Approximations for constructing tree-form structures using specific material with fixed length

Abstract

Substituting for the ordinary objective to minimize the sum of lengths of all edges in some graph structure of a weighted graph, we propose a new problem of constructing certain tree-form structure in same graph, where all edges needed in such a tree-form structure are supposed to be cut from some pieces of a specific material with fixed length. More precisely, we study a new problem defined as follows: a weighted graph \(G=(V,E; w)\), a tree-form structure \(\mathcal{S}\) and some pieces of specific material with length L, where a length function \(w:E\rightarrow Q^+\), satisfying \(w(u,v) \le L\) for each edge uv in G, we are asked how to construct a required tree-form structure \(\mathcal{S}\) as a subgraph \(G'\) of G such that each edge needed in \(G'\) is constructed by a part of a piece of such a specific material, the new objective is to minimize the number of necessary pieces of such a specific material to construct all edges in \(G'\). For this new objective defined, we obtain three results: (1) We present a \(\frac{3}{2}\)-approximation algorithm to construct a spanning tree of G; (2) We design a \(\frac{3}{2}\)-approximation algorithm to construct a single-source shortest paths tree of G; (3) We provide a 4-approximation algorithms to construct a metric Steiner tree of G.