Criticality for multicommodity flows

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For k≥ 1, the k- commodity flow problem is, we are given k pairs of vertices in a graph G, and we ask whether there exist k flows in the graph, where. •the ith flow is between the ith pair of vertices, and has total value one; and•for each edge e, the sum of absolute values of the flows along e is at most one. We prove that for all k there exists n(k) such that if G is connected, and contraction-minimal such that the k-commodity flow problem is infeasible (that is, minimal in the sense that contracting any edge makes the problem feasible) then |V(G)|+|E(G)|≤ n(k). For integers k, p≥ 1, the (k, p)- commodity flow problem is as above, with the extra requirement that the flow value of each flow along each edge is a multiple of 1/p. We prove that if p> 1, and G is connected, and contraction-minimal such that the (k, p)-commodity flow problem is infeasible, then |V(G)|+|E(G)|≤ n(k), with the same n(k) as before, independent of p. In contrast, when p= 1 there are arbitrarily large contraction-minimal instances, even when k = 2. We give some other corollaries of the same approach, including. •a proof that for all k≥0 there exists p>0 such that every feasible k-commodity flow problem has a solution in which all flow values are multiples of 1/p, and•a very simple polynomial-time algorithm to solve the (k, p) multicommodity flow problem when p>1.

Original languageEnglish (US)
Pages (from-to)136-179
Number of pages44
JournalJournal of Combinatorial Theory. Series B
StatePublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


  • Multicommodity flows
  • Polynomial-time
  • Tree-decomposition


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