Michel X. Goemans

Bio: Michel X. Goemans is an academic researcher from Massachusetts Institute of Technology. The author has contributed to research in topics: Approximation algorithm & Matroid. The author has an hindex of 52, co-authored 144 publications receiving 14131 citations. Previous affiliations of Michel X. Goemans include Congress of Racial Equality & Université catholique de Louvain.

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

Abstract: 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 1/2 for MAX CUT and 3/4 or MAX 2SAT. Slight extensions of our analysis lead to a.79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a.758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1/4 and 3/4, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

A General Approximation Technique for Constrained Forest Problems

Abstract: We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems. Our technique produces approximation algorithms that run in $O(n^2\log n)$ time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of $O(n^2\log n)$ time compares favorably with the best strongly polynomial exact algorithms running in $O(n^3)$ time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413--420].

A general approximation technique for constrained forest problems

Abstract: We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems.Our technique produces approximation algorithms that run in O(n2 log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of O(n2 log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n3) time for dense graphs. A similar result is obtained for the 2-matching problem and its variants.We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [4].

Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT

Abstract: It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the Feige-Lovasz (STOC92) semidefinite programming relaxation of one-round two-prover proof systems, together with rounding techniques for the solutions of semidefinite programs, as introduced by Goemans and Williamson (STOC94). As a consequence of our approach, we present improved approximation algorithms for MAX 2SAT and MAX DICUT. The algorithms are guaranteed to deliver solutions within a factor of 0.931 of the optimum for MAX 2SAT and within a factor of 0.859 for MAX DICUT, improving upon the guarantees of 0.878 and 0.796 of Goemans and Williamson (1994). >

Randomized Algorithms

Abstract: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.

Graphical Models, Exponential Families, and Variational Inference

Abstract: The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

Semidefinite programming

Abstract: In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

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

Abstract: 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 1/2 for MAX CUT and 3/4 or MAX 2SAT. Slight extensions of our analysis lead to a.79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a.758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1/4 and 3/4, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.