Multicommodity facility location under group Steiner access cost

• Authors: Laura J. Poplawski;Rajmohan Rajaraman
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

Motivated by publish-subscribe mechanisms in networks, we introduce a new class of multicommodity facility location problems: Multicommodity Group Steiner Facility Location (MGSFL). The input to MGSFL consists of a metric space over a given set of locations, a cost function which provides a building cost for each commodity at each location, a set of clients located at various points in the metric, and the set of commodities that each client is interested in reaching. A solution to MGSFL consists of (a) for each commodity, the locations where facilities are built, and (b) for each client, a tree connecting the client to at least one facility for each commodity in its interest set. The goal is to minimize the sum of the total facility building costs and the metric cost of the client trees. MGSFL is a natural generalization of the well-studied Group Steiner Tree problem, which is equivalent to the special case of MGSFL in which every building cost is either 0 or ∞ and there is only one client. We also note that given the facility locations, the best client tree is an optimal solution to an appropriate Group Steiner Tree instance. Since the Group Steiner Tree problem is hard to approximate to within a factor of Ω(log2−∈ m) times optimum unless NP has quasi-polynomial Las Vegas algorithms, where m is the number of commodities, the same hardness result immediately extends to MGSFL. Our main result is a randomized 2O(√log n log log n)-approximation algorithm for MGSFL, where n is the number of clients. We also present deterministic poly-logarithmic approximations for three special cases. We give an O(log n)-approximation algorithm when the facility building costs differ only by commodity, not by location. We present an O(log4 n log m)-approximation algorithm when the interest sets are laminar --- i.e., for each pair of clients, either their interest sets do not intersect or else one client's interest set is contained within the other client's interest set. We end with an O(log n)-approximation algorithm when there are no building costs but each commodity must be built exactly once.