Dependent rounding and its applications to approximation algorithms

Gandhi, Rajiv; Khuller, Samir; Parthasarathy, Srinivasan; Srinivasan, Aravind

doi:10.1145/1147954.1147956

Cited by 224 publications

(279 citation statements)

References 33 publications

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“…The literature in broadcast scheduling is vast and many variations of the problem have been studied (see [2,4] and references therein). In the variant we are concerned with here, a client request is specified by a time window I and a data type A.…”

Section: Previous and Related Workmentioning

confidence: 99%

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Approximating the Interval Constrained Coloring Problem

Althaus¹,

Canzar²,

Elbassioni³

et al. 2008

Algorithm Theory – SWAT 2008

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We consider the interval constrained coloring problem, which appears in the interpretation of experimental data in biochemistry. Monitoring hydrogen-deuterium exchange rates via mass spectroscopy experiments is a method used in that field to obtain information about protein tertiary structure. The output of these experiments provides data about the exchange rate of residues in overlapping fragments of the protein backbone. These fragments must be re-assembled in order to obtain a global picture of the protein structure. The interval constrained coloring problem is the mathematical abstraction of this re-assembly process.The objective of the interval constrained coloring problem is to assign a color (exchange rate) to a set of integers (protein residues) such that a set of constraints is satisfied. Each constraint is made up of a closed interval (protein fragment) and requirements on the number of elements that belong to each color class (exchange rates observed in the experiments).We show that the problem is NP-complete for arbitrary number of colors and we provide algorithms that given a feasible instance find a coloring that satisfies all the coloring requirements within ±1 of the prescribed value. In light of our first result, this is essentially the best one can hope for. Our approach is based on polyhedral theory techniques and randomized rounding. Furthermore, we develop a quasi-polynomial-time approximation scheme for a variant of our problem where we are asked to find a coloring satisfying as many as fragments as possible.

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Section: Previous and Related Workmentioning

confidence: 99%

“…Let x be a fractional solution in P. We use the scheme of Gandhi et al [4] to round x to an integral solutionx with the following properties: (P3) Every I ∈ I is satisfied with probability greater or equal than…”

Section: A ±1 Guaranteementioning

confidence: 99%

Approximating the Interval Constrained Coloring Problem

Althaus¹,

Canzar²,

Elbassioni³

et al. 2008

Algorithm Theory – SWAT 2008

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show abstract

“…We thus seek to round this fractional matching to an integral matching by preserving marginal probabilities, a well-studied setting discussed e.g. in Gandhi et al (2006), Chekuri et al (2010). A conceptually simple rounding technique would be to use a constructive version of Carathéodory's theorem to express the fractional matching as a convex combination of integral matchings, and then pick randomly one of the integral matchings with probability equal to the corresponding coefficient in the convex combination.…”

Section: An Adaptive Lp-based Routing Approachmentioning

confidence: 99%

“…However, we use the specialized technique as presented e.g. in Gandhi et al (2006), since it provides a very efficient way to round the fractional matching by doing simple local rounding steps on fractional cycles. After having obtained the locations u π for π ∈ Π through the rounding, we send pod π to location u π between time step τ and τ + 1.…”

Section: An Adaptive Lp-based Routing Approachmentioning

confidence: 99%

An adaptive routing approach for personal rapid transit

Schüpbach

Zenklusen

2012

Math Meth Oper Res

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Personal Rapid Transit (PRT) is a public transportation mode, in which small automated vehicles transport passengers on demand. Central control of the vehicles leads to interesting possibilities for optimized routings. The complexity of the involved routing problems together with the fact that routing algorithms for PRT essentially have to run in real-time often leads to the choice of fast greedy approaches. The most common routing approach is arguably a sequential one, where upcoming requests are greedily served in a quickest way without interfering with previously routed vehicles. The simplicity of this approach stems from the fact that a chosen route is never changed later. This is as well the main drawback of it, potentially leading to large detours. It is natural to ask how much one could gain by using a more adaptive routing strategy. This question is the main motivation of this article. In this paper, we first suggest a simple mathematical model for PRT, and then introduce a new adaptive routing algorithm that repeatedly uses solutions to an LP as a guide to route vehicles. Our routing approach incorporates new requests in the LP as soon as they appear, and reoptimizes the routing of all currently used vehicles, contrary to sequential routing. We provide preliminary computational results that give first evidence of the potential gains of an adaptive routing strategy, as used in our algorithm.

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“…Besides the edge coloring step, the key difference from the algorithm of the previous subsection is in the choice of E. For this we use the GKSP procedure of Gandhi et al [11], which we describe next.…”

Section: Weighted Stochastic Matching: Bipartite Graphsmentioning

confidence: 99%

When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

Bansal

Gupta

et al. 2011

Algorithmica

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Consider a random graph model where each possible edge e is present independently with some probability p e . Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most t i times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following:• We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09].• Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.46 approximation for unweighted stochastic matching on general graphs.• We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.

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Dependent rounding and its applications to approximation algorithms

Cited by 224 publications

References 33 publications

Approximating the Interval Constrained Coloring Problem

Approximating the Interval Constrained Coloring Problem

An adaptive routing approach for personal rapid transit

When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

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