Most real world combinatorial optimization problems are affected by noise in the input data. Search algorithms to identify "good" solutions with low costs behave like the dynamics of large disordered particle systems, e.g., random networks or spin glasses. Such solutions to noise perturbed optimization problems are characterized by Gibbs distributions when the optimization algorithm searches for typical solutions by stochastically minimizing costs. The free energy that determines the normalization of the Gibbs distribution balances cost minimization relative to entropy maximization.The problem to analytically compute the free energy of disordered systems has been known as a notoriously difficult mathematical challenge for at least half a century (Talagrand, 2003). We provide rigorous, matching upper and lower bounds on the free energy for two disordered combinatorial optimization problems of random graph instances, the sparse Minimum Bisection Problem (sMBP) and Lawler's Quadratic Assignment Problem (LQAP). These two problems exhibit phase transitions that are equivalent to the statistical behavior of Derrida's Random Energy Model (REM). Both optimization problems can be characterized as parameter rich since individual solutions depend on more parameters than the logarithm of the solution space cardinality would suggest for e.g. a coordinate representation. Noisy Combinatorial OptimizationCombinatorial optimization arises in many real world settings and these problems are often notoriously difficult to solve due to data dependent noise in the parameters. Apart from algorithmic questions -like efficient (stochastic) search for solutions with provable guarantees -more theoretical challenges, such as generalization of solutions and their typicality relate to the * jbuhmann@inf. analytical computation of various macroscopic properties (Frenk et al., 1985) like the free energy and these problems remain largely open. Especially, the free energy of the corresponding Gibbs distribution is one of those most important macroscopic parameters that often arises in the context of combinatorial optimization. For example, Vannimenus and Mézard (1984) explored the free energy properties of the traveling salesman problem. In this paper we compute the free energy for two optimization problems -sparse Minimum Bisection (sMBP) and Lawler's Quadratic Assignment (LQAP).Both, sMBP and LQAP belong to a class of optimization problems that can be formulated as follows: Let n be an integer (e.g., number of vertices in a graph, size of a matrix, number of keys in a digital tree, etc.), and S n a set of objects (e.g., a set of vertices, elements of a matrix, keys, etc). The data X denote a set of random variables that enter into the definition of an instance (e.g., weights of edges in a weighted graph). One often is interested in asymptotic behavior of the optimal values R max or R min defined aswhere C n is a set of all feasible solutions, S n (c) is a set of objects from S n belonging to the c-th feasible solution (e.g., set of edg...
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