Randomized contractions meet lean decompositions

TL;DR

This paper introduces an efficient algorithm for constructing tree decompositions with optimal adhesion and unbreakability properties, enabling improved parameterized algorithms for graph cut problems.

Contribution

It adapts lean decompositions to the parameterized setting, achieving better bounds and faster algorithms for graph separation problems.

Findings

Algorithm constructs tree decompositions with optimal adhesion and unbreakability.

Improved parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut.

Faster running time bounds due to smaller parametric factors.

Abstract

We show an algorithm that, given an $n$-vertex graph $G$ and a parameter $k$, in time $2^{O(k \log k)} n^{O(1)}$ finds a tree decomposition of $G$ with the following properties: * every adhesion of the tree decomposition is of size at most $k$, and * every bag of the tree decomposition is $(i,i)$-unbreakable in $G$ for every $1 \leq i \leq k$. Here, a set $X \subseteq V(G)$ is $(a,b)$-unbreakable in $G$ if for every separation $(A,B)$ of order at most $b$ in $G$, we have $|A \cap X| \leq a$ or $|B \cap X| \leq a$. The resulting tree decomposition has arguably best possible adhesion size boundsand unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.