Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Preliminaries
- 3 Matching and vertex cover in bipartite graphs
- 4 Spanning trees
- 5 Matroids
- 6 Arborescence and rooted connectivity
- 7 Submodular flows and applications
- 8 Network matrices
- 9 Matchings
- 10 Network design
- 11 Constrained optimization problems
- 12 Cut problems
- 13 Iterative relaxation: Early and recent examples
- 14 Summary
- Bibliography
- Index
7 - Submodular flows and applications
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Preliminaries
- 3 Matching and vertex cover in bipartite graphs
- 4 Spanning trees
- 5 Matroids
- 6 Arborescence and rooted connectivity
- 7 Submodular flows and applications
- 8 Network matrices
- 9 Matchings
- 10 Network design
- 11 Constrained optimization problems
- 12 Cut problems
- 13 Iterative relaxation: Early and recent examples
- 14 Summary
- Bibliography
- Index
Summary
Quoting Lovász from his paper “Submodular Functions and Convexity” [94]:
Several recent combinatorial studies involving submodularity fit into the following pattern. Take a classical graph-theoretical result (e.g. the Marriage Theorem, the Max-flow-min-cut Theorem etc.), and replace certain linear functions occurring in the problem (either in the objective function or in the constraints) by submodular functions. Often the generalizations of the original theorems obtained this way remain valid; sometimes even the proofs carry over. What is important here to realize is that these generalizations are by no means l'art pour l'art. In fact, the range of applicability of certain methods can be extended tremendously by this trick.
The submodular flow model is an excellent example to illustrate this point. In this chapter, we introduce the submodular flow problem as a generalization of the minimum cost circulation problem. We then show the integrality of its LP relaxation and its dual using the iterative method. We then discuss many applications of the main result. We also show an application of the iterative method to an NP-hard degree bounded generalization and show some applications of this result as well.
The crux of the integrality of the submodular flow formulations will be the property that a maximal tight set of constraints form a cross-free family. This representation allows an inductive token counting argument to show a 1-element in an optimal extreme point solution. We will see that this representation is precisely the one we will eventually encounter in Chapter 8 on network matrices.
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- Iterative Methods in Combinatorial Optimization , pp. 110 - 130Publisher: Cambridge University PressPrint publication year: 2011