Maximum Internal Spanning Tree Research Articles

Erratum to: Incremental algorithms for the maximum internal spanning tree problem

In the original paper [1], there are two errors on page 5. The rules in the algorithm are not sufficient since we omitted one rule during publication. The following rule should be added to the paper. Rule 3: If P1 = (u, v) and P2 = (x, y) are two maximal leaf-paths both with two vertices in T, where u and x are leaves in T and (v, y) ∈ E(G), we remove (y′, y) and then add (v, y) to T where y′ is the non-leaf neighbor of y in T. The expression of Algorithm 2 is not accurate. It should be modified as follows. Line 3: Exhaustively apply Rules 1–3, and for i ∈ {2, 3} only apply Rule i when none of Rules j where 1 ⩽ j < i is applicable. Line 8: Determine the longest path P among all paths whose endpoints have nonempty neighbors in T \ V(T′).

Algorithms for maximum internal spanning tree problem for some graph classes

For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

A simple linear time algorithm to solve the MIST problem on interval graphs

Motivated by the design of cost-efficient communication networks, the problem about Maximum Internal Spanning Tree that arises in a connected graph, is proposed to find a spanning tree with the maximum number of internal vertices. In 2018, Xingfu Li et al. presented a polynomial algorithm to find a maximum internal spanning tree in a connected interval graph. Based on the structure of normal orderings on interval graphs, we present a simple linear time algorithm that solve the problem when restricted to connected interval graphs in this paper. The proof provides additional insight about the linear time algorithm on interval graphs.

Performance analysis of evolutionary algorithm for the maximum internal spanning tree problem

Performance analysis of evolutionary algorithm for the maximum internal spanning tree problem

Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs

Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor $7/12$. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. We improve the approximation factors for the MaxwIST problem in cubic graphs and claw-free graphs. For cubic graphs we present an algorithm that computes a spanning tree whose total weight of internal vertices is at least $\frac{3}{4}-\frac{3}{n}$ times the total weight of all vertices, where $n$ is the number of vertices of $G$. This ratio is almost tight for large values of $n$. For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least $\frac{3}{5}-\frac{1}{n}$ times the total vertex weight. The degree constraint is necessary as this ratio may not be achievable if we allow vertices of degree less than three. With the above ratios, we immediately obtain better approximation algorithms with factors $\frac{3}{4}-\epsilon$ and $\frac{3}{5}-\epsilon$ for the MaxwIST problem in cubic graphs and claw-free graphs of degree at least three, for any $\epsilon>0$. In addition to improving the approximation factors, the new algorithms are relatively short compared to that of Chen et al.. The new algorithms are fairly simple, and employ a variant of the depth-first search algorithm that selects a relatively-large-weight vertex in every branching step. Moreover, the new algorithms take linear time while previous algorithms for similar problem instances are super-linear.

A [formula omitted]-approximation algorithm for the Maximum Internal Spanning Tree Problem

In this paper, we study the Maximum Internal Spanning Tree Problem (MIST). Given an undirected simple graph G, the task for the Maximum Internal Spanning Tree problem is to find a spanning tree of G with maximum number of internal vertices. We present an approximation algorithm with performance ratio 43, which improves upon the best known performance ratio 32. Our algorithm benefits from a new observation for bounding the number of internal vertices of a spanning tree. We can also give an example to show that the performance ratio 43 is actually tight for this algorithm. Finally, we show that MIST is Max-SNP-hard.

Approximation Algorithms for the Maximum Weight Internal Spanning Tree Problem

Given a vertex-weighted connected graph $G = (V, E)$, the maximum weight internal spanning tree (MwIST for short) problem asks for a spanning tree $T$ of $G$ such that the total weight of the internal vertices in $T$ is maximized. The un-weighted variant, denoted as MIST, is NP-hard and APX-hard, and the currently best approximation algorithm has a proven performance ratio $13/17$. The currently best approximation algorithm for MwIST only has a performance ratio $1/3 - \epsilon$, for any $\epsilon > 0$. In this paper, we present a simple algorithm based on a novel relationship between MwIST and the maximum weight matching, and show that it achieves a better approximation ratio of $1/2$. When restricted to claw-free graphs, a special case been previously studied, we design a $7/12$-approximation algorithm.

An approximation algorithm for maximum internal spanning tree

Given a graph G, the maximum internal spanning tree problem (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of $$\frac{3}{4}$$ . In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, $$\frac{13}{17}$$ ) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.

Solving the maximum internal spanning tree problem on interval graphs in polynomial time

This paper studies the Maximum Internal Spanning Tree problem which is to find a spanning tree with the maximum number of internal vertices on a graph. We prove that the problem can be solved in polynomial time on interval graphs. The idea is based on the observation that the number of internal vertices in a maximum internal spanning tree is at most one less than the number of edges in a maximum path cover on any graph. On an interval graph, we present an O(n2)-algorithm to find a spanning tree in which the number of internal vertices is exactly one less than the number of edges in a maximum path cover of the graph, where n is the number of vertices in the interval graph.

Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree

The maximum internal spanning tree problem asks for a spanning tree of a given graph that has the maximum number of internal vertices among all spanning trees of this graph. In its parameterized version, we are interested in whether the graph has a spanning tree with at least k internal vertices. Fomin et al. (2013) [4] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a 3k-vertex kernel, implying an O⁎(8k)-time parameterized algorithm. Using depth-2 local search, Knauer and Spoerhase (2015) [9] developed a (5/3)-approximation algorithm for the optimization version. We try deeper local search: We conduct a thorough combinatorial analysis on the obtained spanning trees and explore their algorithmic consequences. We first observe that from the spanning tree obtained by depth-3 local search, one can easily find a reducible structure and apply the reduction rule of Fomin et al. This gives an improved kernel of 2k vertices, and as a by-product, a deterministic algorithm running in time O⁎(4k). We then go even deeper by considering the spanning tree obtained by depth-5 local search. It is shown that the number of internal vertices of this spanning tree is at least 2/3 of the maximum number a spanning tree can have, thereby delivering an improved approximation algorithm with ratio 1.5 for the problem.

Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem

We examine the problem of determining a spanning tree of a given graph such that the number of internal nodes is maximum. The best approximation algorithm known so far for this problem is due to Prieto and Sloper and has a ratio of 2. For graphs without pendant nodes, Salamon has lowered this factor to $\frac{7}{4}$ by means of local search. However, the approximative behaviour of his algorithm on general graphs has remained open. In this paper we show that a simplified and faster version of Salamon's algorithm yields a $\frac{5}{3}$ -approximation even on general graphs. In addition to this, we investigate a node weighted variant of the problem for which Salamon achieved a ratio of 2?Δ(G)?3. Extending Salamon's approach we obtain a factor of 3+∈ for any ∈>0. We complement our results with worst case instances showing that our analyses are tight.

A linear vertex kernel for maximum internal spanning tree

We present a polynomial time algorithm that for any graph G and integer k⩾0, either finds a spanning tree with at least k internal vertices, or outputs a new graph GR on at most 3k vertices and an integer k′ such that G has a spanning tree with at least k internal vertices if and only if GR has a spanning tree with at least k′ internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k has a 3k-vertex kernel. Our result is based on an innovative application of a classical min–max result about hypertrees in hypergraphs which states that “a hypergraph H contains a hypertree if and only if H is partition-connected.”

Fast Polynomial-Space Algorithms Using Inclusion-Exclusion

Given a graph with n vertices, k terminals and positive integer weights not larger than c, we compute a minimum Steiner Tree in $\mathcal{O}^{\star}(2^{k}c)$ time and $\mathcal{O}^{\star}(c)$ space, where the $\mathcal{O}^{\star}$ notation omits terms bounded by a polynomial in the input-size. We obtain the result by defining a generalization of walks, called branching walks, and combining it with the Inclusion-Exclusion technique. Using this combination we also give $\mathcal{O}^{\star}(2^{n})$ -time polynomial space algorithms for Degree Constrained Spanning Tree, Maximum Internal Spanning Tree and #Spanning Forest with a given number of components. Furthermore, using related techniques, we also present new polynomial space algorithms for computing the Cover Polynomial of a graph, Convex Tree Coloring and counting the number of perfect matchings of a graph.

Sharp Separation and Applications to Exact and Parameterized Algorithms

Many divide-and-conquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a trade-off between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NP-hard problems. We illustrate that by presenting two applications. Our first application is a O(2 n+o(n))-time algorithm for the Degree Constrained Spanning Tree problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some well-studied problems, among them Hamiltonian Path, Full Degree Spanning Tree, Bounded Degree Spanning Tree, and Maximum Internal Spanning Tree. The second application is a parameterized algorithm with running time O(16 k+o(k)+n O(1)) for the k-Internal Out-Branching problem: here the goal is to compute an out-branching of a digraph with at least k internal nodes. This is a significant improvement over the best previously known parameterized algorithm for the problem by Cohen et al. (J. Comput. Syst. Sci. 76:650–662, 2010), running in time O(49.4k +n O(1)).

A Survey on Algorithms for the Maximum Internal Spanning Tree and Related Problems

Abstract Given an undirected connected graph G we consider the problem of finding a spanning tree of G with a maximum number of internal ( ⩾ 2 degree) vertices. This problem, called the Maximum Internal Spanning Tree problem, is obviously NP-hard since it generalizes the Hamiltonian Path problem. In this paper we aim at giving a survey on recent results about the Maximum Internal Spanning Tree problem including different approaches such as exact exponential algorithms, fixed parameter tractability, and approximation algorithms. We also consider the problem of finding a large ( ⩽ q )-leaf subtree of the input graph for some fixed q.

Approximating the Maximum Internal Spanning Tree problem

Given an undirected connected graph G we consider the problem of finding a spanning tree of G which has a maximum number of internal (non-leaf) vertices among all spanning trees of G . This problem, called Maximum Internal Spanning Tree problem, is clearly NP-hard since it is a generalization of the Hamiltonian Path problem. From the optimization point of view the Maximum Internal Spanning Tree problem is equivalent to the Minimum Leaf Spanning Tree problem. However, the two problems have different approximability properties. Lu and Ravi proved that the latter has no constant factor approximation–unless P = NP–, while Salamon and Wiener gave a linear-time 2-approximation algorithm for the Maximum Internal Spanning Tree problem. In this paper, we improve this approximation ratio by giving an O ( | V ( G ) | 4 ) -time 7 / 4 -approximation algorithm for graphs without pendant vertices. Our approach is based on the successive execution of local improvement steps. We use a linear programming formulation and a primal–dual technique to prove the approximation ratio. We also investigate the vertex-weighted case, that is to find a spanning tree of a vertex-weighted graph G in which the weight sum of internal vertices is maximal among all spanning trees of G . For this problem we present a ( 2 Δ ( G ) − 3 ) -approximation algorithm, where Δ ( G ) is the maximum vertex-degree of G . A slight modification of this algorithm ensures a 2-approximation whenever the input graph is claw-free. Both algorithms run in O ( | V ( G ) | 4 ) time for graphs with no pendant vertices.