Shortest disjoint paths on a grid

The well-known k-disjoint paths problem involves finding pairwise vertex-disjoint paths between k specified pairs of vertices within a given graph if they exist. In the shortest k-disjoint paths problem one looks for such paths of minimum total length. Despite nearly 50 years of active research on the k-disjoint paths problem, many open problems and complexity gaps still persist. A particularly well-defined scenario, inspired by VLSI design, focuses on infinite rectangular grids where the terminals are placed at arbitrary grid points. While the decision problem in this context remains NP-hard, no prior research has provided any positive results for the optimization version. The main result of this paper is a fixed-parameter tractable (FPT) algorithm for this scenario. It is important to stress that this is the first result achieving the FPT complexity of the shortest disjoint paths problem in any, even very restricted classes of graphs where we do not put any restriction on the placements of the terminals

Efficient Algorithms for Finding the Maximum Number of Disjoint Paths in Grids

In a rectangular grid, given two sets of nodes, script capital L sign (sources) and script T sign (sinks), of size N/2 each, the disjoint paths (DP) problem is to connect as many nodes in script capital L sign to the nodes in script T sign using a set of "disjoint" paths. (Both edge-disjoint and vertex-disjoint cases are considered in this paper.) Note that in this DP problem, a node in script capital L sign can be connected to any node in script T sign. Although in general the sizes of script capital L sign and script T sign do not have to be the same, algorithms presented in this paper can also find the maximum number of disjoint paths pairing nodes in script capital L sign and script T sign. We use the network flow approach to solve this DP problem. By exploiting all the properties of the network, such as planarity and regularity of a grid, integral flow, and unit capacity source/sink/flow, we can optimally compress the size of the working grid (to be defined) from O(N2) to U(N1.5) and solve the problem in O(N2.5) time for both the edge-disjoint and vertex-disjoint cases, an improvement over the straightforward approach which takes O(N3) time. © 2000 Academic Press.link_to_subscribed_fulltex