2 research outputs found
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, and other combinatorial problems like
knapsack. We observe that the expected value is inadequate in capturing
different types of {\em risk-averse} or {\em risk-prone} behaviors, and instead
we consider a more general objective which is to maximize the {\em 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 {\em exact} version of
the problem (This is true for the problems mentioned above), we can obtain the
following approximation results for several important classes of utility
functions: (1) If the utility function \uti is continuous, upper-bounded by a
constant and \lim_{x\rightarrow+\infty}\uti(x)=0, we show that we can obtain
a polynomial time approximation algorithm with an {\em additive error}
for any constant . (2) If the utility function \uti is
a concave increasing function, we can obtain a polynomial time approximation
scheme (PTAS). (3) If the utility function \uti 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.Comment: 31 pages, Preliminary version appears in the Proceeding of the 52nd
Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011), This
version contains several new results ( results (2) and (3) in the abstract