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Solving Co-Path/Cycle Packing and Co-Path Packing Faster Than 3^k

Authors Yuxi Liu, Mingyu Xiao

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Author Details

Yuxi Liu
  • University of Electronic Science and Technology of China, Chengdu, China
Mingyu Xiao
  • University of Electronic Science and Technology of China, Chengdu, China

Funding

The work is supported by the National Natural Science Foundation of China, under the grants 62372095, 62172077, and 62350710215.

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Yuxi Liu and Mingyu Xiao. Solving Co-Path/Cycle Packing and Co-Path Packing Faster Than 3^k. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.11

Abstract

The Co-Path/Cycle Packing problem (resp. The Co-Path Packing problem) asks whether we can delete at most k vertices from the input graph such that the remaining graph is a collection of induced paths and cycles (resp. induced paths). These two problems are fundamental graph problems that have important applications in bioinformatics. Although these two problems have been extensively studied in parameterized algorithms, it seems hard to break the running time bound 3^k. In 2015, Feng et al. provided an O^*(3^k)-time randomized algorithms for both of them. Recently, Tsur showed that they can be solved in O^*(3^k) time deterministically. In this paper, by combining several techniques such as path decomposition, dynamic programming, cut & count, and branch-and-search methods, we show that Co-Path/Cycle Packing can be solved in O^*(2.8192^k) time deterministically and Co-Path Packing can be solved in O^*(2.9241^{k}) time with failure probability ≤ 1/3. As a by-product, we also show that the Co-Path Packing problem can be solved in O^*(5^p) time with probability at least 2/3 if a path decomposition of width p is given.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Graph Algorithms
  • Parameterized Algorithms
  • Co-Path/Cycle Packing
  • Co-Path Packing
  • Cut & Count
  • Path Decomposition

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