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Constructing Worst Case Instances for Semidefinite Programming Based Approximation Algorithms

Noga Alon, Benny Sudakov, Uri Zwick
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  ...  Here we further extend their results and show, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible  ...  The MAX CUT approximation algorithm of Goemans and Williamson [GW95] uses a semidefinite programming relaxation of the problem.  ... 
doi:10.1137/s0895480100379713 fatcat:z2i2bgtrejeohllkhj7dz6emma

Tight lower bounds by semidefinite relaxations for the discrete lot-sizing and scheduling problem with sequence-dependent changeover costs [chapter]

Celine Gicquel, Abdel Lisser
2012 Operations Research Proceedings  
In the present paper, we propose to compute a tight lower bound of the optimal solution value by using a semidefinite relaxation of the problem rather than a standard linear relaxation.  ...  Tight lower bounds by semidefinite relaxation for the discrete lot-sizing and scheduling problem with sequence-dependent changeover costs. 9th International ABSTRACT: We study a production planning problem  ...  In particular, semidefinite relaxations were proved to provide tight bounds for some well-known quadratic combinatorial optimization problems such as the max-cut problem, the quadratic assignment problem  ... 
doi:10.1007/978-3-642-29210-1_67 dblp:conf/or/GicquelL11 fatcat:l4u3qofn3jea5h6xduuebswlvq

BiqCrunch

Nathan Krislock, Jérôme Malick, Frédéric Roupin
2017 ACM Transactions on Mathematical Software  
It has been successfully tested on a variety of well-known combinatorial optimization problems, such as Max-Cut, Max-k-Cluster, and Max-Independent-Set.  ...  BiqCrunch: a semidefinite branch-and-bound method for solving binary quadratic problems.  ...  ACKNOWLEDGMENTS The authors are deeply indebted to three anonymous referees whose knowledgable comments and feedback led to major improvements to both this paper and the BiqCrunch code.  ... 
doi:10.1145/3005345 fatcat:pbgqg24elnculiecrr2zfxzl4i

A hybrid constraint programming and semidefinite programming approach for the stable set problem [article]

W.J. van Hoeve
2003 arXiv   pre-print
The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable domain values, based on the solution of a semidefinite relaxation.  ...  This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming.  ...  Several papers on approximation theory following [9] have shown the tightness of semidefinite relaxations.  ... 
arXiv:math/0302307v1 fatcat:qsceyqqeeve2lpyzqkfaxdzbsq

MADAM: A parallel exact solver for Max-Cut based on semidefinite programming and ADMM [article]

Timotej Hrga, Janez Povh
2021 arXiv   pre-print
We present MADAM, a parallel semidefinite based exact solver for Max-Cut, a problem of finding the cut with maximum weight in a given graph.  ...  The algorithm uses branch and bound paradigm that applies alternating direction method of multipliers as the bounding routine to solve the basic semidefinite relaxation strengthened by a subset of hypermetric  ...  Acknowledgements The authors would like to thank the anonymous referees for their careful reading of the paper and for their constructive comments, which are greatly appreciated.  ... 
arXiv:2010.07839v2 fatcat:tu3jus5atfeedaodojgcoujwxi

A Hybrid Constraint Programming and Semidefinite Programming Approach for the Stable Set Problem [chapter]

Willem Jan van Hoeve
2003 Lecture Notes in Computer Science  
The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable domain values, based on the solution of a semidefinite relaxation.  ...  This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming.  ...  Several papers on approximation theory following [11] have shown the tightness of semidefinite relaxations.  ... 
doi:10.1007/978-3-540-45193-8_28 fatcat:yhpdj4gc4nflxbxqd5sltf2isi

New bounds for the max- k -cut and chromatic number of a graph

E.R. van Dam, R. Sotirov
2016 Linear Algebra and its Applications  
We consider several semidefinite programming relaxations for the max-k-cut problem, with increasing complexity.  ...  We prove that the eigenvalue bound for the max-k-cut is tight for several classes of graphs.  ...  A well-known semidefinite programming (SDP) relaxation for the max-cut problem, in dual form, was introduced by Delorme and Poljak [14] .  ... 
doi:10.1016/j.laa.2015.09.043 fatcat:u3krjhylc5fdlpi7kxybabqgyi

Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition

Ilse Fischer, Gerald Gruber, Franz Rendl, Renata Sotirov
2005 Mathematical programming  
Semidefinite Relaxations The formulation (1) of Max-Cut gives rise to the following relaxation, which has attracted a lot of attention due to the famous approximation analysis of Goemans and Williamson  ...  Our approach provides accurate solutions to semidefinite relaxations of the Max-Cut and the Equipartition problem, which are not achievable by direct approaches based only on interior-point methods.  ...  Acknowledgement: We thank two anonymous referees for their constructive comments and suggestions to improve an earlier version of the paper.  ... 
doi:10.1007/s10107-005-0661-9 fatcat:2yxzp7updfgurccimeyganpmva

A Hierarchy of Subgraph Projection-Based Semidefinite Relaxations for Some NP-Hard Graph Optimization Problems

Elspeth Adams, Miguel F. Anjos, Franz Rendl, Angelika Wiegele
2015 INFOR. Information systems and operational research  
We propose a new hierarchy of semidefinite relaxations for classes of such problems that are based on graphs and for which the projection of the problem onto a subgraph shares the same structure as the  ...  This includes the well-studied max-cut and stable-set problems.  ...  This includes the well-studied max-cut and stable-set problems. For max-cut for example, the projection of any cut of the original graph onto a subgraph induces a cut in every subgraph.  ... 
doi:10.3138/infor.53.1.40 fatcat:pespvnzmnnecvha2v3eeducw6a

Page 7483 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
tight semidefinite relaxation of the MAX CUT problem.  ...  Summary: “We obtain a tight semidefinite relaxation of the MAX CUT problem which improves several previous SDP relaxations in the literature.  ... 

Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality [article]

Dimitris Bertsimas, Ryan Cory-Wright, Jean Pauphilet
2021 arXiv   pre-print
By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a cutting-plane method which solves the problem to certifiable optimality at the scale of selecting k=  ...  We also propose a convex relaxation and greedy rounding scheme that provides bound gaps of 1-2% in practice within minutes for p=100s or hours for p=1,000s and is therefore a viable alternative to the  ...  Acknowledgments We are grateful to the three anonymous referees and the associate editor for many valuable comments which improved the paper.  ... 
arXiv:2005.05195v4 fatcat:ma6bsgai3rhw3pyuepc6tpxevm

The Power of Semidefinite Programming Relaxations for MAX-SAT [chapter]

Carla P. Gomes, Willem-Jan van Hoeve, Lucian Leahu
2006 Lecture Notes in Computer Science  
We compare Semidefinite Programming (SDP) based relaxations with LP relaxations for MAX-2-SAT.  ...  SDP allows us to set up to around 80% of the variables without degrading the optimal solution, while setting a single variable based on the LP relaxation generally degrades the global optimal solution  ...  In a seminal paper, Goemans and Williamson [8] used SDP to obtain improved approximations for the Max-Cut and the MAX-2-SAT problem.  ... 
doi:10.1007/11757375_10 fatcat:zf7a7cushnfajpqr5kfawpvm3u

On Khot's unique games conjecture

Luca Trevisan
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 problems and the effectiveness of "semidefinite programming" convex relaxations  ...  In 2002, Subhash Khot formulated the Unique Games Conjecture, a conjecture about the computational complexity of certain optimization problems.  ...  Applying this generic transformation to the quadratic programming formulation of Max Cut, we obtain the following Semidefinite Programming formulation of Max Cut: (12) max ij∈E 1 2 − 1 2 v i , v j subject  ... 
doi:10.1090/s0273-0979-2011-01361-1 fatcat:olwc5dausved3fhbgb5fn6rx5y

Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints

Lijie Bai, John E. Mitchell, Jong-Shi Pang
2013 Optimization Letters  
By solving a semidefinite program, an equivalent QPCC can be obtained whose QP relaxation is as tight as possible.  ...  Following the QCR method, the products of linear equality constraints can also be used to perturb the QPCC objective function, with the goal that the new QP relaxation provides a tighter lower bound.  ...  In order to have the QP relaxation bound as tight as possible, the following max-min problem needs to be solved: max Qµ 0 min x∈Φ q µ (x) ≡ max Qµ 0 min x {q µ (x) | Ax = b, x J ≥ 0} ≡ max " Qµ 0 (β, λ  ... 
doi:10.1007/s11590-013-0647-0 fatcat:5vudpjjz7rbl3kmdbqqaj473aa

Exact Solution Methods for the k-Item Quadratic Knapsack Problem [chapter]

Lucas Létocart, Angelika Wiegele
2016 Lecture Notes in Computer Science  
The purpose of this paper is to solve the 0-1 k-item quadratic knapsack problem (kQKP ), a problem of maximizing a quadratic function subject to two linear constraints.  ...  Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method.  ...  Our new algorithm consists of a branch-and-bound framework using a combination of a semidefinite relaxation and polyhedral cutting planes to obtain tight upper bounds and fast hybrid heuristics [18]  ... 
doi:10.1007/978-3-319-45587-7_15 fatcat:oly6a7dr3ff23cqevlzmf54hr4
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