A Constant Factor Approximation for Orthogonal Order Preserving Layout Adjustment

TL;DR

This paper presents a constant factor approximation algorithm for the Layout Adjustment for Disjoint Rectangles (LADR) problem, which involves repositioning rectangles to minimize bounding box area while preserving their initial order, addressing its computational hardness.

Contribution

The paper introduces the first constant factor approximation algorithm for LADR, improving upon previous heuristics and establishing its APX-hardness.

Findings

LADR is NP-hard and APX-hard.

A constant factor approximation algorithm for LADR is proposed.

The approach guarantees solutions within a constant factor of the optimal.

Abstract

Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e.\ maintains the sorted ordering of the rectangle centers along both $x$-axis and $y$-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles(LADR). It was known that LADR is $\mathbb{NP}$-hard, but only heuristics were known for it. We show that a certain decision version of LADR is $\mathbb{APX}$-hard, and give a constant factor approximation for LADR.