Approximation Algorithms for Reliable Stochastic Combinatorial Optimization

Abstract

We consider optimization problems that can be formulated as minimizing the cost of a feasible solution w T x over an arbitrary combinatorial feasible set F ⊂ {0, 1} n . For these problems we describe a broad class of corresponding stochastic problems where the cost vector W has independent random components, unknown at the time of solution. A natural and important objective that incorporates risk in this stochastic setting is to look for a feasible solution whose stochastic cost has a small tail or a small convex combination of mean and standard deviation. Our models can be equivalently reformulated as nonconvex programs for which no efficient algorithms are known. In this paper, we make progress on these hard problems.Our results are several efficient general-purpose approximation schemes. They use as a black-box (exact or approximate) the solution to the underlying deterministic problem and thus immediately apply to arbitrary combinatorial problems. For example, from an available δ-approximation algorithm to the linear problem, we construct a δ(1 + ǫ)-approximation algorithm for the stochastic problem, which invokes the linear algorithm only a logarithmic number of times in the problem input (and polynomial in 1 ǫ ), for any desired accuracy level ǫ > 0. The algorithms are based on a geometric analysis of the curvature and approximability of the nonlinear level sets of the objective functions.

Practical Route Planning Under Delay Uncertainty: Stochastic Shortest Path Queries

“…In this section, we provide a brief overview of the stochastic shortest-paths problem and known results, extracted from [18].…”

“…This is a non-convex combinatorial problem, which can be solved exactly by enumerating all paths that minimize some positive combination of mean and variance (the latter is a deterministic shortest-path problem with respect to edge lengths equal to the corresponding mean-variance linear combination) and selecting the one with minimal objective (2) (see [18,20] for more details). The number of deterministic shortest-path iterations is at most n O(log n) in the worst case [20].…”

“…This model assumes normally-distributed edge delays in order to obtain a closed-form expression for the probability tail. In particular, when w is Gaussian, the problem is transformed into the following formulation [18]:…”

“…Our Contributions: The algorithm in this paper improves the state of the art on stochastic path planning [18,19,20] and its application to traffic routing [15]. The running time of the state-of-the-art stochastic-shortest-path algorithm is approximately quadratic in the number of nodes, rendering it too slow for real-time performance on city-scale networks.…”

“…We improve upon their running time by using efficient data structures. In contrast to previous work [18], we provide a two-phase algorithm, consisting of preprocessing and query algorithms, better suited for practical implementations. Our preprocessing algorithm runs in quasi-linear time and it does not depend on the origin-destination pair or on the deadline, which is the main theoretical challenge.…”