Utility Maximization under Uncertainty

Motivated by several search and optimization problems over uncertain datasets, we study the stochastic versions of a broad class of combinatorial problems where either the existences or the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs; top-k queries over probabilistic datasets; and other combinatorial problems like knapsack. By noticing that the expected value is inadequate in capturing different types of risk-averse or risk-prone behaviors, we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (which becomes a special case). For weight uncertainty model, we show that we can obtain a polynomial time approximation algorithm with additive error for any > 0, if there is a pseudopolynomial time algorithm for the exact version of the problem1 (This is true for the problems mentioned above). Our result generalizes several prior works on stochastic shortest path and stochastic knapsack. Then we consider a special case, the expected weight minimization with penalty (EWMP) problem in the element uncertainty model, where each element has a fixed weight but its existence is uncertain. In this problem, the objective is to minimize the expected value of the cost which is the weight of the solution if all elements in the chosen solution are present, or to a fixed penalty otherwise. We show that the problem is NP-hard and present a polynomial time approximation scheme (PTAS) provided we can compute an approximate Pareto curve for the problem in polynomial time. This implies PTASs for the EWMP version of shortest path, matching, and spanning tree (or more generally, matroid base). Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.