Combinatorial generation via permutation languages. V. Acyclic orientations

TL;DR

This paper generalizes Gray code constructions for acyclic orientations from graphs to hypergraphs and lattice quotients, enabling efficient enumeration and Hamiltonian paths on related polytopes.

Contribution

It introduces new Gray code algorithms for acyclic orientations of hypergraphs and lattice quotients, unifying previous methods and addressing open questions.

Findings

Gray codes for hypergraph acyclic orientations

Gray codes for lattice quotients of acyclic orientations

Efficient algorithms for Hamilton paths on polytopes

Abstract

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs. Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung-Hoang-M\"utze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.