Rectangle Transformation Problem

TL;DR

This paper introduces the rectangle transformation problem (RTP), analyzes its variants, and provides a polynomial-time approximation algorithm for a specific case, revealing complexity and open questions in rectangle partition transformations.

Contribution

It defines the RTP and SRTP variants, proves the non-existence of solutions for irrational ratios, and presents an approximation algorithm with bounds for the integral case.

Findings

SRTP has no finite solution for irrational side ratios.

ALGSIRTP achieves an approximation within $q/p+O(\sqrt{p})$ for SIRTP.

No constant-factor solution exists for all integer side ratios in SIRTP.

Abstract

In this paper, we propose the rectangle transformation problem (RTP) and its variants. RTP asks for a transformation by a rectangle partition between two rectangles of the same area. We are interested in the minimum RTP which requires to minimize the partition size. We mainly focus on the strict rectangle transformation problem (SRTP) in which rotation is not allowed in transforming. We show that SRTP has no finite solution if the ratio of the two parallel side lengths of input rectangles is irrational. So we turn to its complement denoted by SIRTP, in which case all side lengths can be assumed integral. We give a polynomial time algorithm ALGSIRTP which gives a solution at most $q/p+O(\sqrt{p})$ to SIRTP$(p,q)$ ($q\geq p$), where $p$ and $q$ are two integer side lengths of input rectangles $p\times q$ and $q\times p$, and so ALGSIRTP is a $O(\sqrt{p})$-approximation algorithm for minimum SIRTP$(p,q)$. On the other hand, we show that there is not constant solution to SIRTP$(p,q)$ for all integers $p$ and $q$ ($q>p$) even though the ratio $q/p$ is within any constant range. We also raise a series of open questions for the research along this line.