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11
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 1244 (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 ...
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 110 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
New Results on Monotone Dualization and Generating Hypergraph Transversals
 SIAM JOURNAL ON COMPUTING
, 2002
"... We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NPcompleteness. We present a number of new polynomial time resp. outputpolynomial time results for significant ..."
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Cited by 50 (12 self)
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We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NPcompleteness. We present a number of new polynomial time resp. outputpolynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log² n/log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.
Generating All Vertices of a Polyhedron Is Hard
 DISCRETE COMPUT GEOM (2008 ) 39 : 174–190 175
, 2008
"... We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other ne ..."
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Cited by 38 (6 self)
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We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P = NP. As a corollary, we solve in the negative two wellknown generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains
Abduction and the Dualization Problem
, 2003
"... Computing abductive explanations is an important problem, which has been studied extensively in Artificial Intelligence (AI) and related disciplines. ..."
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Cited by 8 (0 self)
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Computing abductive explanations is an important problem, which has been studied extensively in Artificial Intelligence (AI) and related disciplines.
The Stable Set Problem and the LiftandProject Ranks of Graphs
, 2002
"... We study the liftandproject procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the o ..."
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Cited by 7 (3 self)
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We study the liftandproject procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N 0 , N , or N+ rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N 0  or the N  rank. Hence we obtain the ranks for these graphs, and we also present some graphminor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N+ ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N+ rank of a graph.
On berge multiplication for monotone boolean dualization
 In Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, 2008, LNCS 5125
"... Given the prime CNF representation φ of a monotone Boolean function f: {0, 1} n ↦ → {0, 1}, the dualization problem calls for finding the corresponding prime DNF representation ψ of f. A very simple method (called Berge multiplication [3, Page 52–53]) works by multiplying out the clauses of φ from l ..."
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Cited by 3 (0 self)
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Given the prime CNF representation φ of a monotone Boolean function f: {0, 1} n ↦ → {0, 1}, the dualization problem calls for finding the corresponding prime DNF representation ψ of f. A very simple method (called Berge multiplication [3, Page 52–53]) works by multiplying out the clauses of φ from left to right in some order, simplifying whenever possible using the absorption law. We show that for any monotone CNF φ, Berge multiplication can be done in subexponential time, and for many interesting subclasses of monotone CNF’s such as CNF’s with bounded size, bounded degree, bounded intersection, bounded conformality, Let f: {0, 1} n → {0, 1} be a Boolean function. A function is called monotone (also called positive) if for every pair of vectors x, y ∈ {0, 1} n, x ≤ y (i.e., xi ≤ yi for all i) always implies f(x) ≤ f(y). Any monotone function f has a unique prime conjunctive normal form (CNF) expression
OutputSensitive Algorithms for Enumerating Minimal Transversals for Some Geometric Hypergraphs
"... We give a general framework for the problem of finding all minimal hitting sets of a family of objects in R d by another. We apply this framework to the following problems: (i) hitting hyperrectangles by points in R d; (ii) stabbing connected objects by axisparallel hyperplanes in R d; and (iii) ..."
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Cited by 2 (1 self)
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We give a general framework for the problem of finding all minimal hitting sets of a family of objects in R d by another. We apply this framework to the following problems: (i) hitting hyperrectangles by points in R d; (ii) stabbing connected objects by axisparallel hyperplanes in R d; and (iii) hitting halfplanes by points. For both the covering and hitting set versions, we obtain incremental polynomialtime algorithms, provided that the dimension d is fixed.