math.COcs.DMEmpirical
Symmetric lexicographic symmetric-subset reverse search for the enumeration of circuits, cocircuits, and triangulations up to symmetry
Jul 7, 2026
This paper introduces and implements variants of the symmetric lexicographic symmetric-subset reverse search framework for enumerating symmetric feasible subsets of a finite set, applying it to cocircuits, circuits, and triangulations.
Introduces a new framework for enumerating symmetric feasible subsets of a finite set and applies it to three specific problems, improving computation times for known benchmarks.
Keywords
Before reading this…
Applications
To understand this paper, make sure you know these concepts first:
This paper introduces, analyzes, and applies variants of the enumeration framework symmetric lexicographic symmetric-subset reverse search for the enumeration of symmetric feasible subsets of a finite set up to symmetry. The framework is implemented in detail for three applications: cocircuits, circuits, and triangulations of int configurations. There are two new methods presented and analyzed to check the lexicographic minimality of a subset in its orbit: the critical-element method and the modified switch-table method. Moreover, new application-dependent methods to reduce the number of necessary enumeration nodes are introduced: rank-pruning for cocircuits and lex-pruning for triangulations. With a C++-implementation of the ideas in the software package TOPCOM, in all three applications known benchmarks can be computed faster by a large margin. The following new numbers could be computed for the first time (among others): the number of cocircuits of the 9-cube, the number of circuits of the 8-cube, and the number of all triangulations of the product of a 5- and a 3-simplex, as well as the number of all triangulations of a point configuration in dimension six with 17~points with disconnected flip-graph (constructed by Santos). Moreover, for Santos's triangulation it has computationally been checked that its flip-graph component is indeed purely non-regular. Furthermore, in another instance in dimension five with 26 points (also constructed by Santos), a flaw has been detected: Santos's triangulation can be heuristically flipped to a regular triangulation in the original point configuration. In a mildly modified version of the point configuration, the heuristics cannot flip Santos's triangulation to a regular triangulation anymore.