We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...

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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 max-cut. Other applications include max-min eigenvalue problems and relaxations for the stable set problem.

We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.

We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shock graph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and changes in viewpoint. Keywords: shape representation; shape matching; shock graph; shock graph grammar; subgraph isomorphism. 1 I...

a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NP-hard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Be-and timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�-colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3-colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first non-trivial approximation result as a function of the maximum degree. This result can be generalized to�-colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�-function. 1

This software package is a Matlab implementation of infeasible path-following algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictor-corrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a few hundreds.

Polynomial-time approximation algorithms with non-trivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph into k blocks so as to maximise the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximise the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite relaxation. 1 Introduction Goemans and Williamson [5] have significantly advanced the theory of approximation algorithms. Previous work on approximation algorithms was largely dependent on comparing heuristic solution values to that of a Linear Program (LP) relaxation, either implicitly or explicitly. This was recognised some time ago by Wolsey [11]. (One significant exception to this general rule has been the case of Bin Packing.) The main novelty of [5] is that it uses a Semi-Definite Program (SDP) as a relaxation. To be more precise let...

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This paper deals with a class of primal-dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X 1=2 S \Gamma1=2 . They study three primaldual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno and Yoshise [10]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present: 1) a simplified polynomial convergence proof for one of their short-step path-following ...