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81
The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations
, 1993
"... We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some ..."
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Cited by 155 (7 self)
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We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some ffl ? 0 there exists a polynomialtime algorithm that approximates the optimum within a factor of 2 log 0:5\Gammaffl n , then every NP language can be decided in quasipolynomial deterministic time, i.e., NP ` DTIME(n poly(log n) ). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the `1 norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2 log 1\Gammaffl n or n ffl . Improving the factor 2 log 0:5\Gammaffl n to p dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in `2 norm; an old open problem. Our proofs use reductions from fewpr...
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 104 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
B.: Adaptive service composition in flexible processes
 IEEE Trans. Software Eng
, 2007
"... Abstract—In advanced service oriented systems, complex applications, described as abstract business processes, can be executed by invoking a number of available Web services. End users can specify different preferences and constraints and service selection can be performed dynamically identifying th ..."
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Cited by 78 (4 self)
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Abstract—In advanced service oriented systems, complex applications, described as abstract business processes, can be executed by invoking a number of available Web services. End users can specify different preferences and constraints and service selection can be performed dynamically identifying the best set of services available at runtime. In this paper, we introduce a new modeling approach to the Web service selection problem that is particularly effective for large processes and when QoS constraints are severe. In the model, the Web service selection problem is formalized as a mixed integer linear programming problem, loops peeling is adopted in the optimization, and constraints posed by stateful Web services are considered. Moreover, negotiation techniques are exploited to identify a feasible solution of the problem, if one does not exist. Experimental results compare our method with other solutions proposed in the literature and demonstrate the effectiveness of our approach toward the identification of an optimal solution to the QoS constrained Web service selection problem. Index Terms—Web services, quality of service, service composition, integer programming. Ç 1
On the Approximability of Minimizing Nonzero Variables Or Unsatisfied Relations in Linear Systems
, 1997
"... We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasib ..."
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Cited by 74 (4 self)
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We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, , ? and 6= are considered. While Min RVLS with equations was known to be NPhard in [27], we established in [2, 5] that Min ULR with equalities and inequalities are NPhard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in [8]. In this paper we determine strong bou...
Derandomized graph products
 COMPUTATIONAL COMPLEXITY
, 1995
"... Berman and Schnitger [10] gave a randomized reduction from approximating MAXSNP problems [24] within constant factors arbitrarily close to 1 to approximating clique within a factor of n ɛ (for some ɛ). This reduction was further studied by Blum [11], who gave it the name randomized graph products. ..."
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Cited by 73 (11 self)
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Berman and Schnitger [10] gave a randomized reduction from approximating MAXSNP problems [24] within constant factors arbitrarily close to 1 to approximating clique within a factor of n ɛ (for some ɛ). This reduction was further studied by Blum [11], who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs [1]. The main technical contribution of this paper is in lower bounding the probability that all steps of a random walk stay within a specified set of vertices of a graph. (Previous work was mainly concerned with upper bounding this probability.) This lower bound extends also to the case that different sets of vertices are specified for different time steps of the walk.
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 ..."
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Cited by 71 (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.
On approximating the depth and related problems
 SIAM J. COMPUT
, 2008
"... We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear pr ..."
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Cited by 66 (12 self)
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We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.
Distinguishing string selection problems
 Information and Computation
, 1999
"... This paper presents a collection of string algorithms that are at the core of several biological problems such as discovering potential drug targets, creating diagnostic probes, universal primers or unbiased consensus sequences. All these problems reduce to the task of finding a pattern that, with s ..."
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Cited by 65 (9 self)
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This paper presents a collection of string algorithms that are at the core of several biological problems such as discovering potential drug targets, creating diagnostic probes, universal primers or unbiased consensus sequences. All these problems reduce to the task of finding a pattern that, with some error, occurs in one set of strings (Closest Substring Problem) and does not occur in another set (Farthest String Problem). In this paper, we break down the problem into several subproblems and prove the following results. 1. The following are all NPHard: the Farthest String Problem, the Closest Substring Problem, and the Closest String Problem of finding a string that is close to each string in a set. 2. There is a PTAS for the Farthest � String � Problem based on a linear programming relaxation technique. 43 3. There is a polynomialtime + ɛapproximation algorithm for the Closest String Problem for any small constant ɛ>0. Using this algorithm, we also provide an efficient heuristic algorithm for the Closest Substring Problem. 4. The problem of finding a string that is at least Hamming distance d from as many strings in a set as possible,