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A Tight Max-Flow Min-Cut Duality Theorem for Non-Linear Multicommodity Flows

Matthew Broussard, Bala Krishnamoorthy

TL;DR

A multiple commodity generalization of MAX-FLOW in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in Rk, and shows that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow.

Abstract

The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flow problem, which considers flow of a single commodity. We study a multiple commodity generalization of Max-Flow in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in \R^k. Given the absence of a clear notion of ordering in the multicommodity case, we define the generalized max flow as the feasible region of all flow values. We define a collection of concepts and operations on flows and cuts in the multicommodity setting. We study the mutual capacity of a set of cuts, defined as the set of flows that can pass through all cuts in the set. We present a method to calculate the mutual capacity of pairs of cuts, and then generalize the same to a method of calculation for arbitrary sets of cuts. We show that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow. Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations. We also study more tractable special cases of the multicommodity max flow problem where the objective is to transport a maximum real or integer multiple of a given vector through the network. We devise an augmenting cycle search algorithm that reduces the optimization problem to one with m constraints in at most \R^{(m-n+1)k} space from one that requires mn constraints in \R^{mk} space for a network with n nodes and m edges. We present efficient algorithms that compute eps-approximations to both the ratio and the integer ratio maximum flow problems.

A Tight Max-Flow Min-Cut Duality Theorem for Non-Linear Multicommodity Flows

TL;DR

A multiple commodity generalization of MAX-FLOW in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in Rk, and shows that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow.

Abstract

The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flow problem, which considers flow of a single commodity. We study a multiple commodity generalization of Max-Flow in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in \R^k. Given the absence of a clear notion of ordering in the multicommodity case, we define the generalized max flow as the feasible region of all flow values. We define a collection of concepts and operations on flows and cuts in the multicommodity setting. We study the mutual capacity of a set of cuts, defined as the set of flows that can pass through all cuts in the set. We present a method to calculate the mutual capacity of pairs of cuts, and then generalize the same to a method of calculation for arbitrary sets of cuts. We show that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow. Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations. We also study more tractable special cases of the multicommodity max flow problem where the objective is to transport a maximum real or integer multiple of a given vector through the network. We devise an augmenting cycle search algorithm that reduces the optimization problem to one with m constraints in at most \R^{(m-n+1)k} space from one that requires mn constraints in \R^{mk} space for a network with n nodes and m edges. We present efficient algorithms that compute eps-approximations to both the ratio and the integer ratio maximum flow problems.

Paper Structure

This paper contains 18 sections, 21 theorems, 7 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Previous Work
  4. Problems and Definitions
  5. Definitions on Networks
  6. Flows and Flow-like Objects
  7. Operations
  8. Mutual Capacity: Towards an MFMC Result
  9. Capacity of Two Cuts: Pairwise Capacity
  10. Duality Gap of Pairwise Capacity
  11. Mutual Capacity of Fully Disjoint Networks
  12. General Mutual Capacity and Duality Result
  13. Min-Cut Algorithm
  14. Framework Based on the Work of Krishnan
  15. Construction
  16. ...and 3 more sections

Key Result

Theorem 1.1

The maximal flow value obtainable in a network $N$ is the minimum of $v(D)$ taken over all disconnecting sets $D$.

Figures (15)

  • Figure 1: Example of a 2-commodity network. Capacities of the arcs are the regions shown in red, yellow, and blue.
  • Figure 2: Example of a multicommodity flow in the network shown in Figure \ref{['fig:MultiComNetworkEx']}. The green vectors denote flow values for the two commodities ($x,y$) on each arc.
  • Figure 3: Example of a local flow over a cut (in red) with arc set $\{(s,1),(s,2)\}$.
  • Figure 4: A second example of a local flow on the same network used in Figure \ref{['fig:MCMFLocalFlow']}. Arc set of the single cut is shown in green. Note that arc $(1,2)$ is a backward arc of this cut, and hence subtracts from the local flow's flow value.
  • Figure 5: A local flow over a pair of cuts. This flow is obtained by gluing (see Definition \ref{['def:Gluing']}) together the flows in Figures \ref{['fig:MCMFLocalFlow']} and \ref{['fig:MCMFLocalFlow2']}.
  • ...and 10 more figures

Theorems & Definitions (80)

  • Theorem 1.1: Max-Flow Min-Cut Theorem, Ford and Fulkerson FoFu1956
  • Theorem \ref{thm:MCMFMC}
  • Definition \ref{thm:MCMFMC}: Multicommodity Max Flow Kr20??final
  • Definition \ref{thm:MCMFMC}: Ratio Max Flow LeRa1999
  • Definition \ref{thm:MCMFMC}: Integer Ratio Max Flow
  • Definition \ref{thm:MCMFMC}: Single commodity Network AhMaOr1993
  • Definition \ref{thm:MCMFMC}: $k$-commodity Network
  • Definition \ref{thm:MCMFMC}: Opposite Orientation
  • Example \ref{thm:MCMFMC}
  • Definition \ref{thm:MCMFMC}: Enhanced Network Kr20??final
  • ...and 70 more