A Worst-Case Optimal Algorithm to Compute the Minkowski Sum of Convex Polytopes

Abstract

We propose algorithms to compute the Minkowski sum of a set of convex polytopes in $\mathbb {R}^d$. The input and output of the proposed algorithms are the face lattices of the input and output polytopes respectively. We first present the algorithm for the Minkowski sum of two convex polytopes. The time complexity of this algorithm is $O(d^\omega nm)$ where n and m are the face lattice sizes of the two input polytopes and $\omega$ is the matrix multiplication exponent ($\omega \sim 2.373$). Our algorithm for two summands is worst-case optimal for fixed d. We generalize this algorithm for r convex polytopes, say $P_i$, $1\le i \le r$. The time complexity of this generalization is $O(\min \{d^\omega \!N\!M,d^\omega \!r\prod |P_i|\})$ where $N = \sum |P_i|$ is the total size of the face lattices of the r input polytopes and M is the size of the face lattice of their Minkowski sum $P_1 \oplus \cdots \oplus P_r$. Our algorithm for multiple summands is worst-case optimal for fixed $d \ge 3$ and $r < d$.