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75
On syntactic versus computational views of approximability
 SIAM JOURNAL ON COMPUTING
, 1999
"... We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximabilit ..."
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Cited by 118 (12 self)
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We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is wellunderstood. Our results provide a syntactic characterization of computational classes, and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX, and show that every problem in APX can be “placed ” (i.e. has approximation preserving reduction to a problem) in MAX SNP. Our methods introduce a general technique for creating approximationpreserving reductions which show that any “well ” approximable problem can be reduced in an approximationpreserving manner to a problem which is hard to approximate to corresponding factors. We demonstrate this technique by applying it to the classes RMAX(2) and MIN F+Π2(1)which have the clique problem and the set cover problem, respectively, as complete problems. We use the syntactic nature of MAX SNP to define a general paradigm, nonoblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximation ratios comparable to the bestknown for a variety of specific MAX SNPhard problems. Nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms.
Robust Trainability of Single Neurons
, 1995
"... It is well known that (McCullochPitts) neurons are efficiently trainable to learn an unknown halfspace from examples, using linearprogramming methods. We want to analyze how the learning performance degrades when the representational power of the neuron is overstrained, i.e., if more complex conce ..."
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Cited by 85 (0 self)
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It is well known that (McCullochPitts) neurons are efficiently trainable to learn an unknown halfspace from examples, using linearprogramming methods. We want to analyze how the learning performance degrades when the representational power of the neuron is overstrained, i.e., if more complex concepts than just halfspaces are allowed. We show that the problem of learning a probably almost optimal weight vector for a neuron is so difficult that the minimum error cannot even be approximated to within a constant factor in polynomial time (unless RP = NP); we obtain the same hardness result for several variants of this problem. We considerably strengthen these negative results for neurons with binary weights 0 or 1. We also show that neither heuristical learning nor learning by sigmoidal neurons with a constant reject rate is efficiently possible (unless RP = NP).
The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations
 Theoretical Computer Science
, 1993
"... We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of rela ..."
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Cited by 77 (12 self)
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We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of relations =, , ? and 6=. Various constrained versions of Max FLS, where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We establish the complexity of solving these problems optimally and, whenever they are intractable, we determine their degree of approximability. Max FLS with =, or ? relations is NPhard even when restricted to homogeneous systems with bipolar coefficients, whereas it can be solved in polynomial time for 6= relations with real coefficients. The various NPhard versions of Max FLS belong to different approximability classes depending on the type of relations and the additional constraints. We show that the ran...
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 72 (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...
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 70 (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.
Computational Aspects of Reordering Plans
 Journal of Artificial Intelligence Research
, 1998
"... This article studies the problem of modifying the action ordering of a plan in order to optimise the plan according to various criteria. One of these criteria is to make a plan less constrained and the other is to minimize its parallel execution time. Three candidate definitions are proposed for the ..."
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Cited by 59 (0 self)
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This article studies the problem of modifying the action ordering of a plan in order to optimise the plan according to various criteria. One of these criteria is to make a plan less constrained and the other is to minimize its parallel execution time. Three candidate definitions are proposed for the first of these criteria, constituting a sequence of increasing optimality guarantees. Two of these are based on deordering plans, which means that ordering relations may only be removed, not added, while the third one uses reordering, where arbitrary modifications to the ordering are allowed. It is shown that only the weakest one of the three criteria is tractable to achieve, the other two being NPhard and even difficult to approximate. Similarly, optimising the parallel execution time of a plan is studied both for deordering and reordering of plans. In the general case, both of these computations are NPhard. However, it is shown that optimal deorderings can be computed in polynomial time...
Hardness of Approximating Problems on Cubic Graphs
"... Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APXcomplete even for cubic graphs. This means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bo ..."
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Cited by 45 (0 self)
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Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APXcomplete even for cubic graphs. This means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.
Constraint satisfaction: The approximability of minimization problems
 Proc. 12th Annual Conference on Structure in Complexity Theory, IEEE
, 1997
"... This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a ..."
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Cited by 45 (5 self)
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This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a collection�of “constraints” (i.e., functions����������) and an instance of a problem is constraints drawn from�applied to specified subsets of Boolean variables. We study the two minimization analogs of classes studied in [15]: in one variant, namely MIN CSP�, the objective is to find an assignment to minimize the number of unsatisfied constraints, while in the other, namely MIN ONES�, the goal is to find a satisfying assignment with minimum number of ones. These two classes together capture an entire spectrum of important minimization problems including Min Cut, vertex cover, hitting set with bounded size sets, integer programs with two variables per inequality, graph bipartization, clause deletion in CNF formulae, and nearest codeword. Our main result is that there exists a finite partition of the space of all constraint sets such that for any given�, the approximability of MIN CSP�and MIN ONES� is completely determined by the partition containing it. Moreover, we present a compact set of rules that determines which partition contains a given family�. Our classification identifies the central elements governing the approximability of problems in these classes, by unifying a large collection algorithmic and hardness of approximation results. When contrasted with the work of [15], our results also serve to formally highlight inherent differences between maximization and minimization problems.
When Hamming Meets Euclid: The Approximability of Geometric TSP and MST (Extended Abstract)
, 1997
"... We prove that the Traveling Salesperson Problem (MIN TSP) and the Minimum Steiner Tree Problem (MIN ST) are Max SNPhard (and thus NPhard to approximate within some constant r ? 1) even if all cities (respectively, points) lie in the geometric space R n (n is the number of cities/points) and ..."
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Cited by 43 (2 self)
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We prove that the Traveling Salesperson Problem (MIN TSP) and the Minimum Steiner Tree Problem (MIN ST) are Max SNPhard (and thus NPhard to approximate within some constant r ? 1) even if all cities (respectively, points) lie in the geometric space R n (n is the number of cities/points) and distances are computed with respect to the l 1 (rectilinear) metric. The TSP hardness results also hold for any l p metric, including the Euclidean metric, and in R logn . The running time of Arora's approximation scheme for geometric MIN TSP in R d is doubly exponential in d. Our results imply that this dependance is necessary unless NP has subexponential algorithms. We also prove, as an intermediate step, the hardness of approximating MIN TSP and MIN ST in Hamming spaces. The reduction for MIN TSP uses errorcorrecting codes and random sampling; the reduction for MIN ST uses the integrality property of MINCUT. The only previous nonapproximability results for ...
Logical Definability of NP Optimization Problems
 Information and Computation
, 1994
"... : We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using firstorder formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we ..."
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Cited by 41 (2 self)
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: We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using firstorder formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties. To appear in Information and Computation. Research partially supported by NSF Grants CCR8905038 and CCR9108631. y email addresses: kolaitis@cse.ucsc.edu, thakur@cse.ucsc.edu z supersedes Technical report UCSCCRL9048 1 Introduction and Summary of Results It is well known that optimization problems had a major influence on the development of the theory of NPco...