Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems

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 over probabilistic graphs, 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). We show that we can obtain a polynomial time approximation algorithm with additive error ϵ for any ϵ >; 0, if there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above) and the maximum value of the utility function is bounded by a constant. Our result generalizes 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.

Authors

• Jian Li 0015
• Amol Deshpande

Keywords

• Approximation methods
• Approximation algorithms
• Polynomials
• Optimization
• Vectors
• Random variables
• Fourier series
• Optimization Problem
• Optimal Combination
• Stochastic Optimization
• Combinatorial Problem
• Combinatorial Optimization Problem
• Stochastic Problem
• Stochastic Optimization Problem
• Expected Value
• Exponential Function
• Shortest Path
• Class Of Problems
• Minimum Weight
• Utility Maximization
• Version Of Problem
• Weight Of Elements
• Risk-averse Behavior
• Running Time
• Path Length
• Feasible Solution
• Utility Maximization Problem
• Shortest Path Problem
• Deterministic Problem
• Sum Of Exponentials
• Conditions Hold
• Random Vector
• Exact Algorithm
• Polynomial-time Algorithm
• Estimation Strategy