Growth of Replacements

TL;DR

This paper studies a combinatorial game related to Petri nets and chip-firing, establishing the existence of a growth rate bb, its computation via linear programming, and methods for approximating it.

Contribution

It introduces the concept of the growth rate bb for the game, links it to pseudo-loops and linear programs, and provides algorithms for computing and approximating bb.

Findings

The sequence g(n)/n converges to a growth rate bb.

bb can be computed using a linear program and related algorithms.

Multiple proofs establish the existence and properties of bb.

Abstract

The following game in a similar formulation to Petri nets and chip-firing games is studied: Given a finite collection of baskets, each has an infinite number of balls of the same value. Initially, a ball from some basket is chosen to put on the table. Subsequently, in each step a ball from the table is chosen to be replaced by some $2$ balls from some baskets. Which baskets to take depend only on the ball to be replaced and they are decided in advance. Given some $n$, the object of the game is to find the maximum possible sum of values $g(n)$ for a table of $n$ balls. In this article, the sequence $g(n)/n$ for $n=1,2,\dots$ will be shown to converge to a growth rate $\lambda$. Furthermore, this value $\lambda$ is also the rate of a structure called pseudo-loop and the solution of a rather simple linear program. The structure and the linear program are closely related, e.g. a solution of the linear program gives a pseudo-loop with the rate $\lambda$ in linear time of the number of baskets, and vice versa with the pseudo-loop giving a solution to the dual linear program. A method to test in quadratic time whether a given $\lambda_0$ is smaller than $\lambda$ is provided to approximate $\lambda$. When the values of the balls are all rational, we can compute the precise value of $\lambda$ in cubic time, using the quadratic time rate test algorithm and the binary search with a special condition to stop. Four proofs of the limit $\lambda$ are given: one just uses the relation between the baskets, one uses pseudo-loops, one uses the linear program and one uses Fekete's lemma (the latest proof assumes a condition on the rule of replacements).