Weakly symmetric graphs, elementary landscapes, and the TSP

Weakly symmetric graphs, elementary landscapes, and the TSP

Quasi-elementary Landscapes and Superpositions of Elementary Landscapes

There exist local search landscapes where the evaluation function is an eigenfunction of the graph Laplacian that corresponds to the neighborhood structure of the search space. Problems that display this structure are called “Elementary Landscapes” and they have a number of special mathematical properties. The term “Quasi-elementary landscapes” is introduced to describe landscapes that are “alm...

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 Mi...

On the maximum even factor in weakly symmetric graphs

As a common generalization of matchings and matroid intersection, W.H. Cunningham and J.F. Geelen introduced the notion of path-matchings, then they introduced the more general notion of even factor in weakly symmetric digraphs. Here we give a min-max formula for the maximum cardinality of an even factor. Our proof is purely combinatorial. We also provide a Gallai-Edmonds-type structure theorem...

The theory of elementary landscapes

When joined to a stipulated neighborhood digraph, an objective function deÞned on the solution space of a real combinatorial optimization problem forms a landscape. Grover shows that landscapes satisfying a certain difference equation have properties favorable to local search. Studying only symmetric and regular neighborhood digraphs, Stadler deÞnes elementary landscapes as those which can be r...