On the exact complexity of polyomino packing

Abstract

We show that the problem of deciding whether a collection of polyominoes, each fitting in a 2×O(log⁡n) rectangle, can be packed into a 3×n box does not admit a 2o(n/log⁡n)-time algorithm, unless the Exponential Time Hypothesis fails. We also give an algorithm that attains this lower bound, solving any instance of polyomino packing with total area n in 2O(n/log⁡n) time. This establishes a tight bound on the complexity of POLYOMINO PACKING, even in a very restricted case. In contrast, for a 2×n box, we show that the problem can be solved in strongly subexponential time.