Globally Optimal Dense and Sparse Spanning Trees, and their Applications

Abstract

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.

Data-driven structural analysis of Small Cell Lung Cancer transcription factor network suggests potential subtype regulators and transition pathways

“…In this way, we just consider whether there is an interaction between two nodes or not without weighing their importance, which allows us to analyze a relatively unbiased network structure. Then, we search for the DSTs of the SCLC TF network following the approach of [55]. Upon solving the global optimization problem in The found major and side hubs are not only structurally important but also shown to have biological importance to the identified SCLC subtypes.…”

“…The main idea here is to find the optimal subset(s) of edges E(G) from which the constructed DST has the optimal objective value which is the total distances between the individual nodes. For more mathematical details and possible applications of this approach, we refer the reader to [54,55]. Upon solving the optimization problem (1) via Genetic Algorithm (GA), which is a metaheuristic optimization method that attempts to find the global optimum or at least its good approximation [66], we observed 146,143 DSTs with the same objective value.…”

“…To identify the hubs of the SCLC TF network, we implement a graph theory concept called Dense Spanning Tree (DST, see Box 1), which can be found by solving an optimization problem (Methods section A) [53][54][55]. We initially analyze a relatively unbiased network structure by considering the undirected and unweighted network.…”

Spanning Trees of Recursive Scale-Free Graphs

“…In this class of trees where only rule #'s 2 and 3 are applied (small-world spanning trees), if one chooses the R links to always be adjacent to the largest degree node of the original edge (breaking ties randomly), then one obtains a member of the subset of spanning trees with the smallest diameter, i.e. a dense spanning tree having minimal Wiener index [35,42], shown in Fig. 2 (b).…”

“…Common generalizations of this optimization problem for the undirected and unweighted case include the dense, sparse, and minimum routing cost spanning tree problems. Many other variations of optimal spanning trees exist [15,16,[27][28][29][30][31][32][33][34][35][36], and such trees are relevant to an ever widening diversity of applications from optical network design [14,37] to networked oscillator synchronization [38].…”