[Paper Review] A Note on M-convex Functions on Jump Systems

Kazuo Murota|arXiv (Cornell University)|Jul 14, 2019

Advanced Graph Theory Research17 references4 citations

TL;DR

This paper introduces and characterizes jump M♮-convex functions, a generalization of jump M-convex functions on a broader class of jump systems. It establishes an injective embedding of jump M♮-convex functions in $ n $ variables into jump M-convex functions in $ n+1 $ variables, proving their equivalence and enabling the transfer of known results. The paper further shows that jump M♮-convex functions are closed under key operations such as aggregation, projection, convolution, composition, and network transformation.

ABSTRACT

A jump system is defined as a set of integer points (vectors) with a certain exchange property, generalizing the concepts of matroids, delta-matroids, and base polyhedra of integral polymatroids (or submodular systems). A discrete convexity concept is defined for functions on constant-parity jump systems and it has been used in graph theory and algebra. In this paper we call it "jump M-convexity" and extend it to "jump M-natural-convexity" for functions defined on a larger class of jump systems. By definition, every jump M-convex function is a jump M-natural-convex function, and we show the equivalence of these concepts by establishing an (injective) embedding of jump M-natural-convex functions in n variables into the set of jump M-convex functions in n+1 variables. Using this equivalence we show further that jump M-natural-convex functions admit a number of natural operations such as aggregation, projection (partial minimization), convolution, composition, and transformation by a network.

Motivation & Objective

• To extend the concept of M-convexity from constant-parity jump systems to a broader class of jump systems with a simultaneous exchange property.
• To define and study the properties of jump M♮-convex functions, generalizing M-convex and valuated delta-matroid concepts.
• To establish the equivalence between jump M♮-convex and jump M-convex functions via an injective embedding into one higher dimension.
• To demonstrate that jump M♮-convex functions are closed under fundamental operations such as aggregation, projection, convolution, composition, and network transformation.
• To enable the application of existing results on jump M-convex functions to the broader class of jump M♮-convex functions.

Proposed method

• Introduce the concept of jump M♮-convex functions on jump systems satisfying a simultaneous exchange property, generalizing M-convexity.
• Define an injective embedding of jump M♮-convex functions in $ n $ variables into jump M-convex functions in $ n+1 $ variables, proving equivalence.
• Use the embedding to transfer known structural and algorithmic results from jump M-convex functions to jump M♮-convex functions.
• Analyze operations such as aggregation, projection (partial minimization), convolution, composition, and network transformation using the embedding and known results on jump M-convex functions.
• Prove closure under convolution via Theorem 4.7, showing that the infimal convolution of two jump M♮-convex functions remains jump M♮-convex.
• Establish closure under network transformation via Theorem 4.12, showing that transforming a jump M♮-convex function through a network yields another jump M♮-convex function when the arc cost functions are convex.

Experimental results

Research questions

• RQ1 Is there a way to extend M-convexity to a broader class of jump systems beyond constant-parity systems?
• RQ2 Can jump M♮-convex functions be embedded into jump M-convex functions in a higher-dimensional space?
• RQ3 Are the fundamental operations—aggregation, projection, convolution, composition, and network transformation—preserved under jump M♮-convexity?
• RQ4 Does the equivalence between jump M♮-convexity and jump M-convexity allow transfer of known results from the latter to the former?
• RQ5 Can network transformations preserve the M♮-convexity of functions defined on jump systems?

Key findings

• Jump M♮-convex functions are equivalent to jump M-convex functions via an injective embedding into $ n+1 $ variables, establishing a one-to-one correspondence.
• Every jump M-convex function is a jump M♮-convex function, and the converse holds through the embedding construction.
• Jump M♮-convex functions are closed under aggregation, as shown via the embedding and known results on jump M-convex functions.
• The convolution of two jump M♮-convex functions is itself jump M♮-convex, as proven in Theorem 4.7.
• Network transformation of a jump M♮-convex function by a network with convex arc cost functions results in another jump M♮-convex function, as stated in Theorem 4.12.
• The splitting of a jump M♮-convex function is also jump M♮-convex, extending a result from jump M-convex functions to the broader class.