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⊆U×V), a 2-level drawing (πU,πV) of G is described by a linear ordering πU:U↔{1,…,|U|} of U and linear ordering πV:V↔{1,…,|V|} of V. For a fixed linear ordering πU of U, the OSCM problem seeks to find a linear ordering πV of V that yields a 2-level drawing (πU,πV) 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⁎(1.728n) time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in O⁎(2n) time and polynomial space.