An FPT algorithm for Matching Cut and d-cut
N R Aravind, Roopam Saxena
TL;DR
This work studies the $d$-cut (and its special case MATCHING CUT with $d=1$) problem and provides the first explicit fixed-parameter tractable algorithm parameterized by the maximum edge-cut size $k$, running in $2^{O(k\log k)}n^{O(1)}$ time. The approach integrates a compact, bounded-adhesion tree decomposition with guaranteed unbreakability and a bottom-up dynamic programming framework that internalizes crossing-edge constraints through a memory table $M$ and auxiliary structures. Key techniques include $d$-matched candidate sets, $S$-compatible and $A_s$-restricted $P$-compatible families, and color-coding aided enumeration to bound the combinatorics, achieving explicit parameter dependence. An ETH-based lower bound is also provided, showing that MATCHING CUT cannot be solved in $2^{o(k)}n^{O(1)}$ time unless ETH fails, highlighting a gap between upper and lower bounds and clarifying the problem's complexity landscape.
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.