Combinatorial Algorithms for Capacitated Network Design

TL;DR

This paper develops combinatorial algorithms for the Capacitated Network Design problem, establishing lower bounds and providing poly-logarithmic approximation algorithms for special cases like Connected Cap-SNDP and Unbalanced point-to-point connection.

Contribution

It introduces the first poly-logarithmic approximation algorithms for several special cases of Cap-SNDP, including Connected Cap-SNDP and Unbalanced point-to-point connection, and establishes lower bounds via reduction from Group Steiner tree.

Findings

First poly-logarithmic lower bound for Cap-SNDP.

Poly-logarithmic approximation for Connected Cap-SNDP via reduction to Submodular tree cover.

Logarithmic approximation for Unbalanced point-to-point connection problem.

Abstract

We focus on designing combinatorial algorithms for the Capacitated Network Design problem (Cap-SNDP). 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 problem (GST) 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 when 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 [G+01,SW04]. We solve this problem by reducing it to Submodular tree cover problem, which is a common generalization of Connected Cap-SNDP and Group Steiner tree problem. We generalize the recursive greedy algorithm [CEK] achieving a poly-logarithmic approximation algorithm for Submodular tree cover problem. 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 point-to-point connection problem. Besides its practical applications to shift design problems [EKS], 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.