The Runaway Rectangle Escape Problem

TL;DR

This paper investigates the Rectangle Escape Problem, focusing on algorithms for extending rectangles within a region while controlling maximum density, with complexity results for various cases including NP-hardness.

Contribution

It provides approximation and exact algorithms for fixed density bounds and proves NP-hardness for specific special cases involving unit squares.

Findings

Polynomial algorithms for d=1

NP-hardness for d≥2

NP-hardness persists in grid unit squares case

Abstract

Motivated by the applications of routing in PCB buses, the Rectangle Escape Problem was recently introduced and studied. In this problem, we are given a set of rectangles $\mathcal{S}$ in a rectangular region $R$, and we would like to extend these rectangles to one of the four sides of $R$. Define the density of a point $p$ in $R$ as the number of extended rectangles that contain $p$. The question is then to find an extension with the smallest maximum density. We consider the problem of maximizing the number of rectangles that can be extended when the maximum density allowed is at most $d$. It is known that this problem is polynomially solvable for $d = 1$, and NP-hard for any $d \geq 2$. We consider approximation and exact algorithms for fixed values of $d$. We also show that a very special case of this problem, when all the rectangles are unit squares from a grid, continues to be NP-hard for $d = 2$.