  Studying spin-glass physics through analyzing their ground-state properties
has a long history. Although there exist polynomial-time algorithms for the
two-dimensional planar case, where the problem of finding ground states is
transformed to a minimum-weight perfect matching problem, the reachable system
sizes 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 ground
states for two-dimensional Ising spin glasses with free boundary conditions in
at least one direction. The algorithmic foundations of the method date back to
the work of Kasteleyn from the 1960s for computing the complete partition
function of the Ising model. Using Kasteleyn cities, we calculate exact ground
states for huge two-dimensional planar Ising spin-glass lattices (up to
3000x3000 spins) within reasonable time. According to our knowledge, these are
the largest sizes currently available. Kasteleyn cities were recently also used
by 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 ground
states of planar graphs. Furthermore, we point out that the correctness of
heuristically computed ground states can easily be verified. Finally, we
evaluate the solution quality of heuristic variants of the Bieche et al.
approach.
