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

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

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]

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

2011
*
Discrete Applied Mathematics
*

By Observation 3,

doi:10.1016/j.dam.2010.05.022
fatcat:a5bfxrhd25hf7evulr26swl6oa
*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)

1992
*
Journal of Soviet Mathematics
*

Hoang,

doi:10.1007/bf01097528
fatcat:7jgoc2qlmvaizpqt3kwnidykjq
*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]

2017
*
arXiv
*
pre-print

In a

arXiv:1703.07106v2
fatcat:rg4hztnm7vehfmyar3ro7yly4q
*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]

2021
*
arXiv
*
pre-print

We also demonstrate that

arXiv:2107.02379v1
fatcat:nt4tpzhkyrgpvl5rzmpsagrx3i
*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

2021
*
Annual Reviews in Control
*

We also demonstrate that

doi:10.1016/j.arcontrol.2021.09.001
fatcat:thnnpm2nmnegjp6zzibsak7shy
*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

2008
*
Discrete Mathematics
*

We prove that

doi:10.1016/j.disc.2007.03.044
fatcat:aeft3rd3gvh6rnx25mnlshndcm
*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

2017
*
IEEE/ACM Transactions on Networking
*

For general conflict

doi:10.1109/tnet.2016.2604462
fatcat:kbstz5nfwfcwbinuwbqyyt5jvy
*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

2016
*
Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science - SIGMETRICS '16
*

For general conflict

doi:10.1145/2896377.2901482
dblp:conf/sigmetrics/Houdt16
fatcat:vlmc24f2yfegfektqhkd6r5rj4
*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

2007
*
Proceedings of the 2007 ACM SIGPLAN/SIGBED conference on Languages, compilers, and tools - LCTES '07
*

Still, also because of

doi:10.1145/1254766.1254782
dblp:conf/lctrts/BouchezDR07
fatcat:g67fuwbh6bfurijpe2omijfuqq
*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]

2019
*
arXiv
*
pre-print

Both

arXiv:1801.04339v3
fatcat:jzk3ioszljdgdh7cyxwbenl2kq
*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

2011
*
The international journal of robotics research
*

localization

doi:10.1177/0278364911430419
fatcat:ycwihvu7uzbonbj2kt3r6ifcia
*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

2007
*
SIGPLAN notices
*

Still, also because of

doi:10.1145/1273444.1254782
fatcat:4vdzhuqbezgavgsxs52qcr2nfq
*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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