NDLI: Independence and efficient domination on $P_{6}-free$ graphs

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Independence and efficient domination on $P_{6}-free$ graphs

Content Provider	ACM Digital Library
Author	van Leeuwen, Erik Jan Lokshtanov, Daniel Pilipczuk, Marcin
Abstract	In the Maximum Weight Independent Set problem, the input is a graph G, every vertex has a nonnegative integer weight, and the task is to find a set S of pairwise non-adjacent vertices, maximizing the total weight of the vertices in S. We give an $n^{[EQUATION](log2}$ $^{n)}$ time algorithm for this problem on graphs excluding the path $P_{6}$ on 6 vertices as an induced subgraph. Currently, there is no constant k known for which Maximum Weight Independent Set on $P_{k}-free$ graphs becomes NP-complete, and our result implies that if such a k exists, then k > 6 unless all problems in NP can be decided in (quasi)polynomial time. Using the combinatorial tools that we develop for the above algorithm, we also give a polynomial-time algorithm for Maximum Weight Efficient Dominating Set on $P_{6}-free$ graphs. In this problem, the input is a graph G, every vertex has an integer weight, and the objective is to find a set S of maximum weight such that every vertex in G has exactly one vertex in S in its closed neighborhood, or to determine that no such set exists. Prior to our work, the class of $P_{6}-free$ graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of Maximum Weight Efficient Dominating Set was unknown.
Starting Page	1784
Ending Page	1803
Page Count	20
File Format	PDF
ISBN	9781611974331
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2016-01-10
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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