Maximum Weight Independent Sets in hole- and co-chair-free graphs

doi:10.1016/j.ipl.2011.09.015

Information Processing Letters

Volume 112, Issue 3, 31 January 2012, Pages 67-71

https://doi.org/10.1016/j.ipl.2011.09.015 Get rights and content

Abstract

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be $NP$ -complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying a combination of clique separator and modular decomposition, we obtain a polynomial time solution of MWIS for hole- and co-chair-free graphs (the co-chair consists of five vertices four of which form a clique minus one edge – a diamond – and the fifth has degree one and is adjacent to one of the degree two vertices of the diamond).

Highlights

► The Maximum Weight Independent Set (MWIS) problem on graphs is a frequently studied problem. ► Clique separators and modular decomposition are powerful tools for solving this problem on various graph classes. ► We combine these tools for solving MWIS efficiently on hole- and co-chair-free graphs. ► In particular, we show that prime atoms of those graphs are nearly weakly chordal.

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A. Brandstädt, V. Giakoumakis, F. Maffray, Clique separator decomposition of hole- and diamond-free graphs and...

A. Brandstädt et al.

On clique separators, nearly chordal graphs and the Maximum Weight Stable Set problem

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Combining decomposition approaches for the Maximum Weight Stable Set problem
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The Maximum Weight Stable Set Problem (MWS) is a well-known NP-hard problem. A popular way to study MWS is to detect graph classes for which MWS can be solved in polynomial time. In this context some decomposition approaches have been introduced in the literature, in particular decomposition by homogeneous sets and decomposition by clique separators, which had several applications in the literature. Brandstädt and Hoàng (2007) [6] showed how to combine such two decomposition approaches and stated the following result: If the MWS problem can be solved in polynomial time for every subgraph of a graph G which has no homogeneous set and has no clique separators then so can the problem for G. This result, namely Corollary 9 of [6], was used in various papers.
Unfortunately, it was stated in a successive paper by Brandstädt and Giakoumakis (2015) [5] that “Actually, [6] does not give a proof for this, and it seems that Corollary 9 of [6] promises too much; later attempts for proving it failed, and thus, it is unproven and its use has to be avoided.”
This manuscript introduces a proof of Corollary 9 of [6]. Furthermore a third decomposition approach, namely decomposition by anti-neighborhoods of vertices, is combined together with such two decomposition approaches. Then the various papers in which Corollary 9 of [6] was used would be fixed.
Addendum to: Maximum weight independent sets in hole- and co-chair-free graphs
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In Brandstädt and Giakoumakis [2], we reduce the Maximum Weight Independent Set (MWIS) problem on hole- and co-chair-free graphs to a corresponding problem on prime atoms. Here we refine these results by investigating atoms of prime graphs (avoiding the requirement that the graphs are prime and atoms) and also observe that the MWIS problem is solvable in polynomial time on (odd-hole, co-chair)-free graphs.
The quadratic balanced optimization problem
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We introduce the quadratic balanced optimization problem (QBOP) which can be used to model equitable distribution of resources with pairwise interaction. QBOP is strongly NP-hard even if the family of feasible solutions has a very simple structure. Several general purpose exact and heuristic algorithms are presented. Results of extensive computational experiments are reported using randomly generated quadratic knapsack problems as the test bed. These results illustrate the efficacy of our exact and heuristic algorithms. We also show that when the cost matrix is specially structured, QBOP can be solved as a sequence of linear balanced optimization problems. As a consequence, we have several polynomially solvable cases of QBOP.
On atomic structure of P<inf>5</inf>-free subclasses and Maximum Weight Independent Set problem
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Graph decompositions play a crucial role in structural graph theory and in designing efficient graph algorithms. Among them, clique separator decomposition (a decomposition tree of the graph whose leaves have no clique separator (so-called atoms)) used by Tarjan for solving various optimization problems recently received much attention. In this note, we first derive the atomic structure of two subclasses of $P_{5}$ -free graphs, where $P_{5}$ is a chordless path on five vertices. These results enable us to provide efficient solutions for the Maximum Weight Independent Set problem in these classes of graphs.
Maximum weight independent sets in hole- and dart-free graphs
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Citation Excerpt :
While the complexity of the MWIS problem for hole-free graphs is unknown, it is shown to be solvable in polynomial time for some subclasses of graphs, namely, (hole, diamond)-free graphs [1], (hole, co-chair)-free graphs [2], (hole, paraglider)-free graphs [3], and (hole, apple)-free graphs [6]; see Fig. 1.
The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for hole-free graphs is unknown. In this paper, we first prove that the MWIS problem for (hole, dart, gem)-free graphs can be solved in $O (n^{3})$ -time. By using this result, we prove that the MWIS problem for (hole, dart)-free graphs can be solved in $O (n^{4})$ -time. Though the MWIS problem for (hole, dart, gem)-free graphs is used as a subroutine, we also give the best known time bound for the solvability of the MWIS problem in (hole, dart, gem)-free graphs.
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Maximum Weight Independent Sets in hole- and co-chair-free graphs

Abstract

Highlights

Information Processing Letters

J. Combin. Theory Ser. B

J. Combin. Theory Ser. B

Information Processing Letters

Discrete Math.

Information Processing Letters

Information Processing Letters

Discrete Applied Math.

Discrete Math.

Recognizing weakly triangulated graphs by edge separability

Nordic Journal of Computing

On clique separators, nearly chordal graphs and the Maximum Weight Stable Set problem

Theoretical Computer Science