Finding k-secluded trees faster

We revisit the k-Secluded Tree problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2O(klog⁡k)⋅nO(1), improving on...

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Bibliographic Details
Published in:Journal of computer and system sciences Vol. 138; p. 103461
Main Authors: Donkers, Huib, Jansen, Bart M.P., de Kroon, Jari J.H.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.12.2023
Subjects:
Enumeration algorithm
FPT
Secluded tree
Secluded tree
Enumeration algorithm
FPT
ISSN:0022-0000, 1090-2724
Online Access:Get full text
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Summary:We revisit the k-Secluded Tree problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2O(klog⁡k)⋅nO(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.
ISSN:0022-0000
1090-2724
DOI:10.1016/j.jcss.2023.05.006