[Paper Review] Spanning trees short or small

R. Ravi, R. Sundaram+3 more|arXiv (Cornell University)|Jan 23, 1994

Advanced Graph Theory Research25 references46 citations

TL;DR

This paper investigates the kMST problem—finding a minimum-weight tree spanning at least k nodes in a graph—proving it NP-hard even in the Euclidean plane. It presents approximation algorithms with ratios of 2√k for general edge-weighted graphs and O(k^{1/4}) for Euclidean points, along with exact polynomial-time solutions for treewidth-bounded graphs and points on a convex boundary, while also addressing minimum-diameter k-trees via a simple framework.

ABSTRACT

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2v/ for the general edge-weighted case and O(k1/4) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded graphs, which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees and, more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomiM-time solution for finding k-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

Motivation & Objective

• To address the kMST problem, which seeks a minimum-weight tree spanning at least k nodes in a graph, under the constraint that only k nodes need to be connected.
• To investigate the complexity of the kMST problem, proving it NP-hard even for points in the Euclidean plane.
• To develop efficient approximation algorithms with provable performance guarantees for general and geometric instances of the kMST problem.
• To identify classes of graphs and geometric configurations where the kMST problem admits polynomial-time exact solutions.
• To extend the study to finding short networks, particularly minimum-diameter k-trees, using a framework from T. C. Hu.

Proposed method

• Using a primal-dual approach and tree decomposition techniques, the paper derives a 2√k-approximation algorithm for the general edge-weighted kMST problem.
• For points in the Euclidean plane, the paper applies geometric clustering and spanning tree heuristics to achieve an O(k^{1/4})-approximation ratio.
• For treewidth-bounded graphs, including trees and series-parallel graphs, the paper uses dynamic programming on tree decompositions to solve kMST exactly in polynomial time.
• For points on the boundary of a convex region, the paper exploits geometric convexity and monotonicity properties to design a polynomial-time exact algorithm.
• A simple dynamic programming technique is applied to the minimum-diameter k-tree problem, leveraging the framework of T. C. Hu for network design.
• The paper integrates techniques from network design and combinatorial optimization to unify results on both weight and diameter minimization in k-trees.

Experimental results

Research questions

• RQ1 Is the kMST problem NP-hard in the Euclidean plane, and can it be approximated efficiently?
• RQ2 What approximation ratio can be achieved for the kMST problem in general edge-weighted graphs?
• RQ3 Can exact polynomial-time solutions be found for kMST in special graph classes such as treewidth-bounded graphs?
• RQ4 What is the approximation performance for kMST when the input consists of points in the Euclidean plane?
• RQ5 Can minimum-diameter k-trees be constructed in polynomial time using a structured framework?

Key findings

• The kMST problem is proven to be NP-hard even for points in the Euclidean plane, establishing its computational intractability in geometric settings.
• An approximation algorithm with a performance ratio of 2√k is developed for the general edge-weighted kMST problem.
• For points in the Euclidean plane, the paper achieves an O(k^{1/4})-approximation ratio, improving upon previous bounds.
• Polynomial-time exact algorithms are provided for kMST in treewidth-bounded graphs, including trees and series-parallel graphs.
• An exact polynomial-time algorithm is presented for kMST when nodes lie on the boundary of a convex region in the plane.
• A simple dynamic programming method enables polynomial-time construction of minimum-diameter k-trees, leveraging T. C. Hu's framework for network design.