Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

Abstract: In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.

• The paper introduces an inverse semigroup framework to capture partial symmetries in the perturbation space of minimal valid functions.
• It demonstrates that additive and limit-additive moves yield a finitely presented closure for functions with rational breakpoints.
• The study decomposes the effective perturbation space into finite-dimensional and equivariant components, enhancing integer programming methodologies.

On Perturbation Spaces of Minimal Valid Functions: Inverse Semigroup Theory and Equivariant Decomposition

The paper presents a comprehensive examination of the perturbation spaces of minimal valid functions, focusing on the 1-row Gomory–Johnson infinite group problem. The primary contribution lies in the development of a theoretical framework for understanding how perturbations of these minimal valid functions can be systematically decomposed. This is achieved through the lens of inverse semigroup theory, which is adeptly applied to delineate the partial symmetries inherent in these perturbations.

Overview of Key Contributions

1. Inverse Semigroups and Partial Symmetries: The authors employ inverse semigroup theory to articulate the notion of partial symmetry within the perturbation space of minimal valid functions. This approach extends beyond the traditional group actions by recognizing that symmetries of these functions are often partial and can be effectively captured by the composition and inverse operations of inverse semigroups.
2. Additive and Limit-Additive Moves: The paper introduces the concept of initial move sets, which encompass additive and limit-additive moves. These moves are derived from the subadditivity conditions of the minimal valid functions, and they establish a foundational basis for exploring the perturbation space.
3. Finite Presentation of Moves Closure: The researchers provide a method to compute the closure of move ensembles, demonstrating that for minimal valid functions with rational breakpoints, this closure is finitely presented. This achievement significantly aids in the computation of perturbations, as it allows one to work with a finite set of generating moves.
4. Decomposition of Perturbations: The work delineates a direct sum decomposition of the effective perturbation space into finite-dimensional components and equivariant components. The finite-dimensional part is described by a system of linear equations, while the equivariant part resembles a space of Lipschitz functions defined over fundamental domains.

Implications and Speculations on Future Developments

The implications of this research are notably significant in the field of integer programming, particularly in enhancing the effectiveness of cutting-plane algorithms through robust characterization of cut-generating functions. By shifting focus to a grid-free paradigm, this approach can handle functions with complex breakpoints more efficiently.

As inverse semigroup theory provides a richer structure for understanding symmetries, there is potential for extending this framework to higher-dimension problems or even other domains where partial symmetries play a critical role. Future research might focus on algorithmic implementations that leverage these theoretical results to provide natural proofs of extremality in automated systems, thus broadening the applicability of cutting-plane methods in optimization.

Conclusion

This paper stands as a pivotal contribution to understanding the perturbational aspects of minimal valid functions through a novel algebraic approach. By systematically addressing the partial symmetries and computational complexities of these functions, the authors open new avenues for both theoretical exploration and practical implementation in the field of integer programming and combinatorial optimization.