A survey on combinatorial optimization in dynamic environments

Abstract

This survey presents major results and issues related to the study of NPO problems in dynamic environments, that is, in settings where instances are allowed to undergo some modifications over time. In particular, the survey focuses on two complementary frameworks. The first one is the reoptimization framework, where an instance I that is already solved undergoes some local perturbation. The goal is then to make use of the information provided by the initial solution to compute a new solution. The second framework is probabilistic optimization, where the instance to optimize is not fully known at the time when a solution is to be proposed, but results from a determined Bernoulli process. Then, the goal is to compute a solution with optimal expected value.

Big data algorithms beyond machine learning

“…For further reading, we refer to the textbook of Cygan et al [15] which provides a thorough treatment of multivariate algorithms; the upcoming textbook by Fomin et al [24] will study data reduction by preprocessing in great detail; dynamic algorithms for discrete optimization problems are surveyed by Boria andPaschos [11]; external memory algorithms are covered by Vitter [55].…”

Robust Reoptimization of Steiner Trees

“…P 2 incident to t. We define, for i ∈ {1, 2} , p � i ∶= c(e i ) and Proof By definition of p, we conclude that opt new k ≥ 3p . Using (24), we conclude that (1 + ) ≥ 3( − 1 − ) . Therefore, by (1), we obtain the claimed result.…”

“…Since then, the concept of reoptimization has been investigated for several different problems, including the traveling salesman problem [1,5,7,15,16,30], the minimum latency problem [4,26], the rural postman problem [3], fast reoptimization of the spanning tree problem [23], the knapsack problem [2], covering problems [12], the shortest common superstring problem [10], maximum-weight induced hereditary problems [22], and scheduling [6,21,31]. There are several overviews on reoptimization [4,19,24,33].…”

Route planning in a weakly dynamic undirected graph

“…Several approaches have been developed [5]. Full dynamic algorithms usually deal with problems that can statically be solved in polynomial time.…”

“…Reoptimization algorithms use an initial solution, and are usually approximation algorithms which provide faster results than classical algorithms with the same guaranty on the approximation ratios, or which even guarantee better approximation ratios than classical static algorithms with the same speed performances [5] [10]. Others papers propose meta-heurisics, for example swarm intelligence algorithms, such as ant colony algorithms [2] [3].…”