Approximation algorithms for constructing required subgraphs using stock pieces of fixed length

Abstract

In this paper, we address the problem of constructing required subgraphs using stock pieces of fixed length (CRS-SPFL, for short), which is a new variant of the minimum-cost edge-weighted subgraph (MCEWS, for short) problem. Concretely, for the MCEWS problem Q, it is asked to choose a minimum-cost subset of edges from a given graph G such that these edges can form a required subgraph $G'$. For the CRS-SPFL problem $Q^{\prime }$, these edges in such a required subgraph $G'$ are further asked to be constructed by plus using some stock pieces of fixed length L. The new objective, however, is to minimize the total cost to construct such a required subgraph $G'$, where the total cost is sum of the cost to purchase stock pieces of fixed length L and the cost to construct all edges in such a subgraph $G'$. We obtain the following three main results. (1) Given an $\alpha$-approximation algorithm to solve the MCEWS problem, where $\alpha \ge 1$ (for the case $\alpha =1$, the MCEWS problem Q is solved optimally by a polynomial-time exact algorithm), we design a $2\alpha$-approximation algorithm and another asymptotic $\frac{7\alpha }{4}$-approximation algorithm to solve the CRS-SPFL problem $Q^{\prime }$, respectively; (2) When Q is the minimum spanning tree problem, we provide a $\frac{3}{2}$-approximation algorithm and an AFPTAS to solve the problem $Q^{\prime }$ of constructing a spanning tree using stock pieces of fixed length L, respectively; (3) When Q is the single-source shortest paths tree problem, we present a $\frac{3}{2}$-approximation algorithm and an AFPTAS to solve the problem $Q^{\prime }$ of constructing a single-source shortest paths tree using stock pieces of fixed length L, respectively.