A branch-and-cut algorithm for the minimum branch vertices spanning tree problem

A genetic approach for the 2‐edge‐connected minimum branch vertices problem

“…Otherwise, bridges B in G ′′ are computed (line 9). If no bridge exists, we can use Lemma 4 and remove from G ′ all the edges in 𝛿 G ′ (v), except for the two edges whose other endpoints have highest degree (lines [10][11]; in addition to making v not branch, this choice potentially reduces the number of branch vertices in the neighborhood of v. Otherwise, we remove existing bridges from G ′′ and recompute the connected components in G ′′ : for each found component C i such that a single bridge in B is incident on C i , we remove all the edges in 𝛿 G ′ (v) except the edge (v, z) which maximizes the degree of z (lines [13][14][15][16][17][18][19][20]. This means reducing the number of edges incident on v: when every found component has at most two incident bridges, Lemma 5 holds and only two edges in 𝛿 G ′ (v) are selected; otherwise, we know from Lemma 6 that v is necessarily a branch vertex, thus at least three edges in 𝛿 G ′ (v) are selected.…”

“…Carrabs et al [2] introduced four IP formulations for the MBV problem, Silva et al [17] proposed an edge-swap heuristic algorithm, while Marín [12] experimented both exact and heuristic solutions. A further contribution on the MBV problem has been provided by Silvestri et al [18], who introduced some valid inequalities and a Branch and Cut approach. Furthermore, Cerulli et al [3] presented some heuristic approaches able to quickly find high-quality solutions.…”

Spanning Trees of Recursive Scale-Free Graphs

“…We use the term ghost to indicate that the link is part of the original graph, but not part of the spanning tree, and we represent these links by dashed lines in all the figures. Furthermore, specific trees with optimal characteristics can be selected (with high probability) by heuristically (and/or stochastically) applying the link-based growth rules, solving various optimal spanning tree problems, which are NP-hard in general [14][15][16].…”

“…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].…”

Spanning Trees with Few Branch Vertices

“…Gargano, Hell, Stacho and Vaccaro [16] showed that the problem of finding a spanning tree with the minimum number of branch vertices is NP-hard. Since then, the problem has been investigated by many authors [3,4,5,7,18,19,20,25,26,27,28].…”