A Theory and Algorithms for Combinatorial Reoptimization

Abstract

Abstract. Many real-life applications involve systems that change dynamically over time. Thus, throughout the continuous operation of such a system, it is required to compute solutions for new problem instances, derived from previous instances. Since the transition from one solution to another incurs some cost, a natural goal is to have the solution for the new instance close to the original one (under a certain distance measure). In this paper we develop a general model for combinatorial reoptimization, encompassing classical objective functions as well as the goal of minimizing the transition cost from one solution to the other. Formally, we say that A is an (r, ρ)-reapproximation algorithm if it achieves a ρ-approximation for the optimization problem, while paying a transition cost that is at most r times the minimum required for solving the problem optimally. When ρ = 1 we get an (r, 1)-reoptimization algorithm. Using our model we derive reoptimization and reapproximation algorithms for several important classes of optimization problems. This includes fully polynomial time reapproximation schemes for DP-benevolent problems, a class introduced by Woeginger (Proc. Tenth ACM-SIAM Symposium on Discrete Algorithms, 1999 ), reapproximation algorithms for metric Facility Location problems, and (1, 1)-reoptimization algorithm for polynomially solvable subset-selection problems. Thus, we distinguish here for the first time between classes of reoptimization problems, by their hardness status with respect to minimizing transition costs while guaranteeing a good approximation for the underlying optimization problem.

Fixing improper colorings of graphs

“…TSP (see Ausiello et al [1]), Shortest Common Superstring (see Bilò et al [2]) or Minimum Steiner Tree (see Zych and Bilò [28]). We also refer the reader to the paper of Shachnai et al [27], where the authors describe a general model for combinatorial reoptimization.…”

Cost-Oblivious Reallocation for Scheduling and Planning

“…Shachnai et al [29] explore a form of reallocation for combinatorial optimization. Given an input, an optimal solution for that input, and a modified version of the input, they develop algorithms that find the minimum-cost modification of the optimal solution to the modified input.…”

Introduction to Reconfiguration

“…Work on related problems was unified into a common framework in 2011 [4], leading to a widening interest in the area and the types of problems and approaches considered. Reconfiguration is distinct from other approaches to the solution space of problems, such as local search [5][6][7][8] (given a solution to an instance, find a better solution that is close to the input solution), reoptimization [9,10] (given an instance, an optimal solution, and changes to the instance, find an optimal solution to the changed instance), and incremental problems [11] (given a yes-instance, a witness that it is a yes-instance, and changes to the instance, determine if the changed instance is a yes-instance). The reconfiguration framework is defined in terms of a source problem, an instance of the source problem, a definition of a feasible solution, and a definition of adjacency of feasible solutions.…”