Max-Weight Integral Multicommodity Flow in Spiders and High-Capacity Trees

  • Authors:

  • Jochen Könemann;Ojas Parekh;David Pritchard

  • Affiliations:

  • Dept. of Combinatorics & Optimization, University of Waterloo, Waterloo, Canada N2L 3G1;Math/CS Department, Emory University, Atlanta, USA 30322;Dept. of Combinatorics & Optimization, University of Waterloo, Waterloo, Canada N2L 3G1

  • Venue:
  • Approximation and Online Algorithms
  • Year:
  • 2009

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Abstract

We consider the max-weight integral multicommodity flow problem in trees. In this problem we are given an edge-capacitated tree and weighted pairs of terminals, and the objective is to find a max-weight integral flow between terminal pairs subject to the capacities. This problem was shown to be $\mathcal{\mathsf{APX}}$-hard by Garg, Vazirani and Yannakakis [Algorithmica, 1997], and a 4-approximation was given by Chekuri, Mydlarz and Shepherd [ACM Trans. Alg., 2007]. Some special cases are known to be exactly solvable in polynomial time, including when the graph is a path or a star. First, when every edge has capacity at least μ ≥ 2, we use iterated LP relaxation to obtain an improved approximation ratio of min {3, 1 + 4/μ + 6/(μ 2 − μ)}. We show this ratio bounds the integrality gap of the natural LP relaxation. A complementary hardness result yields a 1 + Θ(1/μ) threshold of approximability (if P ≠ NP). Second, we extend the range of instances for which exact solutions can be found efficiently. When the tree is a spider (i.e. if only one vertex has degree greater than 2) we give a polynomial-time algorithm to find an optimal solution, as well as a polyhedral description of the convex hull of all integral feasible solutions.