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Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

1995
*
Journal of the ACM
*

Homework The homework deals with the paper "

doi:10.1145/227683.227684
fatcat:xmlvxe4worblnpjcpsoq6anseu
*Improved**approximation**algorithms**for**maximum**cut**and**satisfiability**problems**using**semidefinite**programming*", by Goemans*and*Williamson, JACM 1995. ... Verify to yourself that you understand the following: • Definition of the MAX-*CUT**problem*• The*program*(P), why it is a relaxation of MAX-*CUT**and*how to solve it in polynomial time*using**semidefinite*...##
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Randomized Rounding for Semidefinite Programs – Variations on the MAX CUT Example
[chapter]

1999
*
Lecture Notes in Computer Science
*

Their

doi:10.1007/978-3-540-48413-4_20
fatcat:lrsup62rrbghtlhlydfowkx7be
*algorithm*first*uses*a*semidefinite**programming*relaxation of MAX*CUT*that embeds the vertices of the graph on the surface of an n dimensional sphere,*and*then*cuts*the sphere in two at random. ... In this survey we shall review several variations of this*algorithm*which offer*improved**approximation*ratios*for*some special families of instances of MAX*CUT*, as well as*for**problems*related to MAX*CUT*... This indicates that there is still much room*for*research on the*use*of*semidefinite**programs*in*approximation**algorithms*. ...##
###
Efficient approximation algorithms for semidefinite programs arising from MAX CUT and COLORING

1996
*
Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
*

The best known

doi:10.1145/237814.237980
dblp:conf/stoc/KleinL96
fatcat:m5dvzi22ergdzlxw5nf6yuajoe
*approximation**algorithm**for*graph MAX*CUT*, due to Goemans*and*Williamson, first finds the optimal solution a*semidefinite**program**and*then derives a graph*cut*from that solution. ... Building on this result, Karger, Motwani,*and*Sudan gave an*approximation**algorithm**for*graph coloring that also involves solving a*semidefinite**program*. ... Let u be the characteristic vector*for*the greedy solution to the 34AX*CUT**problem*, where ui = 1*for*every node i on one side of the*cut*,*and*uj = -1*for*every node j on the other side of the*cut*. ...##
###
Approximating the Cut-Norm via Grothendieck's Inequality

2006
*
SIAM journal on computing (Print)
*

This concept plays a major role in the design of efficient

doi:10.1137/s0097539704441629
fatcat:bwrr62wswzbjfnuf4ggolfilue
*approximation**algorithms**for*dense graph*and*matrix*problems*. ... Here we show that the*problem*of*approximating*the*cut*-norm of a given real matrix is MAX SNP hard,*and*provide an efficient*approximation**algorithm*. ... Acknowledgment: Part of this work was carried out during a visit of the first author at Microsoft Research, Redmond, WA,*and*he thanks his hosts at Microsoft*for*their hospitality. ...##
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Approximating the cut-norm via Grothendieck's inequality

2004
*
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04
*

This concept plays a major role in the design of efficient

doi:10.1145/1007352.1007371
dblp:conf/stoc/AlonN04
fatcat:ndhbtsbzangl3dri366xthopti
*approximation**algorithms**for*dense graph*and*matrix*problems*. ... Here we show that the*problem*of*approximating*the*cut*-norm of a given real matrix is MAX SNP hard,*and*provide an efficient*approximation**algorithm*. ... Acknowledgment: Part of this work was carried out during a visit of the first author at Microsoft Research, Redmond, WA,*and*he thanks his hosts at Microsoft*for*their hospitality. ...##
###
Semidefinite Programming and Integer Programming
[chapter]

2005
*
Handbooks in Operations Research and Management Science
*

We thank a referee

doi:10.1016/s0927-0507(05)12008-8
fatcat:ez23hvr5znfolnppcugzpevgpu
*for*his careful reading*and*his suggestions that helped*improve*the presentation of this chapter. ... Feige*and*Goemans [77] propose an*improved**approximation**algorithm**for*the*maximum*dicut*problem*analogue to their*improved**approximation**algorithm**for*MAX 2SAT. ... Goemans*and*Wiliamson [94] give an*improved*3 4 -*approximation**algorithm**using*linear*programming*. ...##
###
A multidimensional maximum bisection problem
[article]

2015
*
arXiv
*
pre-print

This work introduces a multidimensional generalization of the

arXiv:1506.07731v1
fatcat:k5yvnke7znc5fnxljjdytvnx7e
*maximum*bisection*problem*. A mixed integer linear*programming*formulation is proposed with the proof of its correctness. ... The numerical tests, made on the randomly generated graphs, indicates that the multidimensional generalization is more difficult to solve than the original*problem*. ...*Improved**approximation**algorithms**for**maximum**cut**and**satisfiability**problems**using**semidefinite**programming*, Journal of the Association*for*Computing Machinery, 42(6):1115-1145. [8] Hager W. ...##
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Simple Approximation Algorithms for Balanced MAX 2SAT
[chapter]

2016
*
Lecture Notes in Computer Science
*

colored

doi:10.1007/978-3-662-49529-2_49
fatcat:czkctpf75bd2dg36pmqyohxmwi
*cut**problem*. ... We provide experimental results showing that this spectral*algorithm*performs well*and*is slightly better than Johnson's*algorithm**and*the Goemans-Williamson*semidefinite**programming**algorithm*on balanced ... Trevisan [20] gave a spectral 0.531-*approximation**algorithm**for*the*maximum**cut**problem*; Soto [19] generalized*and**improved*Trevisan's analysis to a 0.614-*approximation**algorithm**for*the*maximum*colored ...##
###
Semidefinite programming based approaches to the break minimization problem

2006
*
Computers & Operations Research
*

We formulate the break minimization

doi:10.1016/j.cor.2004.09.030
fatcat:pk466ys27bc5fmj4nsbbokj5t4
*problem*as MAX RES*CUT**and*MAX 2SAT,*and*apply Goemans*and*Williamson's*approximation**algorithm**using**semidefinite**programming*. ... both at away or both at home*for*a team. ...*For*MAX RES*CUT*, Goemans*and*Williamson [4] proposed a 0.878-randomized*approximation**algorithm**using**semidefinite**programming*. ...##
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MAX CUT in cubic graphs

2004
*
Journal of Algorithms
*

We present an

doi:10.1016/j.jalgor.2004.06.001
fatcat:cskgkasiwjb4letrepcmtwjbhu
*improved**semidefinite**programming*based*approximation**algorithm**for*the MAX*CUT**problem*in graphs of*maximum*degree at most 3. ... We also observe that results of Hopkins*and*Staton*and*of Bondy*and*Locke yield a simple combinatorial 4 5approximation*algorithm**for*the*problem*. ... Concluding remarks We obtained an*improved**semidefinite**programming*based*approximation**algorithm**for*the MAX*CUT**problem*in graphs of*maximum*degree at most 3. ...##
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On Khot's unique games conjecture

2012
*
Bulletin of the American Mathematical Society
*

The conjecture has inspired a remarkable body of work, which has clarified the computational complexity of several optimization

doi:10.1090/s0273-0979-2011-01361-1
fatcat:olwc5dausved3fhbgb5fn6rx5y
*problems**and*the effectiveness of "*semidefinite**programming*" convex relaxations ... In this paper, which assumes no prior knowledge of computational complexity, we describe the context*and*statement of the conjecture,*and*we discuss in some detail one specific line of work motivated by ...*Semidefinite**Programming**and**approximation**algorithms*. ...##
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Constructing Worst Case Instances for Semidefinite Programming Based Approximation Algorithms

2001
*
SIAM Journal on Discrete Mathematics
*

*Semidefinite*

*programming*based

*approximation*

*algorithms*, such as the Goemans

*and*Williamson

*approximation*

*algorithm*

*for*the MAX

*CUT*

*problem*, are usually shown to have certain performance guarantees

*using*... We also obtain similar results

*for*a related

*problem*. Our approach is quite general

*and*could possibly be applied to some additional

*problems*

*and*

*algorithms*. ... We would like to thank two anonymous referees

*for*many helpful comments. ...

##
###
A geometric approach to betweenness
[chapter]

1995
*
Lecture Notes in Computer Science
*

The

doi:10.1007/3-540-60313-1_146
fatcat:ttajzxsvrbbtlfuno3c3el5t5m
*algorithm*translates the*problem*into a set of quadratic inequalities*and*solves a*semidefinite*relaxation of them in R n . ... The claimed performance guarantee is shown*using*simple geometric properties of the*semidefinite**programming*(SDP) solution. ... We are grateful to Michel Goemans*for*providing*us*with the MAX*CUT*example*and**for*helpful discussions on*semidefinite**programming*. ...##
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Improved Approximation Algorithms for MAX NAE-SAT and MAX SAT
[chapter]

2006
*
Lecture Notes in Computer Science
*

MAX SAT

doi:10.1007/11671411_3
fatcat:6lts5txgsff4pahfcf7yac2s6m
*and*MAX NAE-SAT are central*problems*in theoretical computer science. We present an*approximation**algorithm**for*MAX NAE-SAT with a conjectured performance guarantee of 0.8279. ...*Using*a variant of our MAX NAE-SAT*approximation**algorithm*, combined with other techniques*used*in [Asa03], we obtain an*approximation**algorithm**for*MAX SAT with a conjectured performance guarantee of ... We also*used*the MAX 2-SAT*algorithm*of Lewin Livnat*and*Zwick to obtain an 0.7968-*approximation**algorithm**for*the MAX SAT*problem*. ...##
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A Geometric Approach to Betweenness

1998
*
SIAM Journal on Discrete Mathematics
*

The

doi:10.1137/s0895480195296221
fatcat:pz35rtc7pzbzzp6d45or7xehwm
*algorithm*translates the*problem*into a set of quadratic inequalities*and*solves a*semidefinite*relaxation of them in R n . ... The claimed performance guarantee is shown*using*simple geometric properties of the*semidefinite**programming*(SDP) solution. ... We are grateful to Michel Goemans*for*providing*us*with the MAX*CUT*example*and**for*helpful discussions on*semidefinite**programming*. ...
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