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159
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 658 (5 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
A formal analysis and taxonomy of task allocation in multirobot systems
 INT’L. J. OF ROBOTICS RESEARCH
, 2004
"... Despite more than a decade of experimental work in multirobot systems, important theoretical aspects of multirobot coordination mechanisms have, to date, been largely untreated. To address this issue, we focus on the problem of multirobot task allocation (MRTA). Most work on MRTA has been ad hoc ..."
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Cited by 213 (4 self)
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Despite more than a decade of experimental work in multirobot systems, important theoretical aspects of multirobot coordination mechanisms have, to date, been largely untreated. To address this issue, we focus on the problem of multirobot task allocation (MRTA). Most work on MRTA has been ad hoc and empirical, with many coordination architectures having been proposed and validated in a proofofconcept fashion, but infrequently analyzed. With the goal of bringing objective grounding to this important area of research, we present a formal study of MRTA problems. A domainindependent taxonomy of MRTA problems is given, and it is shown how many such problems can be viewed as instances of other, wellstudied, optimization problems. We demonstrate how relevant theory from operations research and combinatorial optimization can be used for analysis and greater understanding of existing approaches to task allocation, and to show how the same theory can be used in the synthesis of new approaches.
Computational Limitations on Learning from Examples
 Journal of the ACM
, 1988
"... Abstract. The computational complexity of learning Boolean concepts from examples is investigated. It is shown for various classes of concept representations that these cannot be learned feasibly in a distributionfree sense unless R = NP. These classes include (a) disjunctions of two monomials, (b) ..."
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Cited by 192 (10 self)
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Abstract. The computational complexity of learning Boolean concepts from examples is investigated. It is shown for various classes of concept representations that these cannot be learned feasibly in a distributionfree sense unless R = NP. These classes include (a) disjunctions of two monomials, (b) Boolean threshold functions, and (c) Boolean formulas in which each variable occurs at most once. Relationships between learning of heuristics and finding approximate solutions to NPhard optimization problems are given. Categories and Subject Descriptors: F. 1.1 [Computation by Abstract Devices]: Models of Computationrelations among models; F. 1.2 [Computation by Abstract Devices]: Modes of Computationprobabilistic computation; F. 1.3 [Computation by Abstract Devices]: Complexity Classesreducibility and completeness; 1.2.6 [Artificial Intelligence]: Learningconcept learning; induction
Learning in the Presence of Malicious Errors
 SIAM Journal on Computing
, 1993
"... In this paper we study an extension of the distributionfree model of learning introduced by Valiant [23] (also known as the probably approximately correct or PAC model) that allows the presence of malicious errors in the examples given to a learning algorithm. Such errors are generated by an advers ..."
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Cited by 169 (12 self)
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In this paper we study an extension of the distributionfree model of learning introduced by Valiant [23] (also known as the probably approximately correct or PAC model) that allows the presence of malicious errors in the examples given to a learning algorithm. Such errors are generated by an adversary with unbounded computational power and access to the entire history of the learning algorithm's computation. Thus, we study a worstcase model of errors. Our results include general methods for bounding the rate of error tolerable by any learning algorithm, efficient algorithms tolerating nontrivial rates of malicious errors, and equivalences between problems of learning with errors and standard combinatorial optimization problems. 1 Introduction In this paper, we study a practical extension to Valiant's distributionfree model of learning: the presence of errors (possibly maliciously generated by an adversary) in the sample data. The distributionfree model typically makes the idealize...
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 150 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Greedy Facility Location Algorithms analyzed using Dual Fitting with FactorRevealing LP
 Journal of the ACM
, 2001
"... We present a natural greedy algorithm for the metric uncapacitated facility location problem and use the method of dual fitting to analyze its approximation ratio, which turns out to be 1.861. The running time of our algorithm is O(m log m), where m is the total number of edges in the underlying c ..."
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Cited by 104 (13 self)
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We present a natural greedy algorithm for the metric uncapacitated facility location problem and use the method of dual fitting to analyze its approximation ratio, which turns out to be 1.861. The running time of our algorithm is O(m log m), where m is the total number of edges in the underlying complete bipartite graph between cities and facilities. We use our algorithm to improve recent results for some variants of the problem, such as the fault tolerant and outlier versions. In addition, we introduce a new variant which can be seen as a special case of the concave cost version of this problem.
On the Complexity of Teaching
 Journal of Computer and System Sciences
, 1992
"... While most theoretical work in machine learning has focused on the complexity of learning, recently there has been increasing interest in formally studying the complexity of teaching . In this paper we study the complexity of teaching by considering a variant of the online learning model in which a ..."
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Cited by 103 (2 self)
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While most theoretical work in machine learning has focused on the complexity of learning, recently there has been increasing interest in formally studying the complexity of teaching . In this paper we study the complexity of teaching by considering a variant of the online learning model in which a helpful teacher selects the instances. We measure the complexity of teaching a concept from a given concept class by a combinatorial measure we call the teaching dimension. Informally, the teaching dimension of a concept class is the minimum number of instances a teacher must reveal to uniquely identify any target concept chosen from the class. A preliminary version of this paper appeared in the Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 303314. August 1991. Most of this research was carried out while both authors were at MIT Laboratory for Computer Science with support provided by ARO Grant DAAL0386K0171, DARPA Contract N0001489J1988, NSF Gr...
Learning Boolean Concepts in the Presence of Many Irrelevant Features
 Artificial Intelligence
, 1994
"... In many domains, an appropriate inductive bias is the MINFEATURES bias, which prefers consistent hypotheses definable over as few features as possible. This paper defines and studies this bias in Boolean domains. First, it is shown that any learning algorithm implementing the MINFEATURES bias requ ..."
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Cited by 103 (0 self)
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In many domains, an appropriate inductive bias is the MINFEATURES bias, which prefers consistent hypotheses definable over as few features as possible. This paper defines and studies this bias in Boolean domains. First, it is shown that any learning algorithm implementing the MINFEATURES bias requires \Theta( 1 ffl ln 1 ffi + 1 ffl [2 p + p ln n]) training examples to guarantee PAClearning a concept having p relevant features out of n available features. This bound is only logarithmic in the number of irrelevant features. For implementing the MINFEATURES bias, the paper presents five algorithms that identify a subset of features sufficient to construct a hypothesis consistent with the training examples. FOCUS1 is a straightforward algorithm that returns a minimal and sufficient subset of features in quasipolynomial time. FOCUS2 does the same task as FOCUS1 but is empirically shown to be substantially faster than FOCUS1. Finally, the SimpleGreedy, MutualInformationG...
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
, 1999
"... We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a perf ..."
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Cited by 98 (7 self)
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We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most &Delta;, the algorithm achieves a performance ratio of 2  (1  o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For kuniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for kuniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree kuniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general kuniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.
On the learnability of discrete distributions
 In The 25th Annual ACM Symposium on Theory of Computing
, 1994
"... We introduce and investigate a new model of learning probability distributions from independent draws. Our model is inspired by the popular Probably Approximately Correct (PAC) model for learning boolean functions from labeled ..."
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Cited by 95 (11 self)
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We introduce and investigate a new model of learning probability distributions from independent draws. Our model is inspired by the popular Probably Approximately Correct (PAC) model for learning boolean functions from labeled