Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings

TL;DR

This paper introduces a new degree ordering approach to compute and approximate cutwidth and pathwidth of semi-complete digraphs, simplifying proofs and enabling faster algorithms for these width measures.

Contribution

The paper presents a novel degree ordering technique for semi-complete digraphs, leading to simplified proofs, improved approximation algorithms, and fixed-parameter tractable algorithms for width measures.

Findings

Polynomial-time approximation algorithms for cutwidth and pathwidth.

Constant-factor polynomial-time approximation for pathwidth.

Single-exponential fixed-parameter tractable algorithms for width measures.

Abstract

The notions of cutwidth and pathwidth of digraphs play a central role in the containment theory for tournaments, or more generally semi-complete digraphs, developed in a recent series of papers by Chudnovsky, Fradkin, Kim, Scott, and Seymour [2, 3, 4, 8, 9, 11]. In this work we introduce a new approach to computing these width measures on semi-complete digraphs, via degree orderings. Using the new technique we are able to reprove the main results of [2, 9] in a unified and significantly simplified way, as well as obtain new results. First, we present polynomial-time approximation algorithms for both cutwidth and pathwidth, faster and simpler than the previously known ones; the most significant improvement is in case of pathwidth, where instead of previously known O(OPT)-approximation in fixed-parameter tractable time [6] we obtain a constant-factor approximation in polynomial time. Secondly, by exploiting the new set of obstacles for cutwidth and pathwidth, we show that topological containment and immersion in semi-complete digraphs can be tested in single-exponential fixed-parameter tractable time. Finally, we present how the new approach can be used to obtain exact fixed-parameter tractable algorithms for cutwidth and pathwidth, with single- exponential running time dependency on the optimal width.