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Title: Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree
Authors: Li, W
Cao, Y 
Chen, J
Wang, J
Issue Date: 2017
Source: Information and computation, 2017, v. 252, p. 187-200
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.
Keywords: Local search
Maximum internal spanning tree
Parameterized computation
Publisher: Academic Press
Journal: Information and computation 
ISSN: 0890-5401
EISSN: 1090-2651
DOI: 10.1016/j.ic.2016.11.003
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