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Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs

Published online by Cambridge University Press:  10 January 2024

Pengcheng Liu ,
Zhao Zhang Open the ORCID record for Zhao Zhang [Opens in a new window] ,
Yingli Ran  and
Xiaohui Huang
Pengcheng Liu
Affiliation:
College of Computer Science and Artificial Intelligence, Wenzhou University,Wenzhou, Zhejiang, 325035, China
Zhao Zhang*
Affiliation:
School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China
Yingli Ran
Affiliation:
School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China
Xiaohui Huang
Affiliation:
School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China
*
Corresponding author: Zhao Zhang; Email: hxhzz@sina.com
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Abstract

Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC$_{con}$. In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$-time FPT algorithm for MinCkSC$_{con}$ on a graph with treewidth $\omega$, based on which we further design an $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC$_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.

Keywords

Connected-k-subgraph coverapproximation algorithmlocal searchminor-free graphPTASFPT
Type
Special Issue: TAMC 2022
Information
Mathematical Structures in Computer Science , First View , pp. 1 - 13
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

*

This research is supported in part by National Natural Science Foundation of China (U20A2068, 11771013), Zhejiang Provincial Natural Science Foundation of China (LD19A010001).

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