The maximum multiflow problems with bounded fractionality

Abstract

This paper addresses a fundamental issue in the multicommodity flow theory. For an undirected capacitated supply graph (G,c) having commodity graph H, the maximum multiflow problem is to maximize the total flow-value of multicommodity flows with respect to (G,c;H). For a commodity graph H, the fractionality of H is the least positive integer k with property that there exists a 1/k-integral optimal multiflow in the maximum multiflow problem for every integer-capacitated supply graph (G,c) having H as a commodity graph. If such a positive integer k does not exist, then the fractionality is defined to be infinity. Around 1990, Karzanov raised the problem of classifying commodity graphs with finite fractionality, gave a necessary condition (property P) for the finiteness of fractionality, and conjectured that the property P is also sufficient. Our main result affirmatively solves Karzanov's conjecture in algorithmic form: If H has property P, then there exists a 1/24-integral optimal multiflow in maximum multiflow problem for every integer-capacitated supply graph having H as a commodity graph, and there exists a strongly polynomial time algorithm to find it. Our proof is based on a special combinatorial duality relation involving a class of CAT(0) complexes, and on a fractional version of the splitting-off method for finding an optimal multiflow with a bounded denominator.