Hardness of the Maximum Independent Set Problem on Unit-Disk Graphs and Prospects for Quantum Speedups

TL;DR

It is found that quasi-planar instances with Union-Jack-like connectivity can be solved to optimality for up to thousands of nodes within minutes, with both custom and generic commercial solvers on commodity hardware, without any instance-specific fine-tuning.

Abstract

Rydberg atom arrays are among the leading contenders for the demonstration of quantum speedups. Motivated by recent experiments with up to 289 qubits [Ebadi et al., Science 376, 1209 (2022)] we study the maximum independent set problem on unit-disk graphs with a broader range of classical solvers beyond the scope of the original paper. We carry out extensive numerical studies and assess problem hardness, using both exact and heuristic algorithms. We find that quasi-planar instances with Union-Jack-like connectivity can be solved to optimality for up to thousands of nodes within minutes, with both custom and generic commercial solvers on commodity hardware, without any instance-specific fine-tuning. We also perform a scaling analysis, showing that by relaxing the constraints on the classical simulated annealing algorithms considered in Ebadi et al., our implementation is competitive with the quantum algorithms. Conversely, instances with larger connectivity or less structure are shown to display a time-to-solution potentially orders of magnitudes larger. Based on these results we propose protocols to systematically tune problem hardness, motivating experiments with Rydberg atom arrays on instances orders of magnitude harder (for established classical solvers) than previously studied.

Figures (13)

• Figure 1: Schematic illustration of the problem. (a) We consider unit-disk graphs with nodes arranged on a two-dimensional square lattice with lattice spacing $a$ and filling fraction $\varrho \sim 80\%$, and edges connecting all pairs of nodes within a unit distance (illustrated by the circle). For $\sqrt{2}a \leq R_{b} < 2a$ (as considered here), nodes are connected to nearest and next-nearest neighbors resulting in a (quasi-planar) Union-Jack pattern with maximum degree $d_{\mathrm{max}}=8$. (b) Our goal is to solve the MIS problem on this family of instances (as depicted here with nodes colored in red in the right panel) and assess the hardness thereof using both exact and heuristic algorithms.
• Figure 2: Schematic illustration of the (exact) sweeping line algorithm (SLA) as applied to the MIS-UD problem. (a) SLA proceeds by sweeping a fictitious line across the graph and tracking all potentially optimal MIS configurations on this boundary, efficiently exploiting the quasi-planar structure of the UD graph. Processed nodes are shown in gray, boundary nodes in blue and unprocessed nodes in yellow. (b) The light blue node from (a) is added to the boundary, while the bottom left blue node is dropped (as a result of not having any more connections to unprocessed nodes).
• Figure 3: Schematic illustration of the heuristic simulated annealing (SA) solver. The original configuration (a) is overlaid with connectivity statistics in (b): nodes in the set (red), nodes without marked neighbors (white), and nodes with a single neighbor (blue). Potential moves are (i) removal of (red) nodes currently in the set, (ii) addition of currently white nodes to the set, and (iii) swapping a blue node with its red neighbor. Grey nodes have more than one adjacent node in the set and no valid moves. From (b) to (c), one node is added to the set and the statistics are updated accordingly. From (c) to (d) one blue node is swapped with its adjacent red node.
• Figure 4: Time-to-solution (TTS) for the exact solvers. (a) TTS for the exact SLA solver as a function of system size $N$. For every system size $N$, 1000 random UD instances with $\varrho=0.8$ have been considered. The data fit reasonably to $\mathrm{TTS}(N)\approx cN\phi^{\sqrt{N}}$, where the basis of $\phi\approx 1.62$ is the theoretical expectation for the Fibonacci sequence. At larger system sizes ($N > 500$), high memory usage causes slower access times (cache misses), resulting in a substantially larger pre-factor $c'$. (b) TTS for the B&B solver as a function of system size $N$. For every system size $N$, 1000 random UD instances have been considered; see Section \ref{['plot-description']} for box plot description. Problems with hundreds (thousands) of nodes can be solved to optimality in sub-second (minute) timescales. The solid line is the linear regression over instances whose TTS are in the top highest $2\%$. The linear regression minimizes the residual sum of squares of log(TTS).
• Figure 5: Time required to reach 99% success probability ($\mathrm{TTS}_{99}$) for the heuristic SA solver as a function of system size $N$ (i.e., how long the solver should run for a 99% chance of finding the optimal solution). For every system size $N$, 1000 random UD instances at $\varrho=0.8$ filling have been considered; see Section \ref{['plot-description']} for box plot description. The solid line is the linear regression over instances whose TTS are in the top highest $2\%$. The linear regression minimizes the residual sum of squares of log(TTS).