A Theory and Algorithms for Combinatorial Reoptimization

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 ρapproximation 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. ⋆

Adapting Stable Matchings to Evolving Preferences

“…Finally, let us briefly mention that the motivation for our incremental scenario for stable matching is related to similar scenarios in the context of clustering (Charikar et al 2004;Luo et al 2018), coloring (Hartung and Niedermeier 2013), other dynamic versions of parameterized problems (Abu-Khzam et al 2015;Krithika, Sahu, and Tale 2018), and reoptimization (Böckenhauer et al 2018;Schieber et al 2018).…”

Parameterized Dynamic Cluster Editing

“…Beyond parameterized algorithmics and static Cluster editing, dynamic clustering in general has been subject to many studies, mostly in applied computer science [12,16,17,[42][43][44]. We mention in passing that there are also close ties to reoptimization (e.g., [7,8,40]) and parameterized local search (e.g., [21,25,28,30,36]).…”

Approximation algorithms for process systems engineering

“…show that lexicographic optimization imposes optimal substructure for the makespan scheduling problem. Schieber et al (2018) present a framework designing reoptimization algorithms with analytically proven performance guarantees and present a family of fully polynomial-time reapproximation schemes.…”

“…Recovery strategies improve robust solutions by making second-stage decisions after the uncertainty is revealed (Liebchen et al, 2009). TCS approaches derive recovery methods attaining good trade-offs in terms of final solution quality and initial solution transformation cost (Ausiello et al, 2011;Schieber et al, 2018;Skutella and Verschae, 2016).…”