Fault-tolerant <i>k</i> -Supplier with Outliers

We present approximation algorithms for the Fault-tolerant k -Supplier with Outliers ( F k SO ) problem. This is a common generalization of two known problems – k -Supplier with Outliers, and Fault-tolerant k -Supplier – each of which generalize the well-known k -Supplier problem. In the k -Supplier problem the goal is to serve n clients C , by opening k facilities from a set of possible facilities F ; the objective function is the farthest that any client must travel to access an open facility. In F k SO , each client v has a fault-tolerance ℓ v , and now desires ℓ v facilities to serve it; so each client v ’s contribution to the objective function is now its distance to the ℓ v th closest open facility. Furthermore, we are allowed to choose m clients that we will serve, and only those clients contribute to the objective function, while the remaining n − m are considered outliers. Our main result is a (4 t − 1)-approximation for the F k SO problem, where t is the number of distinct values of ℓ v that appear in the instance. At t = 1, i.e. in the case where the ℓ v ’s are uniformly some ℓ, this yields a 3-approximation, improving upon the 11-approximation given for the uniform case by Inamdar and Varadarajan [2020], who also introduced the problem. Our result for the uniform case matches tight 3-approximations that exist for k -Supplier, k -Supplier with Outliers, and Fault-tolerant k -Supplier. Our key technical contribution is an application of the round-or-cut schema to F k SO . Guided by an LP relaxation, we reduce to a simpler optimization problem, which we can solve to obtain distance bounds for the “round” step, and valid inequalities for the “cut” step. By varying how we reduce to the simpler problem, we get varying distance bounds – we include a variant that gives a (2 t + 1)-approximation, which is better for t ∈ {2, 3}. In addition, for t = 1, we give a more straightforward application of round-or-cut, yielding a 3-approximation that is much simpler than our general algorithm.