Incremental algorithms for the maximum internal spanning tree problem

The maximum internal spanning tree (MIST) problem is utilized to determine a spanning tree in a graph G, with the maximum number of possible internal vertices. The incremental maximum internal spanning tree (IMIST) problem is the incremental version of MIST whose feasible solutions are edge-sequences e1, e2, …, en−1 such that the first k edges form trees for all k ∈ [n − 1]. A solution’s quality is measured using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{max}_{k \in [n - 1]}}\frac{{\text{opt}(G,k)}}{{\left| {\text{In}({T_k})} \right|}}$$\end{document} with lower being better. Here, opt(G, k) denotes the number of internal vertices in a tree with k edges in G, which has the largest possible number of internal vertices, and ∣In(Tk)∣ is the number of internal vertices in the tree comprising the solution’s first k edges. We first obtained an IMIST algorithm with a competitive ratio of 2, followed by a 12/7-competitive algorithm based on an approximation algorithm for MIST.