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\begin{document}$$\lambda >0$$\end{document}, an undirected complete graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} with nonnegative edge-weight function obeying the triangle inequality and a depot vertex r∈V\documentclass[12pt]{minimal}
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\begin{document}$$r\in V$$\end{document}, a set {C1,…,Ck}\documentclass[12pt]{minimal}
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\begin{document}$$\{C_1,\ldots ,C_k\}$$\end{document} of cycles is called a λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}-boundedr-cycle cover if V⊆⋃i=1kV(Ci)\documentclass[12pt]{minimal}
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\begin{document}$$V \subseteq \bigcup _{i=1}^k V(C_i)$$\end{document} and each cycle Ci\documentclass[12pt]{minimal}
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\begin{document}$$C_i$$\end{document} contains r and has a length of at most λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}. The Distance Constrained Vehicle Routing Problem with the objective of minimizing the total cost (DVRP-TC) aims to find a λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}-bounded r-cycle cover {C1,…,Ck}\documentclass[12pt]{minimal}
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\begin{document}$$\{C_1,\ldots ,C_k\}$$\end{document} such that the sum of the total length of the cycles and γk\documentclass[12pt]{minimal}
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\begin{document}$$\gamma k$$\end{document} is minimized, where γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} is an input indicating the assignment cost of a single cycle. For DVRP-TC on tree metric, we show a 2-approximation algorithm and give an LP relaxation whose integrality gap lies in the interval [2,52\documentclass[12pt]{minimal}
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\begin{document}$$\frac{5}{2}$$\end{document}]. For the unrooted version of DVRP-TC, we devise a 5-approximation algorithm and give an LP relaxation whose integrality gap is between 2 and 25. For unrooted DVRP-TC on tree metric we develop a 3-approximation algorithm. For unrooted DVRP-TC on line metric we obtain an O(n3)\documentclass[12pt]{minimal}
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\begin{document}$$O(n^3)$$\end{document} time exact algorithm, where n is the number of vertices. Moreover, we give some examples to demonstrate that our results can also be applied to the path-version of (unrooted) DVRP-TC.
