Quantum algorithms for one-sided crossing minimization

We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. Given an n-vertex bipartite graph $G=(U,V,E\subseteq U\times V)$, a 2-level drawing $({\pi }_U,{\pi }_V)$ of G is described by a linear ordering ${\pi }_U:U\leftrightarrow \{1,\dots ,|U\left|\right\}$ of U and linear ordering ${\pi }_V:V\leftrightarrow \{1,\dots ,|V\left|\right\}$ of V. For a fixed linear ordering ${\pi }_U$ of U, the OSCM problem seeks to find a linear ordering ${\pi }_V$ of V that yields a 2-level drawing $({\pi }_U,{\pi }_V)$ of G with the minimum number of edge crossings. We show that OSCM can be viewed as a set problem over V amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. SODA 2019] to devise a QRAM-based algorithm that solves OSCM in $O^{⁎}\left({1.728}^n\right)$ time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in $O^{⁎}\left(2^n\right)$ time and polynomial space.