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Algorithmic developments and complexity results for finding maximum and exact independent sets in graphs
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Algorithmic developments and complexity results for finding maximum and exact independent sets in graphs
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Milanic
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Martin
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Martin Milanic
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Lozin
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Vadim
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Advisory Committee
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Vadim V. Lozin
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Alizadeh
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Farid Alizadeh
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Boros
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Endre Boros
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Gurvich
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Vladimir
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Vladimir Gurvich
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Kahn
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Jeffry
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Jeffry Kahn
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Monnot
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Jérôme
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Advisory Committee
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Jérôme Monnot
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Rutgers University
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Graduate School-New Brunswick
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theses
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2007
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2007
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English
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electronic
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Extent
viii, 139 pages
Abstract
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We consider the maximum independent set and maximum weight independent set problems in graphs. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, that is, in graph classes defined by a set F of forbidden induced subgraphs.
We describe a condition on the set F, which guarantees that the maximum independent set problem remains NP-hard in the class of F-free graphs. The same hardness result remains valid even under the additional restriction that the graphs are planar and of maximum degree at most three.
We exhibit several cases where the condition is violated, and the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the maximum independent set problem in:
-two graph classes that properly contain claw-free graphs (thus generalizing the classical result for claw-free graphs);
-various subclasses of planar and more general graphs;
-weighted graphs in certain subclasses of graphs of bounded vertex degree.
Our algorithms are based on various approaches. In particular, we develop an extension of the method of finding augmenting graphs. We also make extensive use of some other well-known graph decompositions.
We also introduce and study the exact version of the problem, where, instead of finding an independent set of maximum weight, the goal is to find an independent set of given weight. Determining the computational complexity of this problem for line graphs, or for line graphs of bipartite graphs would resolve long standing open problems. Here, we show that:
-The exact weighted independent set problem is strongly NP-complete for cubic bipartite graphs.
-The problem is solvable in pseudo polynomial time for any of the following graph classes:
mK2-free graphs, interval graphs and their generalizations k-thin graphs, circle graphs, chordal graphs, AT-free graphs, (claw, net)-free graphs, distance-hereditary graphs, and graphs of bounded tree- or clique-width.
Finally, we show how modular decomposition can be applied to the exact weighted independent set problem. As a corollary, we obtain pseudo-polynomial solutions for the problem in certain subclasses of P5-free and fork-free graphs.
We describe a condition on the set F, which guarantees that the maximum independent set problem remains NP-hard in the class of F-free graphs. The same hardness result remains valid even under the additional restriction that the graphs are planar and of maximum degree at most three.
We exhibit several cases where the condition is violated, and the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the maximum independent set problem in:
-two graph classes that properly contain claw-free graphs (thus generalizing the classical result for claw-free graphs);
-various subclasses of planar and more general graphs;
-weighted graphs in certain subclasses of graphs of bounded vertex degree.
Our algorithms are based on various approaches. In particular, we develop an extension of the method of finding augmenting graphs. We also make extensive use of some other well-known graph decompositions.
We also introduce and study the exact version of the problem, where, instead of finding an independent set of maximum weight, the goal is to find an independent set of given weight. Determining the computational complexity of this problem for line graphs, or for line graphs of bipartite graphs would resolve long standing open problems. Here, we show that:
-The exact weighted independent set problem is strongly NP-complete for cubic bipartite graphs.
-The problem is solvable in pseudo polynomial time for any of the following graph classes:
mK2-free graphs, interval graphs and their generalizations k-thin graphs, circle graphs, chordal graphs, AT-free graphs, (claw, net)-free graphs, distance-hereditary graphs, and graphs of bounded tree- or clique-width.
Finally, we show how modular decomposition can be applied to the exact weighted independent set problem. As a corollary, we obtain pseudo-polynomial solutions for the problem in certain subclasses of P5-free and fork-free graphs.
Note
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Ph.D.
Note
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Includes bibliographical references (p. 132-138).
Subject
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Topic
Operations Research
Subject
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Topic
Graph algorithms
Subject
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Topic
Graph theory
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.13482
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doi:10.7282/T3K64JH9
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ETD doctoral
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Martin Milanic
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Rutgers University. Graduate School-New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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