Georgia Institute of Technology College of Sciences

Finding ground states of spin glasses, a model of disordered materials, has a deep connection to many hard combinatorial optimization problems, such as satisfiability, maxcut, graph-bipartitioning, and coloring. Much insight has been gained for the combinatorial problems from the intuitive approaches developed in physics (such as replica theory and the cavity method), some of which have been proven rigorously recently. I present a treasure trove of numerical data obtained with heuristic methods that suggest a number conjectures, such as an equivalence between maxcut and bipartitioning for r-regular graphs, a simple relation for their optimal configurations as a function of degree r, and anomalous extreme-value fluctuations in a variety of models, hotly debated in physics currently. For some, such as those related to finite-size effects, not even a physics theory exists, for others theory exists that calls for rigorous methods.