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18
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 207 (17 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
SEPARATING A SUPERCLASS OF COMB INEQUALITIES IN PLANAR GRAPHS
, 2000
"... Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedu ..."
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Cited by 23 (6 self)
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Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗ , nds one or more inequalities in the class which are violated by x , or proves that none exist. Such algorithms are at the core of the highly successful branchandcut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We de ne a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x ∗ e ¿0 is planar. The separation algorithm runs in O(n³) time, where n is the number of vertices in the graph.
Theta Bodies for Polynomial Ideals
, 2008
"... Abstract. Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lo ..."
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Cited by 17 (3 self)
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Abstract. Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals. 1.
Tighter linear and semidefinite relaxations for maxcut based on the LovászSchrijver liftandproject procedure
 SIAM Journal on Optimization
"... Abstract. We study how the liftandproject method introduced by Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] applies to the cut polytope. We show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K5minor. ..."
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Cited by 7 (4 self)
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Abstract. We study how the liftandproject method introduced by Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] applies to the cut polytope. We show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K5minor. Therefore, for a graph G with n ≥ 4 nodes with stability number α(G), n − 4 iterations suffice instead of the m (number of edges) iterations required in general and, under some assumption, n − α(G) − 3 iterations suffice. The exact number of needed iterations is determined for small n ≤ 7 by a detailed analysis of the new relaxations. If positive semidefiniteness is added to the construction, then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by Anjos and Wolkowicz [Discrete Appl. Math., to appear]. We also show how the Lovász–Schrijver relaxations for the stable set polytope of G can be strengthened using the corresponding relaxations for the cut polytope of the graph G ∇ obtained from G by adding a node adjacent to all nodes of G.
Interior point and semidefinite approaches in combinatorial optimization
, 2005
"... Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient ..."
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Cited by 6 (3 self)
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Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primaldual interiorpoint methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NPhard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to nonconvex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include nonconvex potential reduction methods, interior point cutting plane methods, primaldual IPMs and firstorder algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.
On Greedy Construction Heuristics for the MaxCut Problem
 INTERNATIONAL JOURNAL ON COMPUTATONAL SCIENCE AND ENGINEERING
, 2007
"... Given a graph with nonnegative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in di#erent subsets is maximized. This classical NPhard problem finds applications in VLSI design, statistical physics, an ..."
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Cited by 5 (0 self)
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Given a graph with nonnegative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in di#erent subsets is maximized. This classical NPhard problem finds applications in VLSI design, statistical physics, and classification among other fields. This paper compares the performance of several greedy construction heuristics for MAXCUT problem. In particular, a new "worstout" approach is studied and the proposed edge contraction heuristic is shown to have an approximation ratio of at least 1/3. The results of experimental comparison of the worstout approach, the wellknown bestin algorithm, and modifications for both are also included.
Exact Max 2SAT: Easier and faster
 In Current Trends in Theory and Practice of Computer Science (SOFSEM) (2007
"... Prior algorithms known for exactly solving Max 2Sat improve upon the trivial upper bound only for very sparse instances. We present new algorithms for exactly solving (in fact, counting) weighted Max 2Sat instances. One of them has a good performance if the underlying constraint graph has a small ..."
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Cited by 4 (1 self)
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Prior algorithms known for exactly solving Max 2Sat improve upon the trivial upper bound only for very sparse instances. We present new algorithms for exactly solving (in fact, counting) weighted Max 2Sat instances. One of them has a good performance if the underlying constraint graph has a small separator decomposition, another has a slightly improved worst case performance. For a 2Sat instance F with n variables, the worst case running time is Õ(2 (1−1/( ˜ d(F)−1))n), where ˜ d(F) is the average degree in the constraint graph defined by F. The algorithms and bounds actually are valid for any Max 2Csp, whose clauses are over pairs of binary variables. We use strict αgadgets introduced by Trevisan, Sorkin, Sudan, and Williamson to get the same upper bounds for problems like Max 3Sat and Max Cut. We also introduce a notion of strict (α, β)gadget to provide a framework that allows composition of gadgets. This framework allows us to obtain the same upper bounds for Max kSat and Max kLin2.
Fast Planar Correlation Clustering for Image Segmentation
"... Abstract. We describe a new optimization scheme for finding highquality clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lowerbounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We ..."
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Cited by 2 (1 self)
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Abstract. We describe a new optimization scheme for finding highquality clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lowerbounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We demonstrate our algorithm on the problem of image segmentation where this approach outperforms existing global optimization techniques in minimizing the objective and is competitive with the state of the art in producing highquality segmentations. 1 1
A Note on Polyhedral Relaxations for the Maximum Cut Problem
"... We consider three wellstudied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial nu ..."
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Cited by 1 (1 self)
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We consider three wellstudied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial number of inequalities, while the latter two may require exponentially many constraints. We give an alternate proof of a theorem of Barahona that states that the metric polytope of a general graph is a projection of the metric polytope of the complete graph. We then give an alternate proof of a theorem of Poljak that states that for any nonnegative cost function, the optimal objective value over the relaxation of the bipartite subgraph polytope equals the optimal objective value over the metric polytope. Both proofs are based on the same technique: the separation oracle for the metric polytope of a general graph due to Barahona and Mahjoub. These proofs yield a simple, combinatorial method for proving that three wellstudied polyhedral upper bounds on the value of the maximum cut are the same for graphs with nonnegative edge weights. 1