Afrl-afosr-va-tr-2015-0320 Exploiting Elementary Landscapes for Tsp, Vehicle Routing and Scheduling

There are a number of NP-hard optimization problems where the search space can be characterized as an elementary landscape. For these search spaces the evaluation function is an eigenfunction of the Laplacian matrix that describes the neighborhood structure of the search space. Problems such as the Traveling Salesman Problem (TSP), Graph Coloring, the Frequency Assignment Problem, as well as Min-Cut and Max-Cut Graph Partitioning and select simple satisfiability problems all have elementary landscapes. For all elementary landscapes one can compute neighborhood average evaluations without actually evaluating any neighbors. One can also prove that all local optima have an evaluation that is better than the average evaluation over the set of all solutions: there are no arbitrarily poor local optima. And all neighborhoods must contains at least one improving or disimproving move. There are no flat neighborhoods. We have extended this method for k-bound pseudo-Boolean optimization problems so that we can now compute the exact autocorrelation of the search space as well as the exact statistical moments (mean, variance, skew, kurtosis, ....) over generalized exponentially large neighborhoods at Hamming distance d for any arbitrary point in the search space. We have also proven that all k-bounded pseudo-Boolean optimization problems, such as MAXkSAT and NK-landscapes, can be expressed as a superposition of k elementary landscapes. We have also exploited key properties of elementary landscapes to develop new search methods that are capable of tunneling between local optima, as well as filtering local optima. Tunneling means that given two local optima, the algorithm is able to construct two (or more) new local optima without any additional search. Filtering means that the tunneling operator can reach thousands (even millions) of different local optima, but it automatically selects the best of all of the reachable local optima. For the TSP, tunneling and filtering can be done in O(N) time; a new algorithm which displays tunneling and filtering behavior has already been shown to find better TSP solutions faster than the Chained-Lin-Kernighan algorithm. For clustered Traveling Salesman Problems and asymmetric TSP our results improve on the state of the art for problems instances too large to solve with Branch and Bound methods (e.g. the Concorde TSP solver). 1 DISTRIBUTION A: Distribution approved for public release.