We study the approximability of two classes of network routing problems. The first class of problems in our study corre spend to classical multicommodity flow problems of the following form: We are given a network G with integer capacities on its edges, together with source-sink pairs (a, ti), 1 5 i 2 k, such that a positive integer demand di and a positive "profit" t'i is associated with eah pair. A feasible solution is a subset S of the (sir ti) pairs such that demands associated with pairs in S can be fully met through a routing which respects all capacity constraints, and the objective is to maximize the total profit associated with the satisfied pairs. We consider two natural variants: unsplittable flow (USF) where each pair must be satisfied by routing all its demand on a single path, and integml splittable pow (ISF) where the flow satisfying a demand can be split in several paths, each carrying an integral amount of flow. In the special case when all demands, profits, and capacities equal one, both these variants reduce to the classical NP-hard edge-disjoint paths problem (referred to as EDP). An O(&ii) approximation is known for EDP 1151 and has recently been obtained for the more general USF problem [l] under the assumption that d,, 5 urnin, where m denotes the number of edges, d,.. denotes the maximum demand and urnin denotes the minimum edge capacity. While it has been generally believed that these problems are very hard to approximate, only MAX SNP-hardness was known thus far. We prove here the tight result that in directed graphs, for any e > 0, EDP (and hence also USF), is NProximate within m'/'--'. We also give a simple og log m)-approximation algorithm for USF under the arsumption that d,., 5 urnin, and an O(JTiilog3" m) approximation when the only assumption is that d,, is polynomially bounded. Our algocapyn8ht ACM 1999 l-581 13.067.8199105...%5.00 rithms also turn out to be much simpler than the existing ones. The hardness result for EDP trivially implies an identical result for approximating ISF on directed networks. On the algorithmic side, we give a simple greedy algorithm that gives a O(~log2 m)-approximation for ISF. The second class of routing problems in our study is another well known one: the goal here is to find a maximum number of length bounded edge-disjoint paths between given source-sink pairs in a graph G. We refer to this problem as the bounded length edge-disjoint paths (BLEDP) problem. We show that, for any e > 0, BLEDP is hard to approximate within mll'--l even on undirected graphs. On the algorithmic side, we give an O(&)-approximation algorithm for BLEDP. We also consider a special case of BLEDP, which we refer to as (8, t)-BLEDP, in which there is only one sourcesink pair. Without the length restriction, this problem reduces to the polynomial-time solvable max-flow problem. However, with the length restriction, we show that this problem is hard to approximate within m'/'-', for any L > 0, on directed graphs; for undirected graphs we show MAX SNPhardness even when the length...

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We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (s i ,t i), 1 ≤ i ≤ k, and the goal is to find a largest subset of source-sink pairs that can be simultanoeusly connected in an edge-disjoint manner. We show that in directed networks, for any ε > 0, EDP is NP-hard to approximate within m 1/2-ε. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ε > 0, bounded length EDP is hard to approximate within m 1/2-ε even in undirected networks, and give an O(√m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. find the maximum number of paths of bounded length between a given source-sink pair) is hard to approximate within m 1/2-ε , for any ε > 0.

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The kdomination number of a graph G, y k ( G ) , is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k. thenis the smallest cardinality of a k-dominating set of G. The theory of k-domination in graphs has been extensively studied by Fink and Jacobson [l]. Our contribution is to prove the following result. Proof. Suppose that the result is false and that G is a counterexample having the least number of edges. If r = p -yk(G) then, by assumption, and we deduce x ( G ) > kr. Notice that for any independent set U of vertices of G, V -U is k-dominating.By minhality of G, the set S = {u E VI d(u) > k} is independent (or empty).Let T be a maximal independent set containing S. Then V -T is k-dominating and