Exact ground states of large two-dimensional planar Ising spin glasses

Abstract

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 3000;{2} 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 L. Bieche approach.

Partitioning planar graphs: a fast combinatorial approach for max-cut

“…Using the presented approach leads both to a faster and to a less memory-consuming approach. For a comparison between the exact approach and the heuristic variant just mentioned, see [41].…”

“…This work is the full version of an earlier technical report [32]. The proposed max-cut algorithm for arbitrary weighted planar graphs is a generalization of the methods from [47,41]. The latter focused on the determination of minimum cuts in two-dimensional grid graphs for determining ground states of Ising spin glasses in physics.…”

Spinto: High-performance energy minimization in spin glasses

“…While MWPM is poly-time solvable, practical runtime and memory usage do not scale to very large instances. To overcome these limitations, the work in [10] describes a heuristic based on the MWPM reduction. Another special case is that of ferromagnetic (J i,j > 0) GSD instances, which Barahona [3] reduced to (s-t)-min-cut or max-flow.…”

“…Our heuristic scales to a million spins and empirical runtimes closely fit n log(n). We compared the runtime of our local search against that of the MWPM-based heuristic proposed in [10]. The solution quality of this heuristic depends on a particular choice of parameters and does not work on Gaussian-distributed instances.…”

“…In contrast, our heuristic does not have such dependencies and works on all instances. Table IV in [10] shows the runtime and solution quality of the MWPM-based heuristic ±1 grid graphs of size 164×164. This heuristic takes 59s on average producing the optimal value 61% of the time.…”