[Paper Review] On Fundamental Operations for Multimodular Functions

Satoko Moriguchi, Kazuo Murota|arXiv (Cornell University)|May 11, 2018

Complexity and Algorithms in Graphs19 references3 citations

TL;DR

This paper investigates the stability of multimodular functions under fundamental operations in discrete convex analysis, such as projection, convolution, permutation, and scaling. It proves that multimodularity is preserved under projection when minimizing over a consecutive set of variables, but not under convolution—even with separable convex functions—highlighting a key distinction from L♮-convex functions.

ABSTRACT

Multimodular functions, primarily used in the literature of queueing theory, discrete-event systems, and operations research, constitute a fundamental function class in discrete convex analysis. The objective of this paper is to clarify the properties of multimodular functions with respect to fundamental operations such as permutation and scaling of variables, projection (partial minimization) and convolution. It is shown, in particular, that the class of multimodular functions is stable under projection under a certain natural condition on the variables to be minimized, and the convolution of two multimodular functions is not necessarily multimodular, even in the special case of the convolution of a multimodular function with a separable convex function.

Motivation & Objective

• To systematically analyze the behavior of multimodular functions under fundamental operations such as permutation, scaling, projection, and convolution.
• To clarify the stability of multimodular functions under operations that are central to dynamic programming and optimization.
• To compare the closure properties of multimodular functions with other discrete convex function classes, particularly L♮-convex functions.
• To provide a comprehensive reference for the use of multimodular functions in queueing theory, discrete-event systems, and operations research.
• To resolve open questions about whether convolution of multimodular functions remains multimodular, using counterexamples and theoretical analysis.

Proposed method

• The paper uses the characterization that a function f is multimodular if and only if its associated function g(x₀,x₁,…,xₙ) = f(x₁−x₀,…,xₙ−xₙ₋₁) is L♮-convex.
• It analyzes operations via variable transformations, such as permutation, sign inversion, and scaling, and proves invariance under certain transformations using the L♮-convexity of the associated function.
• For projection (partial minimization), it establishes that multimodularity is preserved when minimizing over a consecutive block of variables, using the structure of the L♮-convex representation.
• For convolution, it constructs counterexamples showing that the convolution of two multimodular functions—or a multimodular function with a separable convex function—need not be multimodular.
• It employs the Minkowski sum of sets and the associated indicator functions to demonstrate failure of multimodularity in convolution via L♮-convexity violations.
• Theoretical results are supported by explicit examples and comparisons with known closure properties of other discrete convex function classes.

Experimental results

Research questions

• RQ1 Is the class of multimodular functions closed under projection (partial minimization), and under what conditions?
• RQ2 Does the convolution of two multimodular functions remain multimodular?
• RQ3 Can the convolution of a multimodular function with a separable convex function preserve multimodularity?
• RQ4 How do the closure properties of multimodular functions compare with those of L♮-convex, M♮-convex, and other discrete convex function classes?
• RQ5 What is the role of variable ordering and consecutive variable minimization in preserving multimodularity?

Key findings

• Multimodularity is preserved under permutation of variables, as shown by the invariance of the associated L♮-convex function under variable reordering.
• Scaling of variables preserves multimodularity, provided the scaling factor is a positive integer, due to the invariance of the L♮-convex representation.
• Projection (partial minimization) preserves multimodularity when minimizing over a consecutive block of variables, as formalized in Proposition 4.3.
• The convolution of two multimodular functions is not necessarily multimodular, as demonstrated by a counterexample involving the Minkowski sum of two multimodular sets that fails to be multimodular.
• Even the convolution of a multimodular function with a separable convex function may fail to be multimodular, contradicting potential intuition from L♮-convexity.
• The paper provides a complete comparison table (Table 1) summarizing closure properties across multiple discrete convex function classes, highlighting the distinct behavior of multimodular functions under convolution.