Bounded-degree spanning tree problems: models and new algorithms

Abstract

Spanning trees, Spanning spiders, Branch vertices, Heuristics, Optical networks,

Exact and heuristic solutions for the Minimum Number of Branch Vertices Spanning Tree Problem

“…Silva et al (2011) propose an adaptation of the Iterative Refinement approach to solve MBV, choosing to penalize violating edges and replace them with edges with less violation. They report better computational results than those obtained by Cerulli et al (2009) on a set of small instances. Sundar et al (2012) design two heuristic algorithms, one that starts with an ST and interchanges one tree edge and one edge outside the tree, keeping the acyclic structure and trying to improve the solution, together with an ant colony heuristic.…”

“…Still in this family, there is a sub-family of interesting problems which optimize a measure that depends only on the so-called branch vertices (alternatively branching nodes, branches), those nodes in the tree with a degree greater than two. One of these problems is the Minimum Degree Sum of Branch Vertices STP, whose objective function is the sum of the degrees of all the vertices except those of degrees 1 and 2 (Cerrone et al, 2014;Cerulli, Gentili, & Iossa, 2009;Sundar, Singh, & Rossi, 2012). Finally, the Minimum Number of Branch Vertices STP, MBV, looks for the spanning tree with minimum number of vertices of degree greater than two, which is what we are interested in.…”

“…These switches must then be located in all the branching nodes and the cost of the network depends on their number. Cerulli et al (2009) propose an Integer Programming formulation for the MBV and designed different heuristic strategies. Their formulation guarantees connection by sending a unit of flow from a root to any other node of the graph.…”

“…Sundar et al (2012) design two heuristic algorithms, one that starts with an ST and interchanges one tree edge and one edge outside the tree, keeping the acyclic structure and trying to improve the solution, together with an ant colony heuristic. They use the same IP formulation as in Cerulli et al (2009), and generate similar random instances to make a comparison with the results in Cerulli et al (2009). Xpress solver is used here, and the authors concluded that their heuristics work better than previous ones.…”

Exact solution for branch vertices constrained spanning problems

“…Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is strictly greater than 2. The Minimum Branch Vertices Spanning Tree problem (MBVST) consists in finding a spanning tree of G with the minimum number of branch vertices [CGI09]. All researches on branch vertices constrained spanning problems are based on spanning trees, Conversely the routing do not explicitly impose a sub-graph as solution.…”