Local ratio: A unified framework for approximation algorithms in memoriam: Shimon Even 1935-2004

Abstract

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.