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Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (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 ..."
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Cited by 1231 (13 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (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 ...
Large margin methods for structured and interdependent output variables
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2005
"... Learning general functional dependencies between arbitrary input and output spaces is one of the key challenges in computational intelligence. While recent progress in machine learning has mainly focused on designing flexible and powerful input representations, this paper addresses the complementary ..."
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Cited by 612 (12 self)
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Learning general functional dependencies between arbitrary input and output spaces is one of the key challenges in computational intelligence. While recent progress in machine learning has mainly focused on designing flexible and powerful input representations, this paper addresses the complementary issue of designing classification algorithms that can deal with more complex outputs, such as trees, sequences, or sets. More generally, we consider problems involving multiple dependent output variables, structured output spaces, and classification problems with class attributes. In order to accomplish this, we propose to appropriately generalize the wellknown notion of a separation margin and derive a corresponding maximummargin formulation. While this leads to a quadratic program with a potentially prohibitive, i.e. exponential, number of constraints, we present a cutting plane algorithm that solves the optimization problem in polynomial time for a large class of problems. The proposed method has important applications in areas such as computational biology, natural language processing, information retrieval/extraction, and optical character recognition. Experiments from various domains involving different types of output spaces emphasize the breadth and generality of our approach.
Cones of matrices and setfunctions and 01 optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1991
"... It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection a ..."
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Cited by 353 (7 self)
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It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 01 valued solutions of a system of linear inequalities. An important feature of these approximations is that one can optimize any linear objective function over them in polynomial time. In the special case of the vertex packing polytope, we obtain a sequence of systems of inequalities, such that already the first system includes clique, odd hole, odd antihole, wheel, and orthogonality constraints. In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope. For various classes of graphs, including tperfect graphs, it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets. We also discuss an extension of the method, which establishes a connection with certain submodular functions and the Möbius function of a lattice.
Approximate graph coloring by semidefinite programming
 Proc. 35 th IEEE FOCS, IEEE
, 1994
"... a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NPhard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register ..."
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Cited by 212 (6 self)
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a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NPhard (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. Beand 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 3colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first nontrivial 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 2SAT 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
The Dense kSubgraph Problem
 Algorithmica
, 1999
"... This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (D ..."
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Cited by 205 (12 self)
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This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (DkS) maximization problem, of computing the dense k vertex subgraph of a given graph. That is, on input a graph G and a parameter k, we are interested in finding a set of k vertices with maximum average degree in the subgraph induced by this set. As this problem is NPhard (say, by reduction from Clique), we consider approximation algorithms for this problem. We obtain a polynomial time algorithm that on any input (G; k) returns a subgraph of size k whose average degree is within a factor of at most n ffi from the optimum solution, where n is the number of vertices in the input graph G, and ffi ! 1=3 is some universal constant. Unfortunately, we are unable to present a complementary negati...
A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time
, 1999
"... We give a strongly polynomialtime algorithm minimizing a submodular function f given by a valuegiving oracle. The algorithm does not use the ellipsoid method or any other linear programming method. No bound on the complexity of the values of f is needed to be known a priori. The number of oracle ..."
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Cited by 203 (0 self)
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We give a strongly polynomialtime algorithm minimizing a submodular function f given by a valuegiving oracle. The algorithm does not use the ellipsoid method or any other linear programming method. No bound on the complexity of the values of f is needed to be known a priori. The number of oracle calls is bounded by a polynomial in the size of the underlying set.
Using linear programming to decode binary linear codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor grap ..."
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Cited by 184 (10 self)
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A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor graph or paritycheck representation of the code. The resulting “LP decoder” generalizes our previous work on turbolike codes. A precise combinatorial characterization of when the LP decoder succeeds is provided, based on pseudocodewords associated with the factor graph. Our definition of a pseudocodeword unifies other such notions known for iterative algorithms, including “stopping sets, ” “irreducible closed walks, ” “trellis cycles, ” “deviation sets, ” and “graph covers.” The fractional distance ��— ™ of a code is introduced, which is a lower bound on the classical distance. It is shown that the efficient LP decoder will correct up to ��— ™ P I errors and that there are codes with ��— ™ a @ I A. An efficient algorithm to compute the fractional distance is presented. Experimental evidence shows a similar performance on lowdensity paritycheck (LDPC) codes between LP decoding and the minsum and sumproduct algorithms. Methods for tightening the LP relaxation to improve performance are also provided.