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Abstract:
In the edge(vertex)disjoint path problem we are given a graph $G$ and a set ${\cal T}$ of connection requests. Every connection request in ${\cal T}$ is a vertex pair $(s_i,t_i),$ $1 \leq i \leq K.$ The objective is to connect a maximum number of the pairs via edge(vertex)disjoint paths. The edgedisjoint path problem can be generalized to the multiplesource unsplittable flow problem where connection request $i$ has a demand $\rho_i$ and every edge $e$ a capacity $u_e.$ All these problems are NPhard and have a multitude of applications in areas such as routing, scheduling and bin packing.
Given the hardness of the problem, we study polynomialtime approximation algorithms. In this context, a $\rho$approximation algorithm is able to route at least a $1/\rho$ fraction of the connection requests. Although the edge and vertexdisjoint path problems, and more recently the unsplittable flow generalization, have been extensively studied, they remain notoriously hard to approximate with a bounded performance guarantee. For example, even for the simple edgedisjoint path problem, no $o(\sqrt{E})$approximation algorithm is known. Moreover some of the best existing approximation ratios are obtained through sophisticated and nonstandard randomized rounding schemes.
In this paper we introduce techniques which yield algorithms for a wide range of disjointpath and unsplittable flow problems. For the general unsplittable flow problem, even with weights on the commodities, our techniques lead to the first approximation algorithm and obtain an approximation ratio that matches, to within logarithmic factors, the $O(\sqrt{E})$ approximation ratio for the simple edgedisjoint path problem. In addition to this result and to improved bounds for several disjointpath problems, our techniques simplify and unify the derivation of many existing approximation results.
We use two basic techniques. First, we propose simple greedy algorithms for edge and vertexdisjoint paths and second, we propose the use of a framework based on packing integer programs for more general problems such as unsplittable flow. A packing integer program is of the form maximize $c^{T}\cdot x,$ subject to $Ax \leq b,$ $A,b,c \geq 0.$ As part of our tools we develop improved approximation algorithms for a class of packing integer programs, a result that we believe is of independent interest.
Note:
Revised November 1997.
Bibliographic citation for this report: [plain text] [BIB] [BibTeX] [Refer]
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Stavros G. Kolliopoulos and
Clifford Stein,
"Approximating DisjointPath Problems Using Greedy Algorithms and Packing Integer Programs ."
Dartmouth Computer Science Technical Report PCSTR97325,
October 1997.
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