Quantum Algorithms for One-Sided Crossing Minimization
Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista
TL;DR
The paper tackles quantum approaches to OSCM, a hard graph-drawing problem, by recasting it as a set decision problem over the bottom partition and applying quantum speedups. It presents two complementary quantum algorithms: a QRAM-based quantum dynamic programming method that runs in $\mathcal{O}^*(1.728^n)$ time/space, and a QRAM-free quantum divide-and-conquer method with $\mathcal{O}^*(2^n)$ time and polynomial space, the latter avoiding explicit storage of partial results. The techniques rely on Ambainis et al.'s quantum DP framework for set problems and a divide-and-conquer scheme tailored to OSCM, with a gamma function capturing cross-term interactions between subsets. Compared to classical approaches, the quantum results offer improved time-space tradeoffs, particularly for larger crossing budgets, and they extend to related problems OSSCM and TLCM, highlighting potential practical quantum speedups in graph drawing tasks.
Abstract
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 $(π_U,π_V)$ of $G$ is described by a linear ordering $π_U: U \leftrightarrow \{1,\dots,|U|\}$ of $U$ and linear ordering $π_V: V \leftrightarrow \{1,\dots,|V|\}$ of $V$. For a fixed linear ordering $π_U$ of $U$, the OSCM problem seeks to find a linear ordering $π_V$ of $V$ that yields a $2$-level drawing $(π_U,π_V)$ 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 $O^*(1.728^n)$ time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in $O^*(2^n)$ time and polynomial space.