Approximation algorithms for constructing spanning K-trees using stock pieces of bounded length

Abstract

Given a weighted graph G on $n + 1$ vertices, a spanning K-tree $T_K$ of G is defined to be a spanning tree T of G together with K distinct edges of G that are not edges of T. The objective of the minimum-cost spanning K-tree problem is to choose a subset of edges to form a spanning K-tree with the minimum weight. In this paper, we consider the constructing spanning K-tree problem that is a generalization of the minimum-cost spanning K-tree problem. We are required to construct a spanning K-tree $T_K$ whose $n+K$ edges are assembled from some stock pieces of bounded length L. Let $c_0$ be the sale price of each stock piece of length L and $k(T_K)$ the number of minimum stock pieces to construct the $n+K$ edges in $T_K$. For each edge e in G, let c(e) be the construction cost of that edge e. Our new objective is to minimize the total cost of constructing a spanning K-tree $T_K$, i.e., $\min _{T_K}\{\sum _{e\in T_K} c(e)+ k(T_K)\cdot c_0\}$. The main results obtained in this paper are as follows. (1) A 2-approximation algorithm to solve the constructing spanning K-tree problem. (2) A $\frac{3}{2}$-approximation algorithm to solve the special case for constant construction cost of edges. (3) An APTAS for this special case.