An FPT algorithm for Matching Cut and d-cut

TL;DR

This paper introduces the first fixed parameter tractable algorithm for the d-CUT and MATCHING CUT problems, providing explicit bounds on running time based on the size of the edge cut, advancing the understanding of their computational complexity.

Contribution

The paper presents the first explicit FPT algorithm for d-CUT and MATCHING CUT with a running time of 2^{O(k log k)}n^{O(1)}, improving upon previous indirect approaches.

Findings

First explicit FPT algorithm for d-CUT and MATCHING CUT

Running time of 2^{O(k log k)}n^{O(1)} for the problems

No algorithm with time 2^{o(k)}n^{O(1)} unless ETH fails

Abstract

Given a positive integer $d$, the d-CUT is the problem of deciding if an undirected graph $G=(V,E)$ has a cut $(A,B)$ such that every vertex in $A$ (resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). For $d=1$, the problem is referred to as MATCHING CUT. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for d-CUT parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However, their paper doesn't provide an explicit bound on the running time, as it indirectly relies on a MSOL formulation and Courcelle's Theorem. Motivated by this, we design and present an FPT algorithm for d-CUT for general graphs with running time $2^{O(k\log k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut. This is the first FPT algorithm for the d-CUT and MATCHING CUT with an explicit dependence on this parameter. We also observe that there is no algorithm solving MATCHING CUT in time $2^{o(k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut unless ETH fails.