On the Power of Static Assignment Policies for Robust Facility Location Problems

TL;DR

This paper demonstrates that for a two-stage robust facility location problem with uncertain demand, a static assignment policy can achieve near-optimal solutions with an approximation ratio of O(log k / log log k), and such policies can be efficiently computed.

Contribution

The paper introduces a novel analysis showing static assignment policies are nearly optimal for a robust facility location problem with exponential demand scenarios, providing efficient computation methods.

Findings

Static policies achieve O(log k / log log k)-approximation.

Efficient algorithms compute near-optimal static assignments.

The model handles exponential demand scenarios with fixed assignment decisions.

Abstract

We consider a two-stage robust facility location problem on a metric under an uncertain demand. The decision-maker needs to decide on the (integral) units of supply for each facility in the first stage to satisfy an uncertain second-stage demand, such that the sum of first stage supply cost and the worst-case cost of satisfying the second-stage demand over all scenarios is minimized. The second-stage decisions are only assignment decisions without the possibility of adding recourse supply capacity. This makes our model different from existing work on two-stage robust facility location and set covering problems. We consider an implicit model of uncertainty with an exponential number of demand scenarios specified by an upper bound $k$ on the number of second-stage clients. In an optimal solution, the second-stage assignment decisions depend on the scenario; surprisingly, we show that restricting to a fixed (static) fractional assignment for each potential client irrespective of the scenario gives us an $O(\log k/\log \log k)$-approximation for the problem. Moreover, the best such static assignment can be computed efficiently giving us the desired guarantee.