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Randomized Contractions Meet Lean Decompositions
ACM Transactions on Algorithms ( IF 1.3 ) Pub Date : 2020-12-31 , DOI: 10.1145/3426738
Marek Cygan 1 , Paweł Komosa 1 , Daniel Lokshtanov 2 , Marcin Pilipczuk 1 , Michał Pilipczuk 1 , Saket Saurabh 3 , Magnus Wahlström 4
Affiliation  

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 ⩽ ik . Here, a set XV ( 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 | BX | ⩽ 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.

中文翻译:

随机收缩满足精益分解

我们展示了一个算法,给定一个n-顶点图G和一个参数ķ, 在时间 2 (ķ日志ķ) n (1)找到一个树分解G具有以下性质:-树分解的每个粘附最多具有大小ķ, 并且 - 树分解的每个包都是 (一世,一世)-牢不可破G每 1 ⩽一世ķ. 在这里,一组X(G) 是 (一种,b)-牢不可破G如果对于每次分离(一种,) 最多的顺序bG, 我们有 |一种\帽X| ⩽一种或 |X| ⩽一种. 由此产生的树分解可以说具有最佳的粘附大小界限和不可破坏性保证。此外,运行时间界限中的参数因子明显小于以前的类似结构。这些改进使我们能够为 M 提供参数化算法最低限度截面, S泰纳C犹他州, 和 S泰纳ULTICUT在运行时间范围内具有改进的参数因子。主要的技术见解是适应精益分解Thomas 和随后的 Bellenbaum 和 Diestel 构造算法到参数化设置。
更新日期:2020-12-31
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