LocalRatioApproximationAlgorithms

# LocalRatioApproximationAlgorithms - Local Ratio A Unied...

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Local Ratio: A Unified Framework for Approximation Algorithms In Memoriam : Shimon Even 1935–2004 REUVEN BAR-YEHUDA AND KEREN BENDEL Technion (Israel Institute of Technology) ARI FREUND IBM Haifa Research Lab AND DROR RAWITZ Tel-Aviv University The local ratio technique is a methodology for the design and analysis of algorithms for a broad range of optimization problems. The technique is remarkably simple and elegant, and yet can be applied to several classical and fundamental problems (including covering problems, packing problems, and scheduling problems). The local ratio technique uses elementary math and requires combinatorial insight into the structure and properties of the problem at hand. Typically, when using the technique, one has to invent a weight function for a problem instance under which every “reasonable” solution is “good.” The local ratio technique is closely related to the primal-dual schema, though it is not based on weak LP duality (which is the basis of the primal-dual approach) since it is not based on linear programming. In this survey we, introduce the local ratio technique and demonstrate its use in the design and analysis of algorithms for various problems. We trace the evolution path of the technique since its inception in the 1980’s, culminating with the most recent development, namely, fractional local ratio , which can be viewed as a new LP rounding technique. Categories and Subject Descriptors: F.2.2 [ Analysis of Algorithms and Problem Complexity ]: Nonnumerical Algorithms and Problems— Computations on discrete structures ; Sequencing and scheduling ; G.2.1 [ Discrete Mathematics ]: Combinatorics— Combinatorial algorithms General Terms: Algorithms, Performance, Theory Additional Key Words and Phrases: Approximation algorithms, local ratio technique, fractional local ratio Most of the work on this survey was done while Ari Freund and Dror Rawitz were at the Technion Depart- ment of Computer Science, and part of it while Ari Freund was at the Caesarea Rothschild Institute, Haifa University. Authors’ addresses: R. Bar-Yehuda and K. Bendel, Department of Computer Science, Technion, Haifa 32000, Israel; email: { reuven,bkeren } @cs.technion.ac.il; A. Freund, IBM Haifa Research Lab, Haifa University Cam- pus, Haifa 31905, Israel; email: [email protected]; D. Rawitz, School of Electrical Engineering, Tel-Aviv Uni- versity, Tel-Aviv 69978, Israel; email: [email protected] Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications

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