We study the stochastic versions of a broad class of combinatorial problems where 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, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk-averse or risk-prone behaviors, and instead 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 (expected weight becomes a special case). Under the assumption that there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above), 1 we can obtain the following approximation results for several important classes of utility functions: 1. If the utility function µ is continuous, upper-bounded by a constant and lim x→+∞ µ(x) = 0, we show that we can obtain a polynomial time approximation algorithm with an additive error for any constant > 0. 2. If the utility function µ is a concave increasing function, we can obtain a polynomial time approximation scheme (PTAS). 3. If the utility function µ is increasing and has a bounded derivative, we can obtain a polynomial time approximation scheme. Our results recover or generalize several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. 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.
For general discrete distributions, Li and Deshpande REF obtained an additive PTAS if the deterministic version of the problem can be solved exactly in pseudopolynomial time.
Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems
Mathematics Computer Science