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20
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 173 (24 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
The Approximability of Constraint Satisfaction Problems
 SIAM J. Comput
, 2001
"... We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Di#erent computational problems arise from constraint satisfaction problems depending ..."
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Cited by 68 (2 self)
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We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Di#erent computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used.
Optimal myopic algorithms for random 3SAT
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Let F 3 (n; m) be a random 3SAT formula formed by selecting uniformly, independently, and with replacement, m clauses among all 8 \Gamma n 3 \Delta possible 3clauses over n variables. It has been conjectured that there exists a constant r 3 such that for any ffl ? 0, F 3 (n; (r 3 \Gamma ffl)n ..."
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Cited by 67 (8 self)
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Let F 3 (n; m) be a random 3SAT formula formed by selecting uniformly, independently, and with replacement, m clauses among all 8 \Gamma n 3 \Delta possible 3clauses over n variables. It has been conjectured that there exists a constant r 3 such that for any ffl ? 0, F 3 (n; (r 3 \Gamma ffl)n) is almost surely satisfiable, but F 3 (n; (r 3 + ffl)n) is almost surely unsatisfiable. The best lower bounds for the potential value of r 3 have come from analyzing rather simple extensions of unitclause propagation. Recently, it was shown [2] that all these extensions can be cast in a common framework and analyzed in a uniform manner by employing differential equations. Here, we determine optimal algorithms expressible in that framework, establishing r 3 ? 3:26. We extend the analysis via differential equations, and make extensive use of a new optimization problem we call "maxdensity multiplechoice knapsack". The structure of optimal knapsack solutions elegantly characterizes the choi...
SOFIE: A SelfOrganizing Framework for Information Extraction
 WWW 2009 MADRID! TRACK: SEMANTIC/DATA WEB / SESSION: LINKED DATA
, 2009
"... This paper presents SOFIE, a system for automated ontology extension. SOFIE can parse natural language documents, extract ontological facts from them and link the facts into an ontology. SOFIE uses logical reasoning on the existing knowledge and on the new knowledge in order to disambiguate words to ..."
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Cited by 41 (9 self)
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This paper presents SOFIE, a system for automated ontology extension. SOFIE can parse natural language documents, extract ontological facts from them and link the facts into an ontology. SOFIE uses logical reasoning on the existing knowledge and on the new knowledge in order to disambiguate words to their most probable meaning, to reason on the meaning of text patterns and to take into account world knowledge axioms. This allows SOFIE to check the plausibility of hypotheses and to avoid inconsistencies with the ontology. The framework of SOFIE unites the paradigms of pattern matching, word sense disambiguation and ontological reasoning in one unified model. Our experiments show that SOFIE delivers highquality output, even from unstructured Internet documents.
Inapproximability of combinatorial optimization problems
 Electronic Colloquium on Computational Complexity
, 2004
"... ..."
A New Way to Use Semidefinite Programming with Applications to Linear Equations mod p
 Journal of Algorithms
, 2001
"... We introduce a new method to construct approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as a constellation of vectors in the semidefinite program. When we apply this technique ..."
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Cited by 20 (5 self)
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We introduce a new method to construct approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as a constellation of vectors in the semidefinite program. When we apply this technique to systems of linear equations mod p with at most two variables in each equation, we can show that the problem is approximable within (1 − κ(p))p, whereκ(p)> 0 for all p. Using standard techniques, we also show that it is NPhard to approximate the problem within a constant ratio, independent of p. Warning: Essentially this paper has been published in Journal of Algorithms and is subject to copyright restrictions. In particular it is for personal use only. New Use of Semidefinite Programming 4 List of symbols used: α lower case Greek letter alpha. δ lower case Greek letter delta. ε lower case Greek letter epsilon. κ lower case Greek letter kappa. λ lower case Greek letter lambda. µ lower case Greek letter mu. π lower case Greek letter pi. σ lower case Greek letter sigma. θ lower case Greek letter theta. Θ upper case Greek letter Theta. ϕ lower case Greek letter phi. Φ upper case Greek letter Phi. ζ lower case Greek letter zeta. ξ lower case Greek letter xi. ℓ∫ lower case Latin letter ell. integral sign. summation sign. R bold face upper case Roman letter ar (denotes the set of reals). Z bold face upper case Roman letter zed (denotes the set of integers). ⊥ perpendicular sign. ← left arrow. ↦ → right arrow with small bar (denotes “maps to”). = ⇒ double right arrow (denotes implication). ⇐ ⇒ double leftright arrow (denotes equivalence). ∅ the empty set. ∪ set union. ∩ set intersection. ∈ belongs to sign. ·  absolute value. ‖· ‖ Euclidean norm. 〈·, · 〉 angular brackets (denotes inner product)
Improved Nonapproximability Results for Minimum Vertex Cover with Density Constraints
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 1996
"... We provide new nonapproximability results for the restrictions of the Min Vertex Cover problem to boundeddegree, sparse and dense graphs. We show that for a sufficiently large B, the recent 16/15 lower bound proved by Bellare et al. [5] extends with negligible loss to graphs with bounded degree ..."
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Cited by 16 (0 self)
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We provide new nonapproximability results for the restrictions of the Min Vertex Cover problem to boundeddegree, sparse and dense graphs. We show that for a sufficiently large B, the recent 16/15 lower bound proved by Bellare et al. [5] extends with negligible loss to graphs with bounded degree B. Then, we consider sparse graphs with no dense components (i.e. everywhere sparse graphs), and we show a similar result but with a better tradeoff between nonapproximability and sparsity. Finally we observe that the Min Vertex Cover problem remains APXcomplete when restricted to dense graph and thus recent techniques developed for several Max SNP problems restricted to "dense" instances introduced by Arora et al. [2] cannot be applied.
The approximability of threevalued Max CSP
 SIAM Journal on Computing
"... In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted ca ..."
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Cited by 15 (9 self)
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In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NPhard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, twovalued) Max CSP problem with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APXcomplete (and hence can have no polynomial time approximation scheme unless P = NP). It has been an open problem for several years whether this result can be extended to nonBoolean Max CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for Max CSP over a threeelement domain. Moreover, we present a simple description of all polynomialtime solvable cases of our problem. This description uses the wellknown algebraic combinatorial property of supermodularity. We also show that every hard threevalued Max CSP problem contains, in a certain specified sense, one of the two basic hard Max CSP problems which are the Maximum kcolourable subgraph problems for k = 2, 3.
A tight linear time (1/2)approximation for unconstrained submodular maximization
 in: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, IEEE
"... Abstract—We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f: 2 N → R +, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applicatio ..."
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Cited by 7 (0 self)
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Abstract—We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f: 2 N → R +, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include MaxCut, MaxDiCut, and variants of MaxSAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem. KeywordsSubmodular Functions, Approximation Algorithms I.
MAX CUT in cubic graphs
 In Proceedings of 13th SODA
, 2001
"... We present an improved semidefinite programming based approximation algorithm for the MAX CUT problem in graphs of maximum degree at most 3. The approximation ratio of the new algorithm is at least 0.9326. This improves, and also somewhat simplifies, a result of Feige, Karpinski and Langberg. We a ..."
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Cited by 6 (0 self)
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We present an improved semidefinite programming based approximation algorithm for the MAX CUT problem in graphs of maximum degree at most 3. The approximation ratio of the new algorithm is at least 0.9326. This improves, and also somewhat simplifies, a result of Feige, Karpinski and Langberg. We also observe that results of Hopkins and Staton and of Bondy and Locke yield a simple combinatorial approximation algorithm for the problem. Slightly improved results would appear in the full version of the paper.