Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree

Wenjun Li, Yixin Cao, Jianer Chen, Jianxin Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

28 Citations (Scopus)

Abstract

The maximum internal spanning tree problem asks for a spanning tree of a given graph that has the maximum number of internal vertices among all spanning trees of this graph. In its parameterized version, we are interested in whether the graph has a spanning tree with at least k internal vertices. Fomin et al. (2013) [4] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a 3k-vertex kernel, implying an O⁎(8k)-time parameterized algorithm. Using depth-2 local search, Knauer and Spoerhase (2015) [9] developed a (5/3)-approximation algorithm for the optimization version. We try deeper local search: We conduct a thorough combinatorial analysis on the obtained spanning trees and explore their algorithmic consequences. We first observe that from the spanning tree obtained by depth-3 local search, one can easily find a reducible structure and apply the reduction rule of Fomin et al. This gives an improved kernel of 2k vertices, and as a by-product, a deterministic algorithm running in time O⁎(4k). We then go even deeper by considering the spanning tree obtained by depth-5 local search. It is shown that the number of internal vertices of this spanning tree is at least 2/3 of the maximum number a spanning tree can have, thereby delivering an improved approximation algorithm with ratio 1.5 for the problem.
Original languageEnglish
Pages (from-to)187-200
Number of pages14
JournalInformation and Computation
Volume252
DOIs
Publication statusPublished - 1 Feb 2017

Keywords

  • Local search
  • Maximum internal spanning tree
  • Parameterized computation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Information Systems
  • Computer Science Applications
  • Computational Theory and Mathematics

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