Concepedia

The complexity of finding maximum disjoint paths with length constraints

Alon Itai, Yehoshua Perl, Yossi Shiloach

Networks · 1982 · 152 citations · 6 references

TL;DR

The paper studies the maximum number of disjoint s–t paths of a given length K in a graph, considering vertex‑ or edge‑disjoint variants, equal or bounded lengths, and directed or undirected graphs. For each variant the authors determine the largest K that admits a polynomial‑time solution and, when such a K exists, provide an efficient algorithm. They prove that, except for small values of K, all these problems are NP‑complete.

Abstract

Abstract The following problem is considered: Given an integer K , a graph G with two distinct vertices s and t , find the maximum number of disjoint paths of length K from s to t . The problem has several variants: the paths may be vertex‐disjoint or edge‐disjoint, the lengths of the paths may be equal to K or bounded by K , the graph may be undirected or directed. It is shown that except for small values of K all the problems are NP‐complete. Assuming P ≠ NP, for each problem, the largest value of K for which the problem is not NP‐complete is found. Whenever a polynomial algorithm exists, an efficient algorithm is described.