Approximation Algorithms for Multi-vehicle Stacker Crane Problems

We study a variety of multi-vehicle generalizations of the Stacker Crane Problem (SCP). The input consists of a mixed graph G=(V,E,A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E,A)$$\end{document} with vertex set V, edge set E and arc set A, and a nonnegative integer cost function c on E∪A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \cup A$$\end{document}. We consider the following three problems: (1) k-depot SCP (k-DSCP). There is a depot set D⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subseteq V$$\end{document} containing k distinct depots. The goal is to determine a collection of k closed walks including all the arcs of A such that the total cost of the closed walks is minimized, where each closed walk corresponds to the route of one vehicle and has to start from a distinct depot and return to it. (2) k-SCP. There are no given depots, and each vehicle may start from any vertex and then go back to it. The objective is to find a collection of k closed walks including all the arcs of A such that the total cost of the closed walks is minimized. (3) k-depot Stacker Crane Path Problem (k-DSCPP). There is a depot set D⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subseteq V$$\end{document} containing k distinct depots. The aim is to find k (open) walks including all the arcs of A such that the total cost of the walks is minimized, where each (open) walk has to start from a distinct depot but may end at any vertex. We present the first constant-factor approximation algorithms for all the above three problems. To be specific, we give 3-approximation algorithms for the k-DSCP, the k-SCP and the k-DSCPP. If the costs of the arcs are symmetric, i.e., for every arc there is a parallel edge of no greater cost, we develop better algorithms with approximation ratios max{95,2-12k+1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \{\frac{9}{5},2-\frac{1}{2k+1}\}$$\end{document}, 2, 2, respectively. All the proposed algorithms have a time complexity of O(|V|3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|V|^3)$$\end{document} except that the two 2-approximation algorithms run in O(|V|2log|V|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|V|^2\log |V|)$$\end{document} time.