An approximation algorithm for the k-fixed depots problem A. Giannakos a , M. Hifi a,⇑ , R. Kheffache b,c , R. Ouafi a,b a EPROAD-EA 4669, Université de Picardie Jules Verne, 7 rue du Moulin Neuf, 80000 Amiens, France b AMCD & RO, Dept de RO, Université des Sciences et de la Technologie Houari Boumediene, Algiers, Algeria c Fac des Sciences, Dept. de Mathématiques, Université Mouloud Mammeri, Tizi Ouzou, Algeria article info Article history: Received 17 February 2016 Received in revised form 1 June 2017 Accepted 12 June 2017 Available online 21 June 2017 Keywords: Approximation Cubic Factor-critical k-Depots TSP abstract In this paper, we consider the k-Depots Hamiltonian Path Problem (k-DHPP) of searching k paths in a graph G, starting from k fixed vertices and spanning all the vertices of G. We propose an approximation algorithm for solving the k-DHPP, where the underlying graph is cubic and 2-vertex-connected. Then, we prove the existence of a 5 3 -approximation algorithm that gives a solution with total cost at most 5 3 n  4k2 3   . In this case, the proposed method is based upon searching for a perfect matching, construct- ing an Eulerian graph and finally a k paths solution, following the process of removing/adding edges. We also present an approximation algorithm for finding a shortest tour passing through all vertices in a factor-critical and 2-vertex connected graph. The proposed algorithm achieves a 7 6 -approximation ratio where the principle of the method is based on decomposing the graph into a series of ears. Ó 2017 Published by Elsevier Ltd. 1. Introduction The Traveling Salesman Problem (TSP) can be formulated as fol- lows: given a finite set of vertices and a non-negative ‘‘length” value between each pair of them, find a tour of minimum total length that visits each vertex only once. In other words, the input of the problem is a complete edge-weighted graph and the optimal solution is a Hamiltonian cycle of minimum sum of the weights of its edges. The TSP has been extensively studied over the past dec- ades. It has been shown to be NP-hard, even in the case that a only a constant ratio approximation of the optimal solution is sought (cf., Cook (2012), Lawler et al. (1985)). The TSP or its variants have been used to modelize numerous problems arising in practical fields, and an impressive number of algorithms have been pre- sented to address their solutions. Like all paradigmatic problems, TSP and its variants have become topics of intensive research, with results that remain very interesting in the field of combinatorial optimization. Recall that an easy reduction from the Hamiltonian tour problem shows that in the general case, the TSP is not approximable within a constant ratio in polynomial time, unless P ¼ NP. Whenever the distances form a metric, there is a 3 2 polynomial time approximation algo- rithm; this is a classic result by Christofides (1976). For this case, inapproximability boundshave been shown (see Karpinski, Lampis, & Schmied (2013) for the state-of-the-art results on the subject). Recently, Seb} o and Vygen (2014) established a 7 5 approximation ratio for the metric TSP in the case where the metric is defined by shortest paths on an underlying (connected) graph; their method is based upon a suitable ear-decomposition of the underlying graph that defines the metric. The Multiple TSP (MTSP) is a generalization of the TSP where one seeks a set of k cycles with a common vertex (the so-called depot point) spanning all the vertices of the input graph and such that their total weight is minimum. The MTSP can be further gen- eralized to the case of multiple depot points (MDTSP) or to the case where the input graph is to be spanned by paths instead of cycles (MDHPP). As mentioned above, on the one hand, TSP has been an important topic of study, as one can understand by contemplating the impressive volume of literature on the topic, that keeps accu- mulating over the years. On the other hand, MTSP, MDTSP and their variants have attracted numerous studies in a more or less applied setting (for instance, the reader is referred to the extensive review of Montoya-Torres, López Franco, Isaza, Felizzola Jiménez, & Herazo-Padilla (2015) covering mostly experimentally analyzed exact or approximate algorithms for the vehicle routing problem with multiple depots). However, rigorously formulated, either exact or approximate, solution methods for such problems are presented to a lesser extent in the literature (see for example, the somewhat older review by Laporte (1992) discussing exact solutions but also heuristics for vehicle routing problems; see also Bae & Rathinam http://dx.doi.org/10.1016/j.cie.2017.06.022 0360-8352/Ó 2017 Published by Elsevier Ltd. ⇑ Corresponding author. E-mail address: hifi@laria.u-picardie.fr (M. Hifi). Computers & Industrial Engineering 111 (2017) 50–55 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie