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

Michel X. Goemans, David P. Williamson
1995 Journal of the ACM
Homework The homework deals with the paper "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  ...

### Randomized Rounding for Semidefinite Programs – Variations on the MAX CUT Example [chapter]

Uriel Feige
1999 Lecture Notes in Computer Science
Their 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

Philip Klein, Hsueh-I Lu
1996 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
The best known 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

Noga Alon, Assaf Naor
2006 SIAM journal on computing (Print)
This concept plays a major role in the design of efficient 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.  ...

### Approximating the cut-norm via Grothendieck's inequality

Noga Alon, Assaf Naor
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 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]

Monique Laurent, Franz Rendl
2005 Handbooks in Operations Research and Management Science
We thank a referee for his careful reading and his suggestions that helped improve the presentation of this chapter.  ...  Feige and Goemans  propose an improved approximation algorithm for the maximum dicut problem analogue to their improved approximation algorithm for MAX 2SAT.  ...  Goemans and Wiliamson  give an improved 3 4 -approximation algorithm using linear programming.  ...

### A multidimensional maximum bisection problem [article]

Zoran Maksimovic
2015 arXiv   pre-print
This work introduces a multidimensional generalization of the 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.  Hager W.  ...

### Simple Approximation Algorithms for Balanced MAX 2SAT [chapter]

Alice Paul, Matthias Poloczek, David P. Williamson
2016 Lecture Notes in Computer Science
colored 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  gave a spectral 0.531-approximation algorithm for the maximum cut problem; Soto  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

Ryuhei Miyashiro, Tomomi Matsui
2006 Computers & Operations Research
We formulate the break minimization 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  proposed a 0.878-randomized approximation algorithm using semidefinite programming.  ...

### MAX CUT in cubic graphs

Eran Halperin, Dror Livnat, Uri Zwick
2004 Journal of Algorithms
We present an 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.  ...

### 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 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.  ...

### 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  ...  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 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.  ...

### Improved Approximation Algorithms for MAX NAE-SAT and MAX SAT [chapter]

Adi Avidor, Ido Berkovitch, Uri Zwick
2006 Lecture Notes in Computer Science
MAX SAT 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.  ...