Globally Optimal Dense and Sparse Spanning Trees, and their Applications

Finding spanning trees under various constraints is a classic problem with applications in many fields. Recently, a novel notion of "dense" ("sparse") tree, and in particular spanning tree (DST and SST respectively), is introduced as the structure that have a large (small) number of subtrees, or small (large) sum of distances between vertices. We show that finding DST and SST reduces to solving the discrete optimization problems. New and efficient approaches to find such spanning trees is achieved by imposing certain conditions on the vertex degrees which are then used to define an objective function that is minimized over all spanning trees of the graph under consideration. Solving this minimization problem exactly may be prohibitively time consuming for large graphs. Hence, we propose to use genetic algorithm (GA) which is one of well known metaheuristics methods to solve DST and SST approximately. As far as we are aware this is the first time GA has been used in this context. We also demonstrate on a number of applications that GA approach is well suited for these types of problems both in computational efficiency and accuracy of the approximate solution. Furthermore, we improve the efficiency of the proposed method by using Kruskal's algorithm in combination with GA. The application of our methods to several practical large graphs and networks is presented. Computational results show that they perform faster than previously proposed heuristic methods and produce more accurate solutions. Furthermore, the new feature of the proposed approach is that it can be applied recursively to sub-trees or spanning trees with additional constraints in order to further investigate the graphical properties of the graph and/or network. The application of this methodology on the gene network of a cancer cell led to isolating key genes in a network that were not obvious from previous studies.