Summary: We consider requests for capacity in a given tree network \(T=(V,E)\) where each edge of the tree has some integer capacity \(u_{e}\). Each request consists of an integer demand \(d_{f}\) and a profit \(w_{f}\) which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provide a maximum profit. This generalizes well-known problems including the knapsack and \(b\)-matching problems.
When all demands are \(1\), we have the integer multicommodity flow problem. N. Garg, V. Vazirani, and M. Yannakakis [Algorithmica 18, 3–20 (1997; Zbl 0873.68075)] had shown that this problem is NP-hard and gave a \(2\)-approximation algorithm for the cardinality case (all profits are \(1\)) via a primal-dual algorithm. In this paper we establish for the first time that the natural linear programming relaxation has a constant factor gap, a factor of \(4\), for the case of arbitrary profits.
We then discuss the situation for arbitrary demands. When the maximum demand \(d_{\max}\) is at most the minimum edge capacity \(u_{\min}\), we show that the integrality gap of the LP is at most \(48\). This result is obtained showing that the integrality gap for demand version of such a problem is at most \(12\) times that for the unit demand case. We use techniques of Kolliopoulos and Stein to obtain this. We also obtain, via this method, improved algorithms for the line and ring networks. Applications and connections to other combinatorial problems are discussed.
For the entire collection see [Zbl 1029.00041].