DIMACSWorkshop on Distance Geometry Theory and Applications

1. Tue 26 July, 9-9:30 Marcia Fampa, COPPE, Federal University of Rio de Janeiro, Brazil Modeling the Euclidean Steiner Tree problem In the Euclidean Steiner Tree Problem, the goal is to find a network of minimum length interconnecting a set P of given points in the n-dimensional Euclidean space. Such networks may be represented by a tree T , where the set of nodes is given by the points in P , known as terminals, and possibly by additional points, known as Steiner points. The length of the network is defined as the sum of the Euclidean lengths of the edges inT . We will present mixed integer nonlinear programming formulations for the problem from the literature, and discuss the difficulties involved in solving them by branch-and-bound algorithms. Different techniques to overcome these difficulties are proposed and some numerical results show their impact on the solution of the problem. (Joint work with Claudia D’Ambrosio, Jon Lee, Nelson Maculan, Wendel Melo and Stefan Vigerske.) 2. Tue 26 July, 9:30-10 Antonios Varvitsiotis,NTU Singapore Graph Cores via Universal Completability A framework for a graph G = (V,E), denoted G(p), consists of an assignment of real vectors p = (p1, p2, ..., pn), where n = |V |, to its vertices. A framework G(p) is called universally completable if for any other frameworkG(q) that satisfies pi pj = q⊤ i qj for all i = j and (i, j) inE there exists an isometry U such that Uqi = pi for all i in V . A graph is called a core if all its endomorphisms are automorphisms. In this work we identify a new sufficient condition for showing that a graph is a core in terms of the universal completability of an appropriate framework for the graph. To use this condition we develop a method for constructing universally completable frameworks based on the eigenvectors for the smallest eigenspace of the graph. This allows us to recover the known result that the Kneser graphKn:r and the q-Kneser graph qKn:r are cores for n ≥ 2r+1. Our proof is simple and does not rely on the use of an Erdös-Ko-Rado type result as do existing proofs. Furthermore, we also show that a new family of graphs from the binary Hamming scheme are cores, which was not known before. (Joint work with Chris Godsil, David Roberson, Brendan Rooney and Robert Šámal.)