... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of (A+ 2)/3 for approximating inde-pendent sets in graphs with degree bounded by A. The analysis directs us towards a simple parallel and dis-tributed algorithm with identical performance, which on constant-degree graphs runs in O(log ” n) time us-ing linear number of processors. We also analyze the Greedy algorithm when run in combination with a frac-tional relaxation technique of Nemhauser and Trotter, and obtain an improved (2Z + 3)/5 performance ratio on graphs with average degree ~. Finally, we introduce a generally applicable technique for improving the approximation ratios of independent set algorithms, and illustrate it by improving the per-formance ratio of Greedy for large ∆.

Several important NP-hard combinatorial optimization problems can be posed as packing/covering integer programs; the randomized rounding technique of Raghavan & Thompson is a powerful tool to approximate them well. We present one elementary unifying property of all these integer programs (IPs), and use the FKG correlation inequality to derive an improved analysis of randomized rounding on them. This also yields a pessimistic estimator, thus presenting deterministic polynomial-time algorithms for them with approximation guarantees significantly better than those known. Keywords.Approximation Algorithms, Combinatorial Optimization, Correlation Inequalities, Covering Integer Programs, De-randomization, Integer Programming, Linear Programming, Linear Relaxations, Packing Integer Programs, Positive Correlation, Randomized Rounding, Rounding Theorems. 1 Preliminary versions of this work appeared in the Proc. ACM Symposium on the Theory of Computing, pages 268--276, 1995, and as DIMACS Te...

Several microarray technologies that monitor the level of expression of a large number of genes have recently emerged. Given DNA-microarray data for a set of cells characterized by a given phenotype and for a set of control cells, an important problem is to identify "patterns" of gene expression that can be used to predict cell phenotype. The potential number of such patterns is exponential in the number of genes. In this paper, we propose a solution to this problem based on a supervised learning algorithm, which differs substantially from previous schemes. It couples a complex, non-linear similarity metric, which maximizes the probability of discovering discriminative gene expression patterns, and a pattern discovery algorithm called SPLASH. The latter discovers efficiently and deterministically all statistically significant gene expression patterns in the phenotype set. Statistical significance is evaluated based on the probability of a pattern to occur by chance in ...

An elementary formal system (EFS) is a logic program con- sisting of definite clauses whose arguments have patterns instead of first-order terms. We investigate EFSs for polynomial-time PAC-learnability. A definite clause of an EFS is hereditary if every pattern in the body is a subword of a pattern in the head. With this new notion, we show that H-EFS(ra, k, t, r) is polynomial-time learnable, which is the class of languages definable by EFSs consisting of at most ra hereditary definite clauses with predicate symbols of arity at most r, where k and t bound the number of variable occurrences in the head and the number of atoms in the body, respectively. The class defined by all finite unions of EFSs in H-EFS(ra, k, t, r) is also polynomialtime learnable. We also show an interesting series of NC-learnable classes of EFSs. As hardness results, the class of regular pattern languages is shown not polynomial-time learnable unless RP=NP. Furthermore, the related problem of deciding whether there is a common subsequence which is consistent with given positive and negative examples is shown NP-complete.

In the past few years, there has been significant progress in our understanding of the extent to which near-optimal solutions can be efficiently computed for NP-hard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the design and analysis of approximation algorithms, and in particular, on those results that rely on linear programming and its generalizations.

ABSTRACT. This paper is a short survey of feedback set problems. It will be published in

Abstract. Used for topology control in ad-hoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity) The input consists of a directed complete weighted graph G = (V; c). The power of a vertex u in a directed spanning subgraph H is given by pH(u) = maxuv2E(H) c(uv). The power of H is given by p(H) = P u2V pH(u), Power Assignment seeks to minimize p(H) while H satisfies the given connectivity constraint. We

.<F3.835e+05> We first present an algorithm that uses membership and equivalence queries to exactly identify a discretized geometric concept defined by the union of<F3.474e+05> m<F3.835e+05> axis-parallel boxes in<F3.474e+05><F3.835e+05> d-dimensional discretized Euclidean space where each coordinate can have<F3.474e+05> n<F3.835e+05> discrete values. This algorithm receives at most<F3.474e+05> md<F3.835e+05> counterexamples and uses time and membership queries polynomial in<F3.474e+05> m<F3.835e+05> and log<F3.474e+05> n<F3.835e+05> for any constant<F3.474e+05><F3.835e+05> d. Furthermore, all equivalence queries can be formulated as the union of<F3.474e+05><F3.835e+05><F3.474e+05> O(md<F3.835e+05> log<F3.474e+05><F3.835e+05> m) axis-parallel boxes. Next, we show how to extend our algorithm to e#ciently learn, from<F3.771e+05> only<F3.835e+05> equivalence queries, any discretized geometric concept generated from any number of halfspaces with any number of known (to the learner) slopes...

The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. We consider three classes of NP-hard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results. 1

Despite more than a decade of experimental work in multi-robot systems, important theoretical aspects of multi-robot coordination mechanisms have, to date, been largely untreated. To address this issue, we focus on the problem of multi-robot task allocation (MRTA). Most work on MRTA has been ad hoc and empirical, with many coordination architectures having been proposed and validated in a proof-of-concept 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 domain-independent taxonomy of MRTA problems is given, and it is shown how many such problems can be viewed as instances of other, well-studied, 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 show how the same theory can be used in the synthesis of new approaches.