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On Clique Separators, Nearly Chordal Graphs, and the Maximum Weight Stable Set Problem [chapter]

Andreas Brandstädt, Chính T. Hoàng
2005 Lecture Notes in Computer Science
Clique separators in graphs are a helpful tool used by Tarjan as a divide-and-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Maximum Clique, Graph  ...  We also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion: A graph is nearly Π if for each of its vertices, the subgraph induced by the set of its  ...  The Maximum Weight Stable (or Independent) Set (MWS) Problem asks for a stable set of maximum weight in the given graph G with vertex weight function w.  ...

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

Andreas Brandstädt, Chính T. Hoàng
2007 Theoretical Computer Science
Clique separators in graphs are a helpful tool used by Tarjan as a divide-and-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Maximum Clique, Graph  ...  We also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion: A graph is nearly Π if for each of its vertices, the subgraph induced by the set of its  ...  The Maximum Weight Stable (or Independent) Set (MWS) Problem asks for a stable set of maximum weight in the given graph G with vertex weight function w.  ...

New Applications of Clique Separator Decomposition for the Maximum Weight Stable Set Problem [chapter]

Andreas Brandstädt, Van Bang Le, Suhail Mahfud
2005 Lecture Notes in Computer Science
Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) Problem, Coloring and Minimum Fill-in.  ...  The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its  ...  The Maximum Weight Stable (or Independent) Set (MWS) Problem asks for a stable set of maximum weight sum for a vertex weight function w on V .  ...

On distance-3 matchings and induced matchings

Andreas Brandstädt, Raffaele Mosca
2011 Discrete Applied Mathematics
By Observation 3, the MDkM problem on graph G corresponds to the Maximum Independent Vertex Set problem on graph L(G) k , and the first problem is efficiently solvable on graph G whenever the last problem  ...  The Maximum Distance-k Matching (MDkM) Problem asks for a distancek matching of maximum size (and of maximum weight in the weighted case).  ...  Acknowledgements We are grateful to the anonymous referees for their comments which helped to improve the presentation of our results.  ...

Representations of graphs and networks (coding, layouts and embeddings)

V. P. Kozyrev, S. V. Yushmanov
1992 Journal of Soviet Mathematics
Hoang, On Clique Separators, Nearly Chordal Graphs, and the Maximum Weight Stable Set Problem, Lecture Notes in Computer Science, 3509 (2005) 265-275. Cites [6,10,32] [30] W. Duckworth, D.F.  ...  Le, and S. Mahfud, New applications of clique separator decomposition for the maximum weight stable set problem, Lecture Notes in Computer Carlos J.  ...

Decomposition techniques applied to the Clique-Stable set Separation problem [article]

Nicolas Bousquet, Aurélie Lagoutte, Frédéric Maffray, Lucas Pastor
2017 arXiv   pre-print
In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K ∩ S = ∅, there exists a cut (  ...  Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [17] asked in 1991 the following questions: does every graph  ...  Such a simpler polytope would correspond to a linear program with extra variables to solve the Maximum Weighted Stable Set problem in G.  ...

Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization [article]

Yang Zheng, Giovanni Fantuzzi, Antonis Papachristodoulou
2021 arXiv   pre-print
We also demonstrate that chordal and factor-width decompositions allow for significant computational savings on a range of classical problems from control theory, and on more recent problems from machine  ...  Chordal and factor-width decomposition methods for semidefinite programming and polynomial optimization have recently enabled the analysis and control of large-scale linear systems and medium-scale nonlinear  ...  A.P. was supported in part by the Engineering and Physical Sciences Research Council (EP-SRC) under project EP/M002454/1.  ...

Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

Yang Zheng, Giovanni Fantuzzi, Antonis Papachristodoulou
2021 Annual Reviews in Control
We also demonstrate that chordal and factor-width decompositions allow for significant computational savings on a range of classical problems from control theory, and on more recent problems from machine  ...  Chordal and factor-width decomposition methods for semidefinite programming and polynomial optimization have recently enabled the analysis and control of large-scale linear systems and medium-scale nonlinear  ...  Max-Cut problem The maximum cut (Max-Cut) problem is a classic problem in graph theory (Goemans & Williamson, 1995) .  ...

Stable sets of maximum weight in (P7, banner)-free graphs

Raffaele Mosca
2008 Discrete Mathematics
We prove that on (P 7 , banner)-free graphs the maximum weight stable set problem is solvable in polynomial time.  ...  Acknowledgements I am indebted and grateful to the referees for pointing out different errors and for their helpful comments in general.  ...  Consider the stable sets of G 1 that contain no vertex of A, and letS 1 be one of maximum weight.  ...

Explicit Back-Off Rates for Achieving Target Throughputs in CSMA/CA Networks

Benny Van Houdt
2017 IEEE/ACM Transactions on Networking
For general conflict graphs we nevertheless show how to adapt the back-off rates when a node is added to the network when its interfering nodes form a clique in the conflict graph.  ...  Moreover, these back-off rates are such that the back-off rate of a node only depends on its own target throughput and the target throughput of its neighbors and can be determined in a distributed manner  ...  While the problem of finding the set E of minimal cardinality (known as the minimum fill) or the set E that minimizes the size of the maximum clique (known as minimum treewidth) are NP-complete [16] ,  ...

Explicit Back-off Rates for Achieving Target Throughputs in CSMA/CA Networks

Benny Van Houdt
2016 Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science - SIGMETRICS '16
For general conflict graphs we nevertheless show how to adapt the back-off rates when a node is added to the network when its interfering nodes form a clique in the conflict graph.  ...  Moreover, these back-off rates are such that the back-off rate of a node only depends on its own target throughput and the target throughput of its neighbors and can be determined in a distributed manner  ...  While the problem of finding the set E of minimal cardinality (known as the minimum fill) or the set E that minimizes the size of the maximum clique (known as minimum treewidth) are NP-complete [16] ,  ...

On the complexity of spill everywhere under SSA form

Florent Bouchez, Alain Darte, Fabrice Rastello
2007 Proceedings of the 2007 ACM SIGPLAN/SIGBED conference on Languages, compilers, and tools - LCTES '07
Still, also because of the side effect on power consumption, the minimization of loads and stores overhead (spilling problem) is an important issue.  ...  In particular, heuristics based on tree scan with two separated phases -- one for spilling, then one for coloring/coalescing -- seem good candidates for designing memory-friendly, fast, and competitive  ...  Acknowledgments We would like to thank Christophe Guillon and Sebastian Hack for fruitful discussions.  ...

Estimating the Number of Connected Components in a Graph via Subgraph Sampling [article]

Jason M. Klusowski, Yihong Wu
2019 arXiv   pre-print
Both the construction and the analysis of the proposed methodology rely on combinatorial properties of chordal graphs and identities of induced subgraph counts.  ...  This significantly expands the scope of previous results which have focused on unbiased estimators and special classes of graphs such as forests or cliques.  ...  Let G(N, d, ω) denote the collection of all chordal graphs on N vertices with maximum degree and clique number at most d and ω ≥ 2, respectively.  ...

iSAM2: Incremental smoothing and mapping using the Bayes tree

Michael Kaess, Hordur Johannsson, Richard Roberts, Viorela Ila, John J Leonard, Frank Dellaert
2011 The international journal of robotics research
localization and mapping (SLAM) problem.  ...  Similar to a clique tree, a Bayes tree encodes a factored probability density, but unlike the clique tree it is directed and maps more naturally to the square root information matrix of the simultaneous  ...  Durrant-Whyte for the Victoria Park dataset, D. Haehnel for the Intel dataset and G. Grisetti for the W10000 dataset.  ...

On the complexity of spill everywhere under SSA form

Florent Bouchez, Alain Darte, Fabrice Rastello
2007 SIGPLAN notices
Still, also because of the side effect on power consumption, the minimization of loads and stores overhead (spilling problem) is an important issue.  ...  In particular, heuristics based on tree scan with two separated phases -one for spilling, then one for coloring/coalescing -seem good candidates for designing memory-friendly, fast, and competitive register  ...  Acknowledgments We would like to thank Christophe Guillon and Sebastian Hack for fruitful discussions.  ...
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