Augmenting Ordered $k$-Median with Fault-Tolerance and Robustness

We study fault-tolerant ordered $k$-median and robust ordered $k$-median, both as generalizations of the ordered $k$-median problem. In these problems, we are often given a metric space, and asked to open a set of at most $k$ facilities and provide a certain assignment of these facilities to a set of clients, in order to minimize the ordered weighted sum of the induced service costs from all facility-client assignments. In the fault-tolerant problem, every client $j$ has a requirement $r_j$ and needs to be assigned $r_j$ distinct facilities. The cost of client $j$ is the sum of distances to its assigned facilities. In the robust problem, a parameter $m$ is given and we need to assign some facility to at least $m$ clients. We give polynomial-time constant-factor approximation algorithms for both problems, which use standard sparsification as well as iterative rounding techniques for LP relaxations. We also consider ordered knapsack median and ordered matroid median, and use the iterative rounding framework to obtain constant-factor approximations as well.