Fast Approximation Algorithms for Computing Constrained Minimum Spanning Trees

Abstract

Given an integer $L\in \mathbb {Z}^{+}$ and an undirected graph with a weight and a length associated with every edge, the constrained minimum spanning tree (CMST) problem is to compute a minimum weight spanning tree with total length bounded by L. The problem was shown weakly $\mathcal{NP}$-hard in [1], admitting a PTAS with a runtime $O(n^{O(\frac{1}{\epsilon })}(m\log ^{2}n+n\log ^{3}n))$ due to Ravi and Goemans [13]. In the paper, we present an exact algorithm for CMST, based on our developed bicameral edge replacement which improves a feasible solution of CMST towards an optimal solution. By applying the classical rounding and scaling technique to the exact algorithm, we can obtain a fully polynomial-time approximation scheme (FPTAS), i.e. an approximation algorithm with a ratio $(1+\epsilon )$ and a runtime $O(mn^{5}\frac{1}{\epsilon ^{2}})$, where $\epsilon >0$ is any fixed real number.