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