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Constraint Logic Programming: A Survey
"... Constraint Logic Programming (CLP) is a merger of two declarative paradigms: constraint solving and logic programming. Although a relatively new field, CLP has progressed in several quite different directions. In particular, the early fundamental concepts have been adapted to better serve in differe ..."
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Cited by 771 (23 self)
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Constraint Logic Programming (CLP) is a merger of two declarative paradigms: constraint solving and logic programming. Although a relatively new field, CLP has progressed in several quite different directions. In particular, the early fundamental concepts have been adapted to better serve in different areas of applications. In this survey of CLP, a primary goal is to give a systematic description of the major trends in terms of common fundamental concepts. The three main parts cover the theory, implementation issues, and programming for applications.
Linear programming in linear time when the dimension is fixed
 J. ACM
, 1984
"... Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that i ..."
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Cited by 194 (13 self)
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Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problemscomputations on matrices; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problemsgeometrical problems and computations; sorting and searching; G. 1.6 [Mathematics of Computing]: Optimizationlinear programming
Algorithms for Sequential Decision Making
, 1996
"... Sequential decision making is a fundamental task faced by any intelligent agent in an extended interaction with its environment; it is the act of answering the question "What should I do now?" In this thesis, I show how to answer this question when "now" is one of a finite set of states, "do" is one ..."
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Cited by 175 (8 self)
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Sequential decision making is a fundamental task faced by any intelligent agent in an extended interaction with its environment; it is the act of answering the question "What should I do now?" In this thesis, I show how to answer this question when "now" is one of a finite set of states, "do" is one of a finite set of actions, "should" is maximize a longrun measure of reward, and "I" is an automated planning or learning system (agent). In particular,
On the complexity of solving Markov decision problems
 IN PROC. OF THE ELEVENTH INTERNATIONAL CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 1995
"... Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argu ..."
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Cited by 131 (10 self)
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Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argue that, although MDPs can be solved efficiently in theory, more study is needed to reveal practical algorithms for solving large problems quickly. To encourage future research, we sketch some alternative methods of analysis that rely on the structure of MDPs.
Settling the complexity of twoplayer Nash equilibrium
 In Proc. 47th FOCS
, 2006
"... We prove that the problem of finding a Nash equilibrium in a twoplayer game is PPADcomplete. 1 ..."
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Cited by 106 (5 self)
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We prove that the problem of finding a Nash equilibrium in a twoplayer game is PPADcomplete. 1
Las Vegas algorithms for linear and integer programming when the dimension is small
 J. ACM
, 1995
"... Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algor ..."
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Cited by 103 (2 self)
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Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algorlthm N gwen for integer hnear programmmg. Let p bound the number of bits required to specify the ratmnal numbers defmmg an input constraint or the ob~ective function vector. Let n and d be as before. Then, the algorithm requires expected 0(2d dn + S~dm In n) + dc)’d) ~ in H operations on numbers with O(1~p bits d ~ ~ ~z + ~, where the constant factors do not depend on d or p. The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any gwen input. The techmque can be extended to other convex programming problems. For example, m algorlthm for finding the smallest sphere enclosing a set of /z points m Ed has the same t]me bound
Boosting in the limit: Maximizing the margin of learned ensembles
 In Proceedings of the Fifteenth National Conference on Artificial Intelligence
, 1998
"... The "minimum margin" of an ensemble classifier on a given training set is, roughly speaking, the smallest vote it gives to any correct training label. Recent work has shown that the Adaboost algorithm is particularly effective at producing ensembles with large minimum margins, and theory suggests th ..."
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Cited by 97 (0 self)
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The "minimum margin" of an ensemble classifier on a given training set is, roughly speaking, the smallest vote it gives to any correct training label. Recent work has shown that the Adaboost algorithm is particularly effective at producing ensembles with large minimum margins, and theory suggests that this may account for its success at reducing generalization error. We note, however, that the problem of finding good margins is closely related to linear programming, and we use this connection to derive and test new "LPboosting" algorithms that achieve better minimum margins than Adaboost. However, these algorithms do not always yield better generalization performance. In fact, more often the opposite is true. We report on a series of controlled experiments which show that no simple version of the minimummargin story can be complete. We conclude that the crucial question as to why boosting works so well in practice, and how to further improve upon it, remains mostly open. Some of our ...
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 84 (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).
Interior Methods for Constrained Optimization
 Acta Numerica
, 1992
"... Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, includ ..."
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Cited by 83 (3 self)
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Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomialtime interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of everincreasing size. Interior methods...
Probe Backtrack Search for Minimal Perturbation in Dynamic Scheduling
, 1999
"... . This paper describes an algorithm designed to minimally recongure schedules in response to a changing environment. External factors have caused an existing schedule to become invalid, perhaps due to the withdrawal of resources, or because of changes to the set of scheduled activities. The total s ..."
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Cited by 70 (12 self)
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. This paper describes an algorithm designed to minimally recongure schedules in response to a changing environment. External factors have caused an existing schedule to become invalid, perhaps due to the withdrawal of resources, or because of changes to the set of scheduled activities. The total shift in the start and end times of already scheduled activities should be kept to a minimum. This optimization requirement may be captured using a linear optimization function over linear constraints. However, the disjunctive nature of the resource constraints impairs traditional mathematical programming approaches. The unimodular probing algorithm interleaves constraint programming and linear programming. The linear programming solver handles only a controlled subset of the problem constraints, to guarantee that the values returned are discrete. Using probe backtracking, a complete, repairbased method for search, these values are simply integrated into constraint programming. Unimodular p...