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, …, e_n−1 such that the first k edges form trees for all k ∈ [n − 1]. A solution’s quality is measured using \({\text{max}_{k \in [n - 1]}}\frac{{\text{opt}(G,k)}}{{\left| {\text{In}({T_k})} \right|}}\) 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(T_k)∣ 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.

最大内部生成树（MIST）问题用于确定图G中具有最大可能内部顶点数量的生成树。增量最大内部生成树（IMIST）的问题是其MIST可行解决方案是边缘序列的增量版本ë ₁，ë ₂，...，Ë _{Ñ -1}，使得第一ķ边缘形成用于所有树木ķ ∈[ Ñ - 1]。使用\（{\ text {max} _ {k \ in [n-1]}} \ frac {{\ text {opt}（G，k）}} {{\ left | {\ text {In}（{T_k}）} \ right |}} \），越低越好。在这里，opt（G，k）表示在G中具有k个边的树中的内部顶点数，该树中内部顶点的数量可能最大，而∣In（T _k）∣是包含解的前k个边的树中的内部顶点数。我们首先获得了竞争比率为2的IMIST算法，然后是基于MIST近似算法的12/7竞争算法。