Hardness and approximation of the Probabilistic $p$-Center problem under Pressure

The Probabilistic $p$-Center problem under Pressure ({\tt Min P$p$CP}) is a variant of the usual {\tt Min $p$-Center} problem we recently introduced in the context of wildfire management. The problem is %basically to locate $p$ shelters minimizing the maximum distance people will have to cover %in order to reach one of these shelters to reach the closest accessible shelter in case of fire. The landscape is divided in zones and is modeled as an edge-weighted graph with vertices corresponding to zones and edges corresponding to direct connections between two adjacent zones. The uncertainty associated with fire outbreaks is modeled using a finite set of fire scenarios. Each scenario %defines corresponds to a fire outbreak on a single zone (i.e., on a vertex) with the main consequence of modifying evacuation paths in two ways. First, an evacuation path cannot pass through the vertex on fire. Second, the fact that %somebody someone close to the fire may not take rational decisions when selecting a direction to escape is modeled using new kinds of evacuation paths. In this paper, for a given instance of {\tt Min P$p$CP} defined by an edge-weighted graph $G=(V,E,L)$ and an integer $p$, we characterize the set of feasible solutions of {\tt Min P$p$CP}. We prove that {\tt Min P$p$CP} cannot be approximated with a ratio less than $\frac{56}{55}$ on subgrids (subgraphs of grids) of degree at most 3. Then, we propose some approximation results for {\tt Min P$p$CP}. These results require approximation results for two variants of the (deterministic) {\tt Min $p$-Center} problem called {\tt Min MAC $p$-Center} and {\tt Min Partial $p$-Center}.