Bounded-Degree Cut is Fixed-Parameter Tractable

TL;DR

This paper proves that the bounded-degree cut problem can be solved efficiently using a fixed-parameter tractable algorithm with exponential dependence on the parameter k, advancing the understanding of cut problems with degree constraints.

Contribution

The paper introduces the first single exponential fixed-parameter tractable algorithm for the bounded-degree cut problem, utilizing new lemmas based on important cuts.

Findings

Developed a $2^{18k}|G|^{O(1)}$-time algorithm for the problem.

Established upper bounds on candidate subsets for minimal cuts.

Lemmas can be applied to related problems in fixed-parameter algorithms.

Abstract

In the bounded-degree cut problem, we are given a multigraph $G=(V,E)$, two disjoint vertex subsets $A,B\subseteq V$, two functions $\mathrm{u}_A, \mathrm{u}_B:V\to \{0,1,\ldots,|E|\}$ on $V$, and an integer $k\geq 0$. The task is to determine whether there is a minimal $(A,B)$-cut $(V_A,V_B)$ of size at most $k$ such that the degree of each vertex $v\in V_A$ in the induced subgraph $G[V_A]$ is at most $\mathrm{u}_A(v)$ and the degree of each vertex $v\in V_B$ in the induced subgraph $G[V_B]$ is at most $\mathrm{u}_B(v)$. In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a $2^{18k}|G|^{O(1)}$-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems.