We consider the following maximum disjoint paths problem (MDPP). We are given a large network, and pairs of nodes that wish to communicate over paths through the network -the goal is to simultaneously connect as many of these pairs as possible in such a way that no two communication paths share an edge in the network. This classical problem has been brought into focus recently in papers discussing applications to routing in high-speed networks, where the current lack of understanding of the MDPP is an obstacle to the design of practical heuristics. We consider the class of densely embedded, nearly-Eulerian graphs, which includes the two-dimensional mesh and other planar and locally planar interconnection networks. We obtain a constant-factor approximation algorithm for the maximum disjoint paths problem for this class of graphs; this improves on an O(1og n)-approximation for the special case of the twodimensional mesh due to Aumann-Rabani and the authors. For networks that are not explicitly required to be "high-capacity," this is the first constant-factor approximation for the MDPP in any class of graphs other than trees. We also consider the MDPP in the on-line setting, relevant'to applications in which connection requests anive over time and must be processed immediately. Here we obtain an asymptotically optimal O(1og n)-competitive on-line algorithm for the same class of graphs; this improves on an O(1og n log log n)-competitive algorithm for the special case of the mesh due to Awerbuch, Gawlick, Leighton, and Rabani.
Kleinberg and Tardos give in REF a constant-factor approximation algorithm for MaxEDP in graphs they call densely embedded and nearly eulerian, and which generalize the rectilinear grids.
Disjoint paths in densely embedded graphs
Proceedings of IEEE 36th Annual Foundations of Computer Science
Proceedings of IEEE 36th Annual Foundations of Computer Science
Computer Science