Optimal Stopping Meets Combinatorial Optimization

Abstract

Optimal stopping theory considers the design of online algorithms for stopping a random sequence subject to an optimization criterion. For example, the famous secretary problem asks to identify a stopping rule that maximizes the probability of selecting the maximum element in a sequence presented in uniformly random order. In a similar vein, the prophet inequality of Krengel, Sucheston, and Garling establishes the existence of an online algorithm for selecting one element from a sequence of independent random numbers, such that the expected value of the chosen element is at least half the expectation of the maximum.

A rich set of problems emerges when one combines these models with notions from combinatorial optimization by allowing the algorithm to select multiple elements from the sequence, subject to a combinatorial feasibility constraint on the set selected. A sequence of results during the past ten years have contributed greatly to our understanding of these problems. I will survey some of these developments and their applications to topics in algorithmic game theory.