CAT-generation of ideals

TL;DR

This paper develops a new method to generate all ideals of certain classes of posets efficiently, including interval dimension at most two and locally planar posets, with applications to various combinatorial structures.

Contribution

It introduces a refined tree traversal method to achieve constant amortized time generation for these classes of posets, extending previous known cases.

Findings

Achieves CAT-generation for posets of interval dimension at most two.

Extends CAT-generation to locally planar posets.

Applies results to generate all c-orientations of planar graphs and related combinatorial objects.

Abstract

We consider the problem of generating all ideals of a poset. It is a long standing open problem, whether or not the ideals of any poset can be generated in constant amortized time, CAT for short. We refine the tree traversal, a method introduced by Pruesse and Ruskey in 1993, to obtain a CAT-generator for two large classes of posets: posets of interval dimension at most two and so called locally planar posets. This includes all posets for which a CAT-generator was known before. Posets of interval dimension at most two generalize both, interval orders and 2-dimensional posets. Locally planar posets generalize for example posets with a planar cover graph. We apply our results to CAT-generate all c-orientations of a planar graph. As a special case this is a CAT-generator for many combinatorial objects like domino and lozenge tilings, planar spanning trees, planar bipartite perfect matchings, Schnyder woods, and others.