On fractionality of the path packing problem

TL;DR

This paper proves Karzanov's conjecture that the fractional path packing problem in Eulerian networks has fractionality D equal to 1 or 2, confirming the minimal integer scaling needed for solutions.

Contribution

The paper confirms that in Eulerian networks, the fractionality D for the path packing problem is always 1 or 2, resolving a conjecture by Karzanov.

Findings

Fractionality D is always 1 or 2 in Eulerian networks.

Confirmed Karzanov's conjecture on fractionality.

Provides a complete characterization of fractional solutions.

Abstract

In this paper, we study fractional multiflows in undirected graphs. A fractional multiflow in a graph G with a node subset T, called terminals, is a collection of weighted paths with ends in T such that the total weights of paths traversing each edge does not exceed 1. Well-known fractional path packing problem consists of maximizing the total weight of paths with ends in a subset S of TxT over all fractional multiflows. Together, G,T and S form a network. A network is an Eulerian network if all nodes in N\T have even degrees. A term "fractionality" was defined for the fractional path packing problem by A. Karzanov as the smallest natural number D so that there exists a solution to the problem that becomes integer-valued when multiplied by D. A. Karzanov has defined the class of Eulerian networks in terms of T and S, outside which D is infinite and proved that whithin this class D can be 1,2 or 4. He conjectured that D should be 1 or 2 for this class of networks. In this paper we prove this conjecture.