Optimization problems with incomplete information

Abstract

This thesis is concerned with the design and analysis of algorithms for new variants of optimization problems where the problem instance is not completely known. Specifically, we consider two online problems where the problem instance is revealed over time, and one distributed problem involving many computational units, each of which can access only local information. We measure the performance of algorithms by the worst-case ratio between their objective values and the optimal objective value obtained by algorithms knowing the entire problem instance. Better algorithms have ratios closer to one. For online problems, this ratio is known as the competitive ratio. First, we study a class of generalized online scheduling problems, where the online Weighted Traveling Repairman Problem (WTRP) is a special case. For the online WTRP, we propose a family of parameterized deterministic and randomized online algorithms. For both deterministic and randomized cases, our competitive ratios are the smallest (best) in the literature. For the general setting, our online algorithms achieve similar competitive ratios. Second, we study a distributed version of the Multi-Depot Vehicle Routing Problem. In particular, we divide the space into smaller regions based on the depot configurations through a partition scheme, and assign each region to a vehicle. For partition schemes, we call the aforementioned worst-case ratio their Price of No-Communication (PoNC). We show that the Voronoi partition achieves a PoNC linear in the number of depots. In addition, for two special classes of depot configurations, we design partition schemes with PoNCs sub-linear in the number of depots. Third, we study quantity-based single-resource revenue management problems in a new parameterized online model, which is a combination of the worst-case and the random-order models. When there are only two classes of customers, we develop two online algorithms and show that they achieve the best-possible competitive ratio for a wide range of problem parameters. We also study two problem extensions. In the first extension, online algorithms can observe whether an arriving customer follows the adversarial arrival order. In the second extension, similar to the classical secretary problem, the goal of the online algorithms is to maximize the probability of selecting the highest-valued customer.