An Integer Linear Programming Model for Tilings

TL;DR

This paper introduces an integer linear programming model to analyze aperiodic rhythm tilings, enabling efficient verification of tiling conditions and enumeration of all possible tilings, supported by experimental validation.

Contribution

The paper presents a novel ILP model for rhythm tilings, facilitating efficient checking of tiling conditions and comprehensive enumeration of solutions.

Findings

Efficient verification of the Coven-Meyerowitz (T2) condition.

Algorithm to find all tilings of a given rhythm.

Experimental results demonstrate model's time efficiency.

Abstract

In this paper, we propose an Integer Linear Model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how this model can be used to efficiently check the necessity of the Coven-Meyerowitz's $(T2)$ condition and also to define an iterative algorithm that finds all the possible tilings of the rhythm A. To conclude, we run several experiments to validate the time efficiency of this model.