Given a graph G, the {\em 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 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, 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.

An Approximation Algorithm for Maximum Internal Spanning Tree

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 unweighted 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.

Approximation Algorithms for the Maximum Weight Internal Spanning Tree Problem

Frontiers in Algorithmics : 10th International Workshop, FAW 2016, Qingdao, China, June 30- July 2, 2016, Proceedings

Subtrees and BC-subtrees (subtrees in which the distance between any two leaves is even) are important concepts in the study of complex graphical structures. In this article, we propose a novel generalization called the leaf-distance granular regular α-tree (abbreviated as LDR α-tree for short). This is a tree in which the distance between any two leaves is divisible by α (α is a positive integer). A LDR α-subtree is simply a subtree that is also a LDR α-tree. We present basic properties and generating functions related to the LDR α-subtrees enumeration. Based on those theoretical results we provide efficient algorithms for enumerating various LDR α-subtrees of trees. Our algorithms can serve as multi-distance granularity sifters of a graph to screen all the α-subtree uniformly, and thus provide novel insights into exploring new structural properties from the perspective of multiple leaf-distance granularity. 

On enumerating algorithms of novel multiple leaf-distance granular regular α-subtrees of trees

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. As our recent \(\frac{1}{2}\)-approximation algorithm for the weighted version of the problem shows, the discovered structure may be used to design better algorithms for related problems.

An approximation algorithm for maximum internal spanning tree

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 \(O(n^{2})\)-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.

A Polynomial Time Algorithm for Finding a Spanning Tree with Maximum Number of Internal Vertices on Interval Graphs

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

A Polynomial Time Algorithm for Finding a Spanning Tree with Maximum Number of Internal Vertices on Interval Graphs