Relations, models and a memetic approach for three degree-dependent spanning tree problems

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

“…In Sundar et al (2012), an ant colony heuristic was tested, and reported to perform better than the previous ones. The dual ascent based on an IP formulation of Carrabs et al (2013), the edge-swap heuristic of Silva et al (2014) and the memetic approach of Cerrone et al (2014) close the list of approaches that, to the best of our knowledge, are all the methods published to date.…”

“…Although they revisited one of the previous IP formulations, it was not used in their study. The last paper on MBV problem is Cerrone et al (2014). The three smallest IP formulations were again reconsidered and relaxed, and another heuristic method (a memetic algorithm) proposed.…”

“…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.…”

“…E-mail address: amarin@um.es to the structure of the tree. Some problems within this family are the Minimum Leaf STP (Cerrone, Cerulli, & Raiconi, 2014;Salamon & Wiener, 2008), the Full Degree STP (also called Degree Preserving STP) (Bhatia, Khuller, Pless, & Sussmann, 1999;Gendron, Lucena, Salles da Cunha, & Simonetti, 2013;Khuller, Bhatia, & Pless, 2003), which keeps the maximum number of nodes with the same degree as in the original graph, and the Minimum-Degree STP (Fürer & Raghavachari, 1990), where the maximum degree of the nodes in the tree is to be minimized. 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.…”

OMEGA one multi ethnic genetic approach

“…Consider, for example, the problem of finding a degree constrained spanning tree [4,5] of a graph G(V, E) (refer to figure 3.1). The vertex set V has cardinality |V | and, once all edges have been numerated, we can use as chromosome c any integer vector of size greater than or equal to |V |−1, whose entries are edge identifiers, with possible repetitions.…”