Automated Discharging Arguments for Density Problems in Grids

TL;DR

This paper introduces an algorithmic framework that automates discharging arguments to establish lower bounds on the density of identifying codes in various infinite grids, improving upon previous bounds.

Contribution

The authors develop a linear programming-based method to automate discharging arguments, providing new lower bounds for identifying codes in hexagonal, square, and triangular grids.

Findings

Lower bound of 23/55 (~0.4182) for the hexagonal grid identifying code

Automated framework replaces manual case analysis with linear programming

Sharp bounds found for variations of identifying codes in multiple grids

Abstract

Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of $\frac{3}{7} \approx 0.428571$ and a lower bound of $\frac{5}{12}\approx 0.416666$. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of $\frac{23}{55} \approx 0.418181$ on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.