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131
Hardness of Approximating the Minimum Distance of a Linear Code
, 2003
"... We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the b ..."
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Cited by 51 (7 self)
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We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the block length n of the code. Under the stronger assumption that NPis not contained in RQP (random quasipolynomial time), we show that the minimum distance is not approximable to within the factor 2log 1ffl(n), for any ffl> 0. Our results hold for codes over any finite field, including binary codes. In the process we show that it is hard to findapproximately nearest codewords even if the number of errors exceeds the unique decoding radius d/2 by only an arbitrarily small fraction ffld. We also prove the hardness of the nearestcodeword problem for asymptotically good codes, provided the number of errors exceeds (2
Mechanism Design for Policy Routing
, 2006
"... The Border Gateway Protocol (BGP) for interdomain routing is designed to allow autonomous systems (ASes) to express policy preferences over alternative routes. We model these preferences as arising from an AS’s underlying utility for each route and study the problem of finding a set of routes that ..."
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Cited by 49 (7 self)
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The Border Gateway Protocol (BGP) for interdomain routing is designed to allow autonomous systems (ASes) to express policy preferences over alternative routes. We model these preferences as arising from an AS’s underlying utility for each route and study the problem of finding a set of routes that maximizes the overall welfare (i.e., the sum of all ASes’ utilities for their selected routes). We show that, if the utility functions are unrestricted, this problem is NPhard even to approximate closely. We then study a natural class of restricted utilities that we call nexthop preferences. We present a strategyproof, polynomialtime computable mechanism for welfaremaximizing routing over this restricted domain. However, we show that, in contrast to earlier work on lowestcost routing mechanism design, this mechanism appears to be incompatible with
CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions
, 2005
"... Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NPcomplete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and ..."
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Cited by 48 (8 self)
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Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NPcomplete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and lower bounding (also across components), elaborate and dynamically chosen bidordering heuristics, and a host of structural observations. CABOB attempts to capture structure in any instance without making assumptions about the instance distribution. Experiments against the fastest prior algorithm, CPLEX 8.0, show that CABOB is often faster, seldom drastically slower, and in many cases drastically faster—especially in cases with structure. CABOB’s search runs in linear space and has significantly better anytime performance than CPLEX. We also uncover interesting aspects of the problem itself. First, problems with short bids, which were hard for the first generation of specialized algorithms, are easy. Second, almost all of the CATS distributions are easy, and the run time is virtually unaffected by the number of goods. Third, we test several random restart strategies, showing that they do not help on this problem—the runtime distribution does not have a heavy tail.
Extractors from ReedMuller Codes
 In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, w ..."
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Cited by 39 (5 self)
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Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, researchers had failed to build extractors directly from a good code, without using other tools from pseudorandomness. We succeed in constructing an extractor directly from a ReedMuller code. To do this, we develop a novel proof technique. Furthermore, our construction is the first and only construction with degree close to linear. In contrast, the best previous constructions had brought the log of the degree within a constant of optimal, which gives polynomial degree. This improvement is important for certain applications. For example, it follows that approximating the VC dimension to within a factor of N
Approximating the independence number via the ϑfunction
, 1994
"... We study the relationship between the independence number of a graph and its semidefinite relaxation, the Lov'asz `function. We deduce an improved approximation algorithm for the independence number. If a graph on n vertices has an independence number n=k + m, for some fixed integer k 3 and some ..."
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Cited by 31 (5 self)
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We study the relationship between the independence number of a graph and its semidefinite relaxation, the Lov'asz `function. We deduce an improved approximation algorithm for the independence number. If a graph on n vertices has an independence number n=k + m, for some fixed integer k 3 and some m ? 0, the algorithm finds, in random polynomial time, an independent set of size ~ \Omega\Gamma m 3=(k+1) ). This is the first improvement upon the Ramsey Theory based algorithm of Boppana and Halldorsson that finds an independent set of size\Omega\Gamma m 1=(k\Gamma1) ) in such a graph. The algorithm is based on semidefinite programming, some properties of the `function, and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the `function of an n vertex graph is at least Mn 1\Gamma2=h , for some absolute constant M , we describe another, related algorithm that finds an independent set of size h. Finally, while it is e...
Evolutionary algorithm with the guided mutation for the maximum clique problem
 IEEE Transactions on Evolutionary Computation
, 2005
"... Abstract—Estimation of distribution algorithms sample new solutions (offspring) from a probability model which characterizes the distribution of promising solutions in the search space at each generation. The location information of solutions found so far (i.e., the actual positions of these solutio ..."
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Cited by 30 (10 self)
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Abstract—Estimation of distribution algorithms sample new solutions (offspring) from a probability model which characterizes the distribution of promising solutions in the search space at each generation. The location information of solutions found so far (i.e., the actual positions of these solutions in the search space) is not directly used for generating offspring in most existing estimation of distribution algorithms. This paper introduces a new operator, called guided mutation. Guided mutation generates offspring through combination of global statistical information and the location information of solutions found so far. An evolutionary algorithm with guided mutation (EA/G) for the maximum clique problem is proposed in this paper. Besides guided mutation, EA/G adopts a strategy for searching different search areas in different search phases. Marchiori’s heuristic is applied to each new solution to produce a maximal clique in EA/G. Experimental results show that EA/G outperforms the heuristic genetic algorithm of Marchiori (the best evolutionary algorithm reported so far) and a MIMIC algorithm on DIMACS benchmark graphs. Index Terms—Estimation of distribution algorithms, evolutionary algorithm, guided mutation, heuristics, hybrid genetic algorithm, maximum clique problem (MCP). I.
The probable value of the LovaszSchrijver relaxations for maximum independent set
 SIAM Journal on Computing
, 2003
"... independent set ..."