In this paper we study 2-dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic eld. We s h o w h o w using a technique called branch and cut, the exact ground states of grids of sizes up to 100 100 can be determined in a moderate amount of computation time, and we r e p o r t on extensive computational tests. With our method we produce results based on more than 20 000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other more popular ways to compute the ground state, like M o n te Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determined a priori. Our ground state energy estimation at zero eld is ;1:317.

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In the present paper we deal with equipartition problems for special classes of trees, i.e., spiders, stars, worms, and caterpillars. We prove that the equipartition problem is NP-complete for spiders (and, hence, for general trees); on the other hand, we give efficient polynomial-time algorithms for stars, worms, and caterpillars.