Approximation algorithms for the restricted k-Chinese postman problems with penalties

Given an undirected graph G=(V,E;w,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E;w,p)$$\end{document} with a depot r∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in V$$\end{document} and an integer k∈Z+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in {{\mathbb {Z}}}^{+}$$\end{document}, each edge e∈E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e\in E$$\end{document} has a weight w(e)∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(e)\in {{\mathbb {R}}}^{+}$$\end{document} and a penalty p(e)∈R0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(e)\in {\mathbb R}^{+}_{0}$$\end{document}, where the weights satisfy the triangle inequality, we consider two types of restricted k-Chinese postman problems with penalties. (1) The restricted min–max k-Chinese postman problem with penalties (MM-RPCPP) is asked to find a set of k tours starting from r and collectively covering all vertices, such that the maximum tour weight plus the total penalty paid for the uncovered edges is minimized. (2) The restricted min-sum k-Chinese postman problem with penalties (MS-RPCPP) is asked to find a set of k tours satisfying the constraint mentioned above and that each edge appears in at most one tour and each tour contains at least one edge, such that the sum of weights of all tours plus the total penalty paid for the uncovered edges is minimized. In the paper, we design a combinatorial (72-2k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\frac{7}{2}-\frac{2}{k})$$\end{document}-approximation algorithm to solve the MM-RPCPP. Furthermore, we present a combinatorial 2-approximation algorithm to solve the MS-RPCPP.