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53
A Tight Analysis of the Greedy Algorithm for Set Cover
, 1995
"... We establish significantly improved bounds on the performance of the greedy algorithm for approximating set cover. In particular, we provide the first substantial improvement of the 20 year old classical harmonic upper bound, H(m), of Johnson, Lovasz, and Chv'atal, by showing that the performan ..."
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Cited by 115 (0 self)
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We establish significantly improved bounds on the performance of the greedy algorithm for approximating set cover. In particular, we provide the first substantial improvement of the 20 year old classical harmonic upper bound, H(m), of Johnson, Lovasz, and Chv'atal, by showing that the performance ratio of the greedy algorithm is, in fact, exactly ln m \Gamma ln ln m+ \Theta(1), where m is the size of the ground set. The difference between the upper and lower bounds turns out to be less than 1:1. This provides the first tight analysis of the greedy algorithm, as well as the first upper bound that lies below H(m) by a function going to infinity with m. We also show that the approximation guarantee for the greedy algorithm is better than the guarantee recently established by Srinivasan for the randomized rounding technique, thus improving the bounds on the integrality gap. Our improvements result from a new approach which might be generally useful for attacking other similar problems. ...
On spectrum sharing games
 In proc. of PODC 2004
, 2004
"... Each access point (AP) in a WiFi network must be assigned a channel for it to service users. There are only finitely many possible channels that can be assigned. Moreover, neighboring access points must use different channels so as to avoid interference. Currently these channels are assigned by admi ..."
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Cited by 76 (3 self)
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Each access point (AP) in a WiFi network must be assigned a channel for it to service users. There are only finitely many possible channels that can be assigned. Moreover, neighboring access points must use different channels so as to avoid interference. Currently these channels are assigned by administrators who carefully consider channel conflicts and network loads. Channel conflicts among APs operated by different entities are currently resolved in an ad hoc manner or not resolved at all. We view the channel assignment problem as a game, where the players are the service providers and APs are acquired sequentially. We consider the price of anarchy of this game, which is the ratio between the total coverage of the APs in the worst Nash equilibrium of the game and what the total coverage of the APs would be if the channel assignment were done by a central authority. We provide bounds on the price of anarchy depending on assumptions on the underlying network and the type of bargaining allowed between service providers. The key tool in the analysis is the identification of the Nash equilibria with the solutions to a maximal coloring problem in an appropriate graph. We relate the price of anarchy of these games to the approximation factor of local optimization algorithms for the maximum�colorable subgraph problem. We also study the speed of convergence in these games.
Assignment of orthologous genes via genome rearrangement
 IEEE/ACM Transactions on Computational Biology and Bioinformatics
, 2005
"... Abstract—The assignment of orthologous genes between a pair of genomes is a fundamental and challenging problem in comparative genomics. Existing methods that assign orthologs based on the similarity between DNA or protein sequences may make erroneous assignments when sequence similarity does not cl ..."
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Cited by 46 (5 self)
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Abstract—The assignment of orthologous genes between a pair of genomes is a fundamental and challenging problem in comparative genomics. Existing methods that assign orthologs based on the similarity between DNA or protein sequences may make erroneous assignments when sequence similarity does not clearly delineate the evolutionary relationship among genes of the same families. In this paper, we present a new approach to ortholog assignment that takes into account both sequence similarity and evolutionary events at a genome level, where orthologous genes are assumed to correspond to each other in the most parsimonious evolving scenario under genome rearrangement. First, the problem is formulated as that of computing the signed reversal distance with duplicates between the two genomes of interest. Then, the problem is decomposed into two new optimization problems, called minimum common partition and maximum cycle decomposition, for which efficient heuristic algorithms are given. Following this approach, we have implemented a highthroughput system for assigning orthologs on a genome scale, called SOAR, and tested it on both simulated data and real genome sequence data. Compared to a recent ortholog assignment method based entirely on homology search (called INPARANOID), SOAR shows a marginally better performance in terms of sensitivity on the real data set because it is able to identify several correct orthologous pairs that are missed by INPARANOID. The simulation results demonstrate that SOAR, in general, performs better than the iterated exemplar algorithm in terms of computing the reversal distance and assigning correct orthologs. Index Terms—Ortholog, paralog, gene duplication, genome rearrangement, reversal, comparative genomics. 1
Approximation of kSet Cover by SemiLocal Optimization
 In Proc. 29th STOC
, 1997
"... We define a powerful new approximation technique called semilocal optimization. It provides very natural heuristics that are distinctly more powerful than those based on local optimization. With an appropriate metric, semilocal optimization can still be viewed as a local optimization, but it has t ..."
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Cited by 38 (0 self)
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We define a powerful new approximation technique called semilocal optimization. It provides very natural heuristics that are distinctly more powerful than those based on local optimization. With an appropriate metric, semilocal optimization can still be viewed as a local optimization, but it has the advantage of making global changes to an approximate solution. Semilocal optimization generalizes recent heuristics of Halldorsson for 3Set Cover, Color Saving, and kSet Cover. Greatly improved performance ratios of 4/3 for 3Set Cover and 6/5 for Color Saving in graphs without independent sets of size 4 are obtained and shown to be the best possible with semilocal optimization. Also, based on the result for 3Set Cover and a restricted greedy phase for big sets, we can improve the performance ratio for kSet Cover to H k \Gamma 1=2. In Color Saving, when larger independent sets exist, we can improve the performance ratio to .
On Local Search for Weighted kSet Packing
, 1997
"... Given a collection of sets of cardinality at most k, with weights for each set, the maximum weighted packing problem is that of finding a collection of disjoint sets of maximum total weight. We study the worst case behavior of the tlocal search heuristic for this problem proving a tight bound of k ..."
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Cited by 30 (3 self)
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Given a collection of sets of cardinality at most k, with weights for each set, the maximum weighted packing problem is that of finding a collection of disjoint sets of maximum total weight. We study the worst case behavior of the tlocal search heuristic for this problem proving a tight bound of k \Gamma 1 + 1 t . As a consequence, for any given r ! 1 k\Gamma1 we can compute in polynomial time a solution whose weight is at least r times the optimal. 1 Introduction Maximum packing problems are among the most often studied in combinatorial optimization: Given a collection X 1 ; : : : ; X q of ksets, find a largest collection of pairwise disjoint sets among them. One of the most fundamental packing problems is that of finding a maximum matching in a graph; this problem is polynomially solvable. However, many other packing problems are NPhard, including maximum 3dimensional matching, maximum triangle packing, maximum H matching, and maximum independent sets of axis parallel recta...
Approximating kSet Cover and Complementary Graph Coloring
"... We consider instances of the Set Cover problem where each set is of small size. For collections of sets of size at most three, we obtain improved performance ratios of 1.4 + ffl, for any constant ffl? 0. Similar improvements hold also for collections of larger sets. A corollary of this result is an ..."
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Cited by 28 (0 self)
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We consider instances of the Set Cover problem where each set is of small size. For collections of sets of size at most three, we obtain improved performance ratios of 1.4 + ffl, for any constant ffl? 0. Similar improvements hold also for collections of larger sets. A corollary of this result is an improved performance ratio of 4/3 for the problem of minimizing the unused colors in a graph coloring.
Nonoverlapping Local Alignments (Weighted Independent Sets of Axis Parallel Rectangles)
, 1996
"... We consider the following problem motivated by an application in computational molecular biology. We are given a set of weighted axisparallel rectangles such that for any pair of rectangles and either axis, the projection of one rectangle does not enclose that of the other. Define a pair to be inde ..."
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Cited by 24 (0 self)
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We consider the following problem motivated by an application in computational molecular biology. We are given a set of weighted axisparallel rectangles such that for any pair of rectangles and either axis, the projection of one rectangle does not enclose that of the other. Define a pair to be independent if their projections in both axes are disjoint. The problem is to find a maximumweight independent subset of rectangles. We show that the problem is NPhard even in the uniform case when all the weights are the same. We analyze the performance of a natural localimprovement heuristic for the general problem and prove a performance ratio of 3.25. We extend the heuristic to the problem of finding a maximumweight independent set in (d+1) claw free graphs, and show a tight performance ratio of d\Gamma1+ 1 d . A performance ratio of d 2 was known for the heuristic when applied to the uniform case. Our contributions are proving the hardness of the problem and providing a tight anal...
Target Tracking with Distributed Sensors: The Focus of Attention Problem
, 2004
"... In this paper, we consider the problem of assigning sensors to track targets so as to minimize the expected error in the resulting estimation for target locations. Specifically, we are interested in how disjoint pairs of bearing or range sensors can be best assigned to targets in order to minimize t ..."
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Cited by 21 (1 self)
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In this paper, we consider the problem of assigning sensors to track targets so as to minimize the expected error in the resulting estimation for target locations. Specifically, we are interested in how disjoint pairs of bearing or range sensors can be best assigned to targets in order to minimize the expected error in the estimates. We refer to this as the focus of attention (FOA) problem. In its
Differential approximation algorithms for some combinatorial optimization problems
 THEORETICAL COMPUTER SCIENCE
, 1998
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