$ such that $j<i$). Thus, a tournament is acyclic if and only if its degreewidth is zero. Additionally, the class of sparse tournaments defined by Bessy et al. [ESA 2017] is exactly the class of tournaments with degreewidth one. We first study computational complexity of finding degreewidth. Namely, we show it is NP-hard and complement this result with a $3$-approximation algorithm. We also provide a cubic algorithm to decide if a tournament is sparse. Finally, we study classical graph problems \textsc{Dominating Set} and \textsc{Feedback Vertex Set} parameterized by degreewidth. We show the former is fixed parameter tractable whereas the latter is NP-hard on sparse tournaments. Additionally, we study \textsc{Feedback Arc Set} on sparse tournaments.