Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation

  • Authors:

  • A. Ridha Mahjoub;S. Thomas McCormick

  • Affiliations:

  • Université Paris-Dauphine, LAMSADE, CNRS Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France;University of British Columbia, Sauder School of Business, V6T 1Z2, Vancouver, BC, Canada

  • Venue:
  • Mathematical Programming: Series A and B - Series B - Special Issue: Combinatorial Optimization and Integer Programming
  • Year:
  • 2010

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Abstract

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm.