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.