Randomized Contractions Meet Lean Decompositions

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 ⩽ i ⩽ k . Here, a set X ⊆ 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 | ⩽ a or | B ∩ X | ⩽ a . The resulting tree decomposition has arguably best possible adhesion size bounds and 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 M INIMUM B ISECTION , S TEINER C UT , and S TEINER M ULTICUT 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.