We consider requests for capacity in a given tree network T = ( V , E ) where each edge e of the tree has some integer capacity u e . Each request f is a node pair with 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 provides 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. Garg et al. [1997] 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. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with 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 by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed.

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The celebrated Hajnal-Szemerédi Theorem states: For every positive integer r, every graph with maximum degree at most r has an equitable coloring with r + 1 colors. We show that this coloring can be obtained in O(rn 2 ) time, where n is the number of vertices.

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Let q be a positve integer, and G be a q-partite simple graph on qn vertices, with n vertices in each vertex class. Let δ = kq kq +1 , where k q = q + O(log q). If each vertex of G is adjacent to at least δn vertices in each of the other vertex classes, q is bounded and n is large enough, then G has a K q -factor.

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f a labeling of its vertices with positive integers; denote by S(v) the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering if S(u) < S(v) for each arc (u, v) of the digraph. The problem asks to find the minimum number k for which D has a topological additive numbering with labels belonging to {1, . . . , k}, denoted by η t (D).We characterize when a digraph has topological additive numberings, give a lower bound for η t (D), and provide an integer programming formulation for our problem, characterizing when its coefficient matrix is totally unimodular. We also present some families for which η t (D) can be computed in polynomial time. Finally, we prove that this problem is N P-Hard even when its input is restricted to planar bipartite digraphs.