Triangle packing in (sparse) tournaments: approximation and kernelization

TL;DR

This paper investigates the computational complexity and kernelization of the triangle packing problem in tournaments, providing approximation algorithms for sparse instances and establishing kernel size bounds.

Contribution

It introduces a (1+6/(c-1)) approximation algorithm for sparse tournaments with length constraints and shows kernelization bounds, including an O(k) vertices kernel for certain instances.

Findings

No PTAS exists unless P=NP for sparse tournaments.

A (1+6/(c-1)) approximation algorithm is provided for specific sparse tournaments.

C_3-Pakcing-T admits an O(m) vertices kernel, and an O(k) kernel for sparse instances.

Abstract

Given a tournament T and a positive integer k, the C_3-Pakcing-T problem asks if there exists a least k (vertex-)disjoint directed 3-cycles in T. This is the dual problem in tournaments of the classical minimal feedback vertex set problem. Surprisingly C_3-Pakcing-T did not receive a lot of attention in the literature. We show that it does not admit a PTAS unless P=NP, even if we restrict the considered instances to sparse tournaments, that is tournaments with a feedback arc set (FAS) being a matching. Focusing on sparse tournaments we provide a (1+6/(c-1)) approximation algorithm for sparse tournaments having a linear representation where all the backward arcs have "length" at least c. Concerning kernelization, we show that C_3-Pakcing-T admits a kernel with O(m) vertices, where m is the size of a given feedback arc set. In particular, we derive a O(k) vertices kernel for C_3-Pakcing-T when restricted to sparse instances. On the negative size, we show that C_3-Pakcing-T does not admit a kernel of (total bit) size O(k^{2-\epsilon}) unless NP is a subset of coNP / Poly. The existence of a kernel in O(k) vertices for C_3-Pakcing-T remains an open question.