# Globally Optimal Dense and Sparse Spanning Trees, and their Applications

• Mustafa Ozen New Jersey Institute of Technology, Newark, NJ, 07102, USA
• Goran Lesaja Georgia Southern University, USA
• Hua Wang Georgia Southern University, Statesboro, Georgia
Keywords: Graphs, dense and sparse spanning trees, degree sequence, global optimization, genetic algorithm, discrete optimization, Kruskal’s algorithm.

### 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.

### Author Biographies

Mustafa Ozen, New Jersey Institute of Technology, Newark, NJ, 07102, USA
Ph.D student at the New Jersey Institute of Technology. Expected to graduate in May 2020.
Goran Lesaja, Georgia Southern University, USA
Professor in the Department of mathematical Sciences at Georgia Southern University.
Hua Wang, Georgia Southern University, Statesboro, Georgia
Professor  in the Department of Mathematical Sciences at Georgia Southern University.

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Published
2020-05-27
How to Cite
Ozen, M., Lesaja, G., & Wang, H. (2020). Globally Optimal Dense and Sparse Spanning Trees, and their Applications. Statistics, Optimization & Information Computing, 8(2), 328-345. https://doi.org/10.19139/soic-2310-5070-855
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Section
Research Articles