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Studying spin-glass physics through analyzing their ground-state propertieshas a long history. Although there exist polynomial-time algorithms for thetwo-dimensional planar case, where the problem of finding ground states istransformed to a minimum-weight perfect matching problem, the reachable systemsizes have been limited both by the needed CPU time and by memory requirements.In this work, we present an algorithm for the calculation of exact groundstates for two-dimensional Ising spin glasses with free boundary conditions inat least one direction. The algorithmic foundations of the method date back tothe work of Kasteleyn from the 1960s for computing the complete partitionfunction of the Ising model. Using Kasteleyn cities, we calculate exact groundstates for huge two-dimensional planar Ising spin-glass lattices (up to3000x3000 spins) within reasonable time. According to our knowledge, these arethe largest sizes currently available. Kasteleyn cities were recently also usedby Thomas and Middleton in the context of extended ground states on the torus.Moreover, they show that the method can also be used for computing groundstates of planar graphs. Furthermore, we point out that the correctness ofheuristically computed ground states can easily be verified. Finally, weevaluate the solution quality of heuristic variants of the Bieche et al.approach.

Exact Ground States of Large Two-Dimensional Planar Ising Spin Glasses