Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions

We use surrogate analysis and constraint pairing in multidimensional knapsack problems to fix some variables to zero and to separate the rest into two groups – those that tend to be zero and those that tend to be one, in an optimal integer solution. Using an initial feasible integer solution, we generate logic cuts based on our analysis before solving the problem with branch and bound. Computational testing, including the set of problems in the OR-library and our own set of difficult problems, shows our approach helps to solve difficult problems in a reasonable amount of time and, in most cases, with a fewer number of nodes in the search tree than leading commercial software.     

New Results on the Queens_n 2 Graph Coloring Problem

For the Queens_n ² graph coloring problems no chromatic numbers are available for n > 9 except where n is not a multiple of 2 or 3. In this paper we propose an exact algorithm that takes advantage of the particular structure of these graphs. The algorithm works on the independent sets of the graph rather than on the vertices to be colored. It combines branch and bound, for independent set assignment, with a clique based filtering procedure. A first experimentation of this approach provided the coloring number values ranging for n = 10 to n = 14.     

A New Method for Enumerating Independent Sets of a Fixed Size in General Graphs

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin [7] holds for all but finitely many graphs. We also use our method to prove special cases of a conjecture of Kahn [13]. In addition, we show that our method is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and we argue that it can be used in many other cases when dealing with graphs that have numerous structural restrictions.     

Graph Optimization Approaches for Minimal Rerouting in Symmetric Three Stage Clos Networks

We consider routing in symmetrical three stage Clos networks. Especially we search for the routing of an additional connection that requires the least rearrangements, i.e. the minimal number of changes of already routed connections. We describe polynomial methods, based on matchings and edge colorings. The basic idea is to swap colors along alternating paths. The paths need to be maximal, and the shortest of these maximal paths is chosen, since it minimizes the rerouting that needs to be done. Computational tests confirm the efficiency of the approach.