Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle Union

TL;DR

This paper establishes new computational hardness results for polygon placement problems, showing that increasing degrees of freedom in placement lead to higher complexity bounds under the k-SUM conjecture, and provides an improved algorithm for specific cases.

Contribution

The paper proves natural hardness results for polygon placement problems that grow with each added degree of freedom, and offers an improved algorithm for constant-sized polygons.

Findings

Hardness of n^{2-o(1)} under 3SUM for x-translation placement

Hardness of n^{2-o(1)} under 4SUM for arbitrary translation

An O((pq)^{2.5})-time algorithm for orthogonal polygons with p and q vertices

Abstract

We revisit the classical problem of determining the largest copy of a simple polygon $P$ that can be placed into a simple polygon $Q$. Despite significant effort, known algorithms require high polynomial running times. (Barequet and Har-Peled, 2001) give a lower bound of $n^{2-o(1)}$ under the 3SUM conjecture when $P$ and $Q$ are (convex) polygons with $\Theta(n)$ vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized $P$ or $Q$. In this paper, we affirmatively answer these questions under the $k$SUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, $x$-translation, $y$-translation, rotation): (1) Finding the largest copy of $P$ that can be $x$-translated into $Q$ requires time $n^{2-o(1)}$ under the 3SUM conjecture. (2) Finding the largest copy of $P$ that can be arbitrarily translated into $Q$ requires time $n^{2-o(1)}$ under the 4SUM conjecture. (3) The above lower bounds are almost tight when one of the polygons is of constant size: we obtain an $\tilde O((pq)^{2.5})$-time algorithm for orthogonal polygons $P,Q$ with $p$ and $q$ vertices, respectively. (4) Finding the largest copy of $P$ that can be arbitrarily rotated and translated into $Q$ requires time $n^{3-o(1)}$ under the 5SUM conjecture. We are not aware of any other such natural $($degree of freedom $+ 1)$-SUM hardness for a geometric optimization problem.