Limit theory of discrete mathematics problems

TL;DR

This paper introduces limit theory as a powerful tool for asymptotic analysis of discrete mathematics problems, demonstrating its application on three related problems including Alpern's Caching Game and two conjectures.

Contribution

It develops and applies limit theory to analyze complex discrete problems, providing new insights and partial solutions, especially for Alpern's Caching Game and related conjectures.

Findings

Optimal hiding strategies involve digging less than 1 in total with positive probability.

Limit problem analysis yields partial solutions to the Caching Game.

Progress made on the Manickam–Miklós–Singhi and Kikuta–Ruckle conjectures.

Abstract

We show a general problem-solving tool called limit theory. This is an advanced version of asymptotic analysis of discrete problems when some finite parameter tends to infinity. We will apply it on three closely related problems. Alpern's Caching Game (for 2 nuts) is defined as follows. The hider caches 2 nuts into one or two of $n$ potential holes by digging at most 1 depth in total. The goal of the searcher is to find both nuts in a limited time $h$, otherwise the hider wins. We will show that if $h$ and $n/h$ are large enough, then very counterintuitively, any optimal hiding strategy should dig less than 1 in total, with positive probability. We will prove it by defining and analyzing a limit problem. Then we will partially solve the entire problem. We will also have significant progress with two other problems: the Manickam--Mikl\'os--Singhi Conjecture and the Kikuta--Ruckle Conjecture.