Degree-Constrained Network Flows

A d-furcated flow is a network flow whose support graph has maximum outdegree d. Take a single-sink multicommodity How problem on any network and with any set of routing demands. Then we show that; the existence of feasible fractional flow with node congestion one implies the existence of a d-furcated How with congestion at most :1+1/d-1, for d >= 2. This result is tight, and so the congestion gap for d-furcated flows is bounded and exactly equal to 1 + 1/d-1, for the case d = 1. (confluent flows), it is known that the congestion gap is unbounded, namely circle minus(logn). Thus, allowing single-sink multicommodity network flows to increase their maximum outdegree from one to two virtually eliminates this previously observed congestion gap. As a corollary we obtain a factor 1 + 1/d-1-approximation algorithm for the problem of finding a minimum congestion d-furcated flow; we also prove that this problem is maxSNP-hard. Using known techniques these results also extend to degree-constrained unsplittable routing, where each individual demand must be routed along a unique path.