A Theory and Algorithms for Combinatorial Reoptimization

• Published in: LATIN 2012: Theoretical Informatics pp. 618 - 630
• Main Authors: Shachnai, Hadas, Tamir, Gal, Tamir, Tami
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Facility Location Problem; Knapsack Problem; Multiobjective Optimization; Polynomial Time; Travel Salesman Problem
• ISBN: 3642293433; 9783642293436
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-29344-3_52

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}$\end{document} 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.