Approximation Algorithms for Robust Covering Problems with Chance Constraints

Two common approaches to model uncertainty in optimization problems are to either explicitly enumerate all the realizations of the uncertain parameters or to specify them implicitly by a probability distribution over the uncertain parameters. We study robust covering problems where the demand or the right hand side of the covering constraints is uncertain, in both models of uncertainty and introduce chance constraints in the formulation. We consider problems with only one stage of decision making as well as a two-stage problem where there is a second recourse stage to augment the first stage decisions. In a chance-constrained model, we are given a reliability parameter ℘ (in addition to the other inputs as in the robust problem) and the goal is to construct a solution that satisfies at least a ℘ fraction of the scenarios; the robust objective implies that the worst case cost over these scenarios is to be minimized. Here the parameter ℘ achieves a trade-off between the reliability and the cost of the solution; when ℘ = 1, we arrive back at the classical robust model. This model can thus be thought of as a method for automatic detection of outlier scenarios. As an example, given ℘ = k/n in an n-node graph, the one-stage chance constrained shortest path problem from a root with every node as a singleton scenario occurring with probability 1/n is to find a rooted k-MST. While it is easy to obtain bi-criteria approximation algorithms for the chance-constrained problems that violate the chance-constraint by a small factor, we consider the problem of satisfying the chance constraint strictly. We show that in the explicit scenario model (with more than one element in all the scenarios), both one-stage and two-stage problems are at least as hard to approximate as the dense k-subgraph (DkS) problem. The hardness arises particularly due to the requirement that the chance constraint be satisfied strictly and does not hold if we relax that condition. For the special case when each scenario has a single element, while the one-stage problem directly reduces to a weighted partial covering problem, we show that many two-stage problems (including set cover, facility location etc) reduce to a weighted partial covering problem via a guess and prune method. We consider the two-stage shortest path problem which can not be reduced to a partial covering version and is closely related to the weighted k-MST problem where the weight function is submodular. We give an O(log k)-approximation for this problem using a charging argument. We also consider the model of uncertainty where scenarios (possibly an exponential number) are specified implicitly by a probability distribution. In particular, we consider an independent distribution where each demand occurs with a given probability independently of others. While it is not even clear if we can succinctly specify the ℘ fraction of selected scenarios for the two-stage problem, we show that the one-stage problem in this model can be reduced to a weighted partial covering problem. We also extend these results to obtain a logarithmic approximation for the one-stage problem when the demand uncertainty is specified by a general probability distribution such that the cumulative probability of any demand-scenario can be computed efficiently and is strictly-monotone with respect to set inclusion.