Given a weighted graph G on n+1\documentclass[12pt]{minimal}
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\begin{document}$$n + 1$$\end{document} vertices, a spanning K-treeTK\documentclass[12pt]{minimal}
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\begin{document}$$T_K$$\end{document} 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 TK\documentclass[12pt]{minimal}
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\begin{document}$$T_K$$\end{document} whose n+K\documentclass[12pt]{minimal}
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\begin{document}$$n+K$$\end{document} edges are assembled from some stock pieces of bounded length L. Let c0\documentclass[12pt]{minimal}
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\begin{document}$$c_0$$\end{document} be the sale price of each stock piece of length L and k(TK)\documentclass[12pt]{minimal}
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\begin{document}$$k(T_K)$$\end{document} the number of minimum stock pieces to construct the n+K\documentclass[12pt]{minimal}
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\begin{document}$$n+K$$\end{document} edges in TK\documentclass[12pt]{minimal}
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\begin{document}$$T_K$$\end{document}. 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 TK\documentclass[12pt]{minimal}
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\begin{document}$$T_K$$\end{document}, i.e., minTK{∑e∈TKc(e)+k(TK)·c0}\documentclass[12pt]{minimal}
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\begin{document}$$\min _{T_K}\{\sum _{e\in T_K} c(e)+ k(T_K)\cdot c_0\}$$\end{document}. 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 32\documentclass[12pt]{minimal}
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\begin{document}$$\frac{3}{2}$$\end{document}-approximation algorithm to solve the special case for constant construction cost of edges. (3) An APTAS for this special case.
