Polytopes of alternating sign matrices with dihedral-subgroup symmetry

2026-02-20

AI Summary

This study investigates the high-dimensional geometry and associated optimization challenges of the convex hull of alternating sign matrices (ASMs) possessing dihedral subgroup symmetries. The authors propose a unified “core–assembly” framework that reduces symmetric ASMs to lower-dimensional core polytopes via core positions and affine assembly maps. By integrating prefix-sum representations, polyhedral combinatorics, and Chvátal–Gomory cutting planes, they provide the first polynomial-size linear inequality descriptions for six classes of symmetric ASMs, fully characterizing their dimensions and facial structures. For quarter-turn symmetric ASMs, structured cutting planes are designed to eliminate fractional vertices. The approach enables efficient minimum-cost computation and establishes a direct link between the combinatorial structure of symmetric ASMs and polyhedral optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Social Networks and Social Media: Computational social science

Abstract

We investigate the convex hulls of the eight dihedral symmetry classes of $n \times n$ alternating sign matrices, i.e., ASMs invariant under a subgroup of the symmetry group of the square. Extending the prefix-sum description of the ASM polytope, we develop a uniform core--assembly framework: each symmetry class is encoded by a set of core positions and an affine assembly map that reconstructs the full matrix from its core. This reduction transfers polyhedral questions to lower-dimensional core polytopes, which are better suited to the tool set of polyhedral combinatorics, while retaining complete information about the original symmetry class. For the vertical, vertical--horizontal, half-turn, diagonal, diagonal--antidiagonal, and total symmetry classes, we give explicit polynomial-size linear inequality descriptions of the associated polytopes. In these cases, we also determine the dimension and provide facet descriptions. The quarter-turn symmetry class behaves differently: the natural relaxation admits fractional vertices, and we need to extend the system with a structured family of parity-type Chv\'atal--Gomory inequalities to obtain the quarter-turn symmetric ASM polytope. Our framework leads to efficient algorithms for computing minimum-cost ASMs in each symmetry class and provides a direct link between the combinatorics of symmetric ASMs and tools from polyhedral combinatorics and combinatorial optimization.

Problem

• alternating sign matrices
• dihedral symmetry
• polytopes
• convex hull
• symmetry classes

Innovation

• alternating sign matrices
• dihedral symmetry
• core–assembly framework
• polyhedral combinatorics
• Chvátal–Gomory inequalities