A survey on covering supermodular functions

A. Frank, Tamás Király

Research output: Chapter

10 Citations (Scopus)

Abstract

In this survey we present recent advances on problems that can be described as the construction of graphs or hypergraphs that cover certain set functions with supermodular or related properties. These include a wide range of network design and connectivity augmentation and orientation problems, as well as some results on colourings and matchings. In the first part of the paper we survey results that follow from the totally dual integral (TDI) property of various systems defined by supermodular-type set functions. One of the aims of the survey is to emphasize the importance of relaxing the supermodularity property to include a wider range of set functions. We show how these relaxations lead to a unified understanding of different types of applications. The second part is devoted to results that, according to our current knowledge, cannot be explained using total dual integrality. We would like to demonstrate that an extensive theory independent of total dual integrality has been developed in the last 15 years, centered around various connectivity augmentation problems. Our survey concentrates on the theoretical foundations, and does not include every detail on applications, since the majority of these applications are described in detail in another survey paper by the first author (Frank 2006). The comprehensive book "Combinatorial Optimization: Polyhedra and Efficiency" by Schrijver (2003) is also a rich resource of results related to submodular functions. It should be noted that sub- and supermodularity have several applications in areas not discussed in this paper. In particular, we should mention the book "Submodular Functions and Optimization" by Fujishige (2005) and the book "Discrete Convex Analysis" by Murota (2003). The former explains the foundations of the theory of submodular functions and describes the methods of submodular analysis, while the latter presents a unified framework for nonlinear discrete optimization by extending submodular function theory using ideas from continuous optimization. Our survey focuses on topics not discussed in detail in those books.

Original languageEnglish
Title of host publicationResearch Trends in Combinatorial Optimization: Bonn 2008
PublisherSpringer Berlin Heidelberg
Pages87-126
Number of pages40
ISBN (Print)9783540767954
DOIs
Publication statusPublished - 2009

Fingerprint

Submodular Function
Covering
Supermodularity
Integrality
Augmentation
Submodularity
Set Cover
Convex Analysis
Network Connectivity
Continuous Optimization
Discrete Optimization
Combinatorial Optimization
Network Design
Nonlinear Optimization
Hypergraph
Polyhedron
Range of data
Connectivity
Resources
Optimization

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Frank, A., & Király, T. (2009). A survey on covering supermodular functions. In Research Trends in Combinatorial Optimization: Bonn 2008 (pp. 87-126). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_6

A survey on covering supermodular functions. / Frank, A.; Király, Tamás.

Research Trends in Combinatorial Optimization: Bonn 2008. Springer Berlin Heidelberg, 2009. p. 87-126.

Research output: Chapter

Frank, A & Király, T 2009, A survey on covering supermodular functions. in Research Trends in Combinatorial Optimization: Bonn 2008. Springer Berlin Heidelberg, pp. 87-126. https://doi.org/10.1007/978-3-540-76796-1_6
Frank A, Király T. A survey on covering supermodular functions. In Research Trends in Combinatorial Optimization: Bonn 2008. Springer Berlin Heidelberg. 2009. p. 87-126 https://doi.org/10.1007/978-3-540-76796-1_6
Frank, A. ; Király, Tamás. / A survey on covering supermodular functions. Research Trends in Combinatorial Optimization: Bonn 2008. Springer Berlin Heidelberg, 2009. pp. 87-126
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