Discrete OptimizationThe Steiner Traveling Salesman Problem with online edge blockages
Introduction
The Traveling Salesman Problem (TSP) is one of the fundamental combinatorial optimization problems that have numerous applications in operational research. In its classic version, we are given a complete edge-weighted graph with the optimization goal to find a Hamiltonian cycle that achieves the minimum weight (or makespan) (Garey and Johnson, 1979). The edge weights are non-negative, which can be used to model various traversing cost such as distance or traveling time between two locations. In a Hamiltonian cycle, every vertex of the graph appears exactly once. TSP is NP-hard (Garey and Johnson, 1979) and APX-hard even if all edge-weights are 1 or 2 (Papadimitriou and Yannakakis, 1993). When the edge weights satisfy the triangle inequality (a.k.a. metric TSP), it can be approximated within a factor of 1.5 (Christofides, 1976).
In many applications, the edge-weighted graph modeling a real instance might neither be complete, nor all vertices need to be visited by the salesman. This inspired the study of the Steiner Traveling Salesman Problem (Cornuéjols, Fonlupt, Naddef, 1985, Fleischmann, 1985, Ratliff, Rosenthal, 1983), denoted as sTSP in this paper, which is a special case of the General Routing Problem introduced by Orloff (1974). Formally, in sTSP, we are given an edge-weighted graph G = (V, E) and a set D⊆V of destination (or client) vertices, with the optimization goal to find a minimum weight closed tour that visits every destination vertex at least once.
Ratliff and Rosenthal (1983) studied a very special case of sTSP, abstracted from a background application of order-picking in a rectangular warehouse; the rectangular warehouse is composed of a number of aisles and there are certain orders to be picked up from the aisles such that the total travel time is minimized. The edge-weighted graph modeling the warehouse is series-parallel. Ratliff and Rosenthal (1983) presented a linear time algorithm for this special case. Cornuéjols et al. (1985) examined more structural properties of sTSP on series-parallel graphs and extended the linear time algorithm to all series-parallel graphs. De Koster, Le-Duc, and Roodbergen (2007) surveyed a number of works on sTSP variants for order-picking in different kinds of warehouses. For another application of sTSP on road networks, Fleischmann (1985) presented a cutting plane procedure to solve instances containing as many as 292 cities. Recently, Letchford, Nasiri, and Theis (2013) presented several new compact polynomial-size formulations for the sTSP instances and demonstrated that the integer programming branch-and-bound solver in IBM ILOG CPLEX Optimization Studio can solve the instances in these formulations faster; in the best of their formulations, the integer programming branch-and-bound solver can solve instances with over 200 cities. Another important sTSP variation is called Graphical Traveling Salesman Problem, denoted as gTSP, in which D = V (Cornuéjols et al., 1985). The optimization goal is the same—to find a minimum weight closed tour that visits every vertex at least once. For the edge-weighted version, Fonlupt and Nachef (1993) presented polynomial time dynamic programming algorithms for certain special classes of graphs; for the edge-unweighted version, several interesting results on approximating gTSP were obtained in the last three years: Gharan, Saberi, and Singh (2011) presented a randomized algorithm with an expected worst-case approximation ratio 1.5 − ε, where ε is a positive constant in the order of O(10−12); Momke and Svensson (2011) made a significant improvement by decreasing the approximation ratio to 1.461; and the currently best 1.4-approximation algorithm is by Sebő and Vygen (2014).
In this paper we consider sTSP on general edge-weighted graphs, for another important application in package delivery services accompanying the booming e-businesses, online shopping in particular. In a sizable city as large as Xi’an, China, every day hundreds of vehicles deliver tens to hundreds of thousands of packages to clients to complete the last step of online transactions. Without a guaranteed fast package delivery service, online shopping would not be as attractive as it is nowadays. Besides establishing warehouses in multiple well distributed places, one way to improve the delivery service is to design effective routing strategies for service vehicles in the urban traffic network, which can be modeled as an edge-weighted graph. Furthermore, such a sizable traffic network is apt to road blockages due to traffic hours, accidents and other sudden road surface changes such as sinking. Some road blockages are predictable and perhaps can be forecasted, while some other are unpredictable. In real-time sTSP, the salesman (or the package delivery vehicle driver) has the real-time edge blockage information, that is, whenever an edge in the graph fails, he knows about it instantly (for example, through traffic radio). We further assume that the blocked edges are non-recoverable to the salesman but the remainder graph remains connected such that a feasible closed tour can always be achieved. We design algorithms for the salesman on how to plan for a provably good closed tour to visit all the destination vertices, while facing the online edge blockages.
In the literature of online TSP, some variations have been studied, for example the sequence of requested vertices (i.e. the destinations in our case) comes in the online fashion (Ausiello, Feuerstein, Leonardi, Stougie, Talamo, 2001, Jaillet, Lu, 2011, Jaillet, Wagner, 2006, Jaillet, Wagner, 2008). There are also works on uncertain traversal time between two vertices. The interested readers might refer to the review by Pillac, Gendreau, Guéret, and Medaglia (2013) for more details. A related routing problem with online edge blockages is the Canadian Traveler Problem (CTP), introduced by Papadimitriou and Yannakakis (1991). In CTP, the goal is to find a shortest path from a source vertex s to a destination vertex t in a given edge-weighted graph G = (V, E), while edges could be blocked during the traversal from s to t. It is PSPACE-complete to find an online algorithm with a constant competitive ratio for CTP (Papadimitriou and Yannakakis, 1991). When the number of blocked edges is at most k, the variation is denoted as k-CTP. Bar-Noy and Schieber (1991) studied k-CTP and several other CTP variations from the worst-case perspective. Zhu, Xu, and Liu (2003) and later Westphal (2008) independently proved a tight lower bound of 2k + 1 on the competitive ratio, and both presented an optimal online algorithm for k-CTP. Xu, Hu, Su, Zhu, and Zhu (2009) presented another greedy and more practical online algorithm for k-CTP. When there are multiple travelers, Zhang, Xu, and Qin (2013) proved a lower bound on the competitive ratio for a k-CTP variation in which the objective is to minimize the maximum traversal time among the travelers; they designed two online algorithms with provable performance. Recently, Liao and Huang (2014) considered another variation of online TSP, called covering Canadian Traveler Problem (CCTP), which is the closest to our target problem sTSP. In CCTP, a complete edge-weighted graph satisfying triangle inequality is given, and the traveler needs to visit all vertices and then return to the origin; in CCTP the blocked edges are generated by the adversary in a different way than in sTSP—all blocked edges incident at a vertex are revealed to the traveler when he arrives the vertex, and no more edge incident at the vertex can be blocked afterward. Liao and Huang (2014) presented an efficient touring strategy when the number of blockages is up to k.
In this paper, we investigate sTSP with real-time online edge blockages, in the perspective of competitive analysis (Borodin and El-Yaniv, 1998). The rest of the paper is organized as follows. In Section 2 we formally define the real-time sTSP problem, and introduce some basic notations and assumptions. We present a lower bound on the competitive ratio for real-time sTSP in Section 3, and an optimal online algorithm in Section 4. While this optimal algorithm has non-polynomial running time, we present in Section 5 another online polynomial-time near optimal algorithm for the problem. We discuss the implementation of the online polynomial-time algorithm, and the experimental results in Section 6. We conclude in Section 7 with some discussion on the achieved results and some possible future work.
Section snippets
Preliminaries
We use an undirected, connected, edge-weighted graph G = (V, E, D) to model the traffic network, where the edge weight w(u, v) represents the traversal time from vertex u to vertex v for edge {u, v} ∈ E. We assume without loss of generality that, if edge {u, v} ∈ E then its weight w(u, v) is the shortest traversal time from vertex u to vertex v in the network G. Clearly, such edge weights satisfy the triangle inequality. The destination vertices, including the package depot s, form the set D⊆V.
A competitive ratio lower bound
Recall that we consider an instance of sTSP denoted as a quadruple I = (V, E, D, δ), where V is the vertex set, E is the edge set and the edge weight w(u, v) is the shortest traversal time from vertex u to vertex v in the network G = (V, E, D) for edge {u, v} ∈ E, D⊆V is the set of destination vertices including the service depot or source vertex s, and δ = 〈e1, e2, …, ek〉 denotes the sequence of online blocked edges that will be revealed to the salesman during the traversal. Also recall that
An exponential-time optimal online algorithm
Recall that in analyzing the competitive ratio for an online algorithm, we examine the worst-case performance and thus we assume without loss of generality in the sequel that the salesman knows about a blocked edge e = {u, v} only when he arrives at vertex u or vertex v. In this section we present an exponential-time online algorithm and show that it is (k + 1)-competitive, which is optimal considering the lower bound in Theorem 1.
A polynomial-time near optimal online algorithm
The algorithm Discover presented above is optimal, in the sense that its worst-case competitive ratio matches the lower bound of k + 1 in Theorem 1. However, the running time of the algorithm Discover is exponential, as it calls an exact TSP algorithm O(k) times. We next present a polynomial-time online algorithm and show that it is (k + 4)-competitive.
Computational results and discussion
We have implemented the online algorithm Piecemeal in Python to examine its efficiency and effectiveness. We also programmed in OPL language the sTSP instances as integer programming instances using the single-commodity flow formulation by Letchford et al. (2013), then used the integer programming branch-and-bound solver in IBM ILOG CPLEX Optimization Studio (64 bit, V12.6.0) to compute the optimal solutions. In the sequel, this is referred to as the offline optimal algorithm. All experiments
Conclusions
In this paper, we investigated the online Steiner TSP, in which the online edge blockages are real-time information. We presented a lower bound on the competitive ratios, and an optimal online algorithm Discover for the problem. While the running time of the algorithm Discover is non-polynomial, we presented a polynomial-time algorithm Piecemeal and showed that it is near optimal. Our experimental results on thousands of instances show that the algorithm Piecemeal is efficient, taking only a
Acknowledgements
Zhang and Xu would like to acknowledge the financial support of NSFC Grants No. 71071123 and No. 61221063 and PCSIRT Grant No. IRT1173. Zhang is also supported by Grant No. 2014M550503 from China Postdoctoral Science Foundation. Zhang, Tong and Lin are supported by NSERC. Lin is also supported by a visiting professorship from the Zhejiang Sci-Tech University.
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