FAST: Kernelization based on Graph Modular Decomposition

TL;DR

This paper introduces a novel linear kernelization algorithm for the unweighted feedback arc set on tournament problem using graph modular decomposition, advancing kernelization theory.

Contribution

It presents the first linear kernel for the unweighted feedback arc set on tournament, leveraging properties of graph modular decomposition.

Findings

First linear kernel for unweighted feedback arc set on tournament

Utilizes graph modular decomposition to improve kernel size

Advances understanding of kernelization for weighted and unweighted problems

Abstract

Kernelization algorithms, usually a preprocessing step before other more traditional algorithms, are very special in the sense that they return (reduced) instances, instead of final results. This characteristic excludes the freedom of applying a kernelization algorithm for the weighted version of a problem to its unweighted instances. Thus with only very few special cases, kernelization algorithms have to be studied separately for weigthed and unweighted versions of a single problem. {\sc feedback arc set on tournament} is currently a very popular problem in recent research of parameterized, as well as approximation computation, and its wide applications in many areas make it appear in all top conferences. The theory of graph modular decompositions is a general approach in the study of graph structures, which only had its surfaces touched in previous work on kernelization algorithms of {\sc feedback arc set on tournament}. In this paper, we study further properties of graph modular decompositions and apply them to obtain the first linear kernel for the unweighted {\sc feedback arc set on tournament} problem, which only admits linear kernel in its weighted version, while quadratic kernel for the unweighted.