Adaptive Local Ratio 3039

Local ratio is a well-known paradigm for designing approximation algorithms for combinatorial optimization problems. At a very high level, a local-ratio algorithm first decomposes the input weight function w into a positive linear combination of simpler weight functions or models. Guided by this process, a solution S is constructed such that S is α-approximate with respect to each model used in the decomposition. As a result, S is α-approximate under w as well. These models usually have a very simple structure that remains " unchanged " throughout the execution of the algorithm. In this work we show that adaptively choosing a model from a richer spectrum of functions can lead to a better local ratio. Indeed, by turning the search for a good model into an optimization problem of its own, we get improved approximations for a data migration problem. 1. Introduction. The local-ratio technique and the primal-dual schema are two well-known paradigms for designing approximation algorithms for combinatorial optimization problems. Over the years a clear connection between the two paradigms was observed as researchers found primal-dual interpretations for local-ratio algorithms [12, 3] and vice versa [8, 5]. This culminated with the work of Bar-Yehuda and Rawitz [10] showing their equivalence under a fairly general and encompassing definition of primal-dual and local-ratio algorithms. For a survey of results and a historical account of the local-ratio technique, see [7]; for a survey of the primal-dual schema, see [20, 30]. At a very high level, a local-ratio algorithm consists of two steps. First, the input weight function w is decomposed into a positive linear combination of modelsˆw i , that is, w = 1 ˆ w 1 + · · · + n ˆ w k and i ≥ 0. Then, guided by this process, a solution S is constructed such thatˆw i (S) ≤ α ˆ w i (A) for any feasible solution A for all i. We refer to α as the local ratio ofˆw i. By the local-ratio theorem [8], it follows that S is α-approximate with respect to w. Typically the models used in local-ratio approximation algorithms are 0-1 functions or simple aggregates of structural features of the problem at hand. (In primal-dual parlance this corresponds to increasing some dual variables uniformly when constructing the dual solution.) Furthermore, the structure of the models remains " unchanged " throughout the execution of the algorithm. For example, consider the vertex …