Almost Disjoint Paths and Separating by Forbidden Pairs

TL;DR

This paper introduces the almost disjoint paths problem and the separating by forbidden pairs problem, analyzing their duality gap, complexity, and providing efficient algorithms for special cases in directed graphs.

Contribution

It establishes the complexity of ADP and SFP, showing NP-completeness and Sigma_2P-completeness, and offers polynomial algorithms for fixed k and acyclic graphs.

Findings

ADP is NP-complete when k is variable.

SFP is Sigma_2P-complete even for acyclic graphs.

Polynomial algorithms exist for fixed k and acyclic graphs.

Abstract

By Menger's theorem the maximum number of arc-disjoint paths from a vertex s to a vertex t in a directed graph equals the minumum number of arcs needed to disconnect s and t, i.e., the minimum size of an s-t-cut. The max-flow problem in a network with unit capacities is equivalent to the arc-disjoint paths problem. Moreover the max-flow and min-cut problems form a strongly dual pair. We relax the disjointedness requirement on the paths, allowing them to be almost disjoint, meaning they may share up to one arc. The resulting almost disjoint paths problem (ADP) asks for k s-t-paths such that any two of them are almost disjoint. The separating by forbidden pairs problem (SFP) is the corresponding dual problem and calls for a set of k arc pairs such that every s-t-path contains both arcs of at least one such pair. In this paper, we explore these two problems, showing that they have an unbounded duality gap in general and analyzing their complexity. We prove that ADP is NP-complete when k is part of the input and that SFP is Sigma_2P-complete, even for acyclic graphs. Furthermore, we efficiently solve ADP when k<=2 is fixed and present a polynomial time algorithm based on dynamic programming for ADP when k is constant and the considered graphs are acyclic.