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 k-cover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NP-hard. 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 k-cover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .

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 distribution-free 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 NP-hard optimization problems are given. Categories and Subject Descriptors: F. 1.1 [Computation by Abstract Devices]: Models of Computation-relations among models; F. 1.2 [Computation by Abstract Devices]: Modes of Computation-probabi-listic computation; F. 1.3 [Computation by Abstract Devices]: Complexity Classes-reducibility and completeness; 1.2.6 [Artificial Intelligence]: Learning-concept learning; induction

In this paper we study an extension of the distribution-free 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 worst-case 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 distribution-free model of learning: the presence of errors (possibly maliciously generated by an adversary) in the sample data. The distribution-free model typically makes the idealize...

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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 NP-complete 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 high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

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.

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 on-line 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 303--314. 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 DAAL03-86-K-0171, DARPA Contract N00014-89-J-1988, NSF Gr...

In many domains, an appropriate inductive bias is the MIN-FEATURES 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 MIN-FEATURES bias requires \Theta( 1 ffl ln 1 ffi + 1 ffl [2 p + p ln n]) training examples to guarantee PAC-learning 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 MIN-FEATURES bias, the paper presents five algorithms that identify a subset of features sufficient to construct a hypothesis consistent with the training examples. FOCUS-1 is a straightforward algorithm that returns a minimal and sufficient subset of features in quasi-polynomial time. FOCUS-2 does the same task as FOCUS-1 but is empirically shown to be substantially faster than FOCUS-1. Finally, the Simple-Greedy, Mutual-Information-G...

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

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 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 k-uniform 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 k-uniform 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 k-uniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general k-uniform 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.