Combinatorial Algorithms for Capacitated Network Design

We focus on designing combinatorial algorithms for the CAPACITATED NETWORK DESIGN problem (CAPSNDP). The CAP-SNDP is the problem of satisfying connectivity requirements when edges have costs and hard capacities. We begin by showing that the GROUP STEINER TREE is a special case of CAP-SNDP even when there is connectivity requirement between only one source-sink pair. This implies the first poly-logarithmic lower bound for the CAP-SNDP. We next provide combinatorial algorithms for several special cases of this problem. The CAP-SNDP is equivalent to its special case where every edge has either zero cost or infinite capacity. We consider a special case, called CONNECTED CAP-SNDP, where all infinite-capacity edges in the solution are required to form a connected component containing the sinks. This problem is motivated by its similarity to the Connected Facility Location problem [20, 31]. We solve this problem by reducing it to SUBMODULAR TREE COVER, which is a common generalization of CONNECTED CAP-SNDP and GROUP STEINER TREE. We generalize the recursive greedy algorithm [10] achieving a poly-logarithmic approximation algorithm for SUBMODULAR TREE COVER. This result is interesting in its own right and gives the first poly-logarithmic approximation algorithms for Connected hard capacities set multi-cover and Connected source location. We then study another special case of CAP-SNDP called UNBALANCED-P2P. Besides its practical applications to shift design problems [13], it generalizes many problems such as k-MST, Steiner Forest and Point-to-Point Connection. We give a combinatorial logarithmic approximation algorithm for this problem by reducing it to degree-bounded SNDP. ∗University of Maryland, College Park, MD 20742, U.S.A., E-mail: [email protected]. The author is also with AT&T Labs– Research, Florham Park, NJ 07932, U.S.A. †IBM T.J. Watson research center, Hawthorne, NY 10532, U.S.A., E-mail: [email protected]. ‡Rutgers University–Camden, NJ 08102, U.S.A., E-mail: [email protected]. Research partially supported by NSF grant 0819959. §The Open University of Israel. Raanana, Israel. E-mail:[email protected]. ar X iv :1 10 8. 11 76 v1 [ cs .D S] 4 A ug 2 01 1