The edge-disjoint paths problem in Eulerian graphs and 4-edge-connected graphs

Abstract

In the edge-disjoint paths problem, we are given a graph and a set of k pairs of vertices, and we have to decide whether or not the graph has k edge-disjoint paths connecting given pairs of terminals. Robertson and Seymour's graph minor project gives rise to a polynomial time algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project. We give a faster algorithm and a much simpler proof of the correctness for the edge-disjoint paths problem. Our results can be summarized as follows: 1.If an input graph is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤3-edge-cuts. (iii) Excluding large clique minors.2.When an input graph is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be a non-trivially super constant number, up to k=O((log log logn)½??) for any ? > 0. In addition, if an input graph is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((logn½??) for any ? > 0.3.We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow the Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, our algorithm is faster than Robertson and Seymour's.