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

Abstract

We revisit the classical problem of determining the largest copy of a simple polygon PP that can be placed into a simple polygon QQ. Despite significant effort, known algorithms require high polynomial running times. (Barequet and Har-Peled, 2001) give a lower bound of n2o(1)n^{2-o(1)} under the 3SUM conjecture when PP and QQ are (convex) polygons with Θ(n)\Theta(n) vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized PP or QQ. In this paper, we affirmatively answer these questions under the kkSUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, xx-translation, yy-translation, rotation): (1) Finding the largest copy of PP that can be xx-translated into QQ requires time n2o(1)n^{2-o(1)} under the 3SUM conjecture. (2) Finding the largest copy of PP that can be arbitrarily translated into QQ requires time n2o(1)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 O~((pq)2.5)\tilde O((pq)^{2.5})-time algorithm for orthogonal polygons P,QP,Q with pp and qq vertices, respectively. (4) Finding the largest copy of PP that can be arbitrarily rotated and translated into QQ requires time n3o(1)n^{3-o(1)} under the 5SUM conjecture. We are not aware of any other such natural ((degree of freedom +1)+ 1)-SUM hardness for a geometric optimization problem

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