Approximation algorithms for combinatorial optimization under uncertainty

Abstract

Combinatorial optimization problems arise in many fields of industry and technology, where they are frequently used in production planning, transportation, and communication network design. Whereas in the context of classical discrete optimization it is usually assumed that the problem inputs are known, in many real-world applications some of the data may be subject to an uncertainty, often because it represents information about the future. In the field of stochastic optimization uncertain parameters are usually represented as random variables that have known probability distributions. In this thesis we study a number of different scenarios of planning under uncertainty motivated by applications from robotics, communication network design and other areas. We develop approximation algorithms for several NP-hard stochastic combinatorial optimization problems in which the input is uncertain - modeled by probability distribution - and the goal is to design a solution in advance so as to minimize expected future costs or maximize expected future profits. We develop techniques for dealing with certain probabilistic cost functions making it possible to derive combinatorial properties of an optimum solution. This enables us to make connections with already well-studied combinatorial optimization problems and apply some of the tools developed for them. The first problem we consider is motivated by an application from AI, in which a mobile robot delivers packages to various locations. The goal is to design a route for robot to follow so as to maximize the value of packages successfully delivered subject to an uncertainty in the robot's lifetime.

(cont.) We model this problem as an extension of the well-studied Prize-Collecting Traveling Salesman problem, and develop a constant factor approximation algorithm for it, solving an open question along the way. Next we examine several classical combinatorial optimization problems such as bin-packing, vertex cover, and shortest path in the context of a "preplanning" framework, in which one can "plan ahead" based on limited information about the problem input, or "wait and see" until the entire input becomes known, albeit incurring additional expense. We study this time-information tradeoff, and show how to approximately optimize the choice of what to purchase in advance and what to defer. The last problem studied, called maybecast is concerned with designing a routing network under a probabilistic distribution of clients using locally available information. This problem can be modeled as a stochastic version of the Steiner tree problem. However probabilistic objective function turns it into an instance of a challenging optimization problem with concave costs.