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.

Abstract. The max-cut problem asks for partitioning the nodes V of a graph G = (V, E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NPhard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. [45] and that of Berman et al. [9]. The running time of the former can be bounded by O(|V | In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by O(|V | 3 2 log |V |), similar to the bound achieved by [45]. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 10 6 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of [45] and [9]. It turns out that our algorithm is considerably faster in practice than [45]. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of [9]. However, whereas the procedure of generating the expanded graph in [9] is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task.

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A minimum equicut of an edge-weighted graph is a partition of the nodes of the graph into two sets of equal size such that the sum of the weights of edges joining nodes in different partitions is minimum. We compare basic linear and semidefinite relaxations for the equicut problem, and find that linear bounds are competitive with the corresponding semidefinite ones but can be computed much faster. Motivated by an application of equicut in theoretical physics, we revisit an approach by Brunetta et al. and present an enhanced branch-and-cut algorithm. Our computational results suggest that the proposed branch-andcut algorithm has a better performance than the algorithm of Brunetta et al.. Further, it is able to solve to optimality in reasonable time several instances with more than 200 nodes from the physics application.