Kernels for Feedback Arc Set In Tournaments

TL;DR

This paper presents a new linear kernelization algorithm for the k-Feedback Arc Set in Tournaments problem, significantly reducing the problem size from quadratic to linear in parameter k.

Contribution

It introduces a polynomial-time kernelization method that reduces instances to O(k) vertices, improving previous bounds and applying to a subclass of tournaments.

Findings

Achieved a linear vertex kernel for k-FAST.

Reduced kernel size from O(k^2) to O(k).

Provided polynomial-time solutions for a subclass of tournaments.

Abstract

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.