Routing problems: A bibliography

Annals of Operations Research 61(1995)227-262 227 Routing problems: A bibliography Gilbert Laporte Centre de Recherche sur les Transports, Universitd de Montreal, C.P. 6128, succursale Centre-ville, Montrdal, Canada H3C 3J7 Ibrahim H. Osman Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent CT2 7NF, United Kingdom This bibliography contains 500 references on four classical routing problems: the TravelingSalesmanProblem,the VehicleRoutingProblem,the ChinesePostmanProblem, and the Rural PostmanProblem.Referencesare presentedalphabeticallyunder a number of subheadings. Keywords: Bibliography, traveling salesman, vehicle routing, Chinese postman, rural postman, node routing, arc routing. I. Introduction Modem logistics is described by Christopher [103] as "the process of strategically managing the movement or storage of materials, parts and finished, inventory from suppliers, through the firm and on to the consumers". The efficient movement of goods or workers in order to increase and meet market demands is often emphasized in the business strategy of companies. It is estimated that distribution costs account for approximately 10% of the firms' revenues (LaLonde and Zinszer [274]) and to more than 45% of the total logistics costs (Institute of Logistics and Distribution Management [242]). In some cases, like in the soft drink industry, they represent approximately 70% of the value added costs of goods (Golden and WasiI [219]). The potential for savings is therefore considerable. Several successful implementations of computerized routing software have been documented in the literature. These successes can be attributed in part to algorithmic advances in the field of vehicle routing and also to the development of new software and computer technologies. Vehicle routing is truly one of the great success stories of operations research. Researchers, particularly newcomers to the field, often have difficulty finding their way through the abundant and somewhat disorganized literature on vehicle © J.C. Baltzer AG, Science Publishers 228 G. Laporte, I.H. Osman, A bibliogaphy of routing problems routing. The purpose of this paper is to draw up a list of some of the main bibliographic references on the subject, within a simple and relatively broad classification scheme. As it is impractical and also rather unhelpful to provide an exhaustive bibliography, we have concentrated on those publications we consider the most useful or significant, namely references of general or historical nature, classical articles, and more recent work describing potentially significant developments. Interested readers should be able to fill some of the gaps by consulting the quoted survey articles. We now introduce some basic notation common to all problems covered by this bibliography. Additional notation will be defined for each case. Let G = (V, E U A) be a graph where V= {ul ..... un} is a vertex set, A = {(% vj) : i ~ j , Vi, DjE V} is a set of directed arcs, and E = {(ui, vj) : i < j , vi, vj ~ V} is a set of undirected edges. In most problems, either A or E is empty. Problems containing only arcs are said to be directed or asymmetrical; problems containing only edges are undirected or symmetrical. Problems containing a combination of arcs and edges are mixed. With each arc or edge (vi, uj) is associated a cost or distance Cij. Various problems can be defined on G by specifying an objective and a set of constraints. Each problem exists in its deterministic or its stochastic form. In the latter case, some of the problem data are random variables. Such problems call for a different solution concept and a different solution methodology (see, e.g., Laporte and Louveaux [287]). The following bibliography covers four classical routing problems, presented in sections 2 to 5: The Traveling Salesman Problem, the Vehicle Routing Problem, the Chinese Postman Problem, the Rural Postman Problem. In each section, references are presented alphabetically under a number of subheadings. 2. The Traveling Salesman Problem (TSP) The TSP is defined on a graph G = (V, A) or G = (V, E) and consists of determining a shortest Hamiltonian circuit or cycle in G, depending on whether G is directed or not. This is probably the most studied problem in combinatorial optimization. Its origins are somewhat obscure. According to Mtiller-Merbach [346], it is stated in a 1831 paper by Voigt [487], but the first mention of the expression "Traveling Salesman Problem" is found in Menger [333]. The first major algorithmic study of the problem is that of Dantzig, Fulkerson and Johnson [114]. An interesting history of the TSP is provided in Hoffman and Wolfe [239]. The TSP is NP-hard [192] and its study has given rise to several theoretical and algorithmic results, some having far reaching effects in other areas of combinatorial optimization. Polyhedral theory [222,367] is probably the most significant of these and has led to the development of powerful exact algorithms. Recently, new families of fast heuristic algorithms have been proposed, making use of sophisticated data structures, restricted neighbourhoods and constructive schemes combined with local reoptimization [198,252]. The book by Lawler, Lenstra, Rinnooy Kan and Shmoys [303] contains an account of the main results on the TSP until 1985. For a more recent survey, see Laporte [283]. G. Laporte, I.H. Osman, A bibliogaphy of routing problems 229 Several variants of the TSP have been defined. We only describe the most common cases. In the TSP with Time Windows, V contains a depot and each remaining vertex vi must be visited within a time window [ai, bi]. In the Generalized TSP, V is a union of clusters V~..... Vk and the problem is to determine a Hamiltonian tour passing through each cluster exactly once or at least once. In the Clustered TSP, V contains a depot and the remaining vertices are partitioned into clusters. A precedence relation < is defined on the partition. The problem consists of determining a least cost tour on G starting and ending at the depot, and such that if Vp < Vq, then all vertices of Vp are visited before those of Vq. Other variants of the TSP are obtained by associating a profit Pi with each vertex vi. In the Selective TSP, one seeks a tour of maximal profit through a subset of V having a length not exceeding a preset bound. The Prize Collecting TSP is the reverse problem: here one must determine a least cost tour through a subset of V having a profit at least equal to a given lower bound. 2.1. GENERALAND SURVEY WORK Arthur and Frendewey [14], Freedman and Johnson, Geoch and Ostreimer [l 89], Hoffman and Wolfe [239], Laporte [282], Lawler, Lenstra, Rinnooy Kan and Shmoys [303], Menger [333], Miiller-Merbach [346], Reinelt [406], Voigt [487]. 2.2. APPLICATIONS Bland and Shallcross [55], Eiselt and Laporte [156]. 2.3. FORMULATIONSAND EXACT ALGORITHMS FOR THE ASYMMETRICALTSP Carpaneto, Fischetti and Toth [83], Desrochers and Laporte [122], Finke, Claus and Gunn [168], Fischetti and Toth [171], Langevin, Soumis and Desrosiers [279], Miller and Pekny [336], Padberg and Sung [371], Pekny and Miller [384], Wong [494]. 2.4. FORMULATIONSAND EXACT ALGORITHMS FOR THE SYMMETRICALTSP Cornurjols, Fonlupt and Naddef [108], Fischetti, Gonzalez and Toth [169], Fleischmann [181], Grrtschel and Holland [221], Grrtschel and Padberg [222], Land [277], Laporte [280], Martin [329], Naddef and Rinaldi [348], Padberg and Grrtschel [367], Padberg and Rinaldi [368,369,370], Padberg and Sung [371]. 2.5. HEURISTICSFOR THE TSP Bartholdi and Platzman [29, 30], Bentley [45], Bonomi and Lutton [64], Braun [71 ], Carlier and Villon [81 ], Codenotti and Margara [ 104], Fiechter [ 167], Gendreau, Hertz and Laporte [198], Glover [208], Gu and Huang [225], Jeong and Kim [251], Johnson [252], JiJnger, Reinelt and Thienel [254], Kinderwater and Savelsbergh 230 G. Laporte, LH. Osrnan, A bibliogaphy of routing problems [261], Knox [262, 263], Lee and Choi [304], Mak and Morton [323], Malek [325], Malek, Guruswamy, Owens and Panday [326], Margot [328], Martin, Otto and Felten [330], Ong and Huang [360], Papadimitriou [374], Potvin [388,389], Reinelt [405], Renaud, Boctor and Laporte [407], Rossier, Troyon and Liebling [418], Schnetzler [439], Sun, Meakin and Jossang [462], Ulder, Aarts, Bandelt, van Laarhoven and Pesch [479]. 2.6. THE TSP WITH TIME WINDOWS Baker [20], Daganzo [112], Desrosiers, Sauv6 and Soumis [130], Dumas, Desrosiers, G61inas and Solomon [145], Gendreau, Laporte and Solomon [204], Kubo and Kasugai [271], Tsitsiklis [477]. 2.7. GENERALIZEDAND CLUSTEREDTSPs. TSPs WITH PRECEDENCECONSTRAINTS Bianco, Mingozzi, Ricciardelli and Spadoni [51 ], Gendreau, Hertz and Laporte [200], Kubo and Kasugai [271 ], Laporte and Nobert [296], Noon and Bean [354, 355]. 2.8. SELECTIVEAND PRIZE COLLECTING TSPs Balas [23], Fischetti and Toth [172], Golden, Levy and Vohra [216], Golden, Wang and Liu [218], Keller [258], Laporte and Martello [291], Laporte, Mercure and Nobert [292], Leifer and Rosenwein [305], Mittenthal and Noon [343], Ramesh and Brown [403], Ramesh, Yong-Seok and Karwan [404], Tsiligirides [476]. 2.9. OTHERVARIANTSOF DETERMINISTICTSPs Bianco, Mingozzi and Ricciardelli [52], Cornu6jols, Fonlupt and Naddef [107], Fischetti, Laporte and Martello [170], Gavish and Srikanth [195], Lucena [316], Potvin, Lapalme and Rousseau [394], Russell [423], Semet and Lowenton [441]. 2.10. STOCHASTICTSPs Bertsimas, Jaillet, and Odoni [47], Jaillet [247,248] Laporte, Louveaux and Mercure [290]. 3. The vehicle routing problem (VRP) The "classical VRP" is also defined on a graph G = (V, A U E). Vertex 1)I is a depot at which are based identical vehicles of capacity Q. The remaining vertices represent customers. The problem is then to determine a set of least cost vehicle routes starting and ending at the depot, such that each vertex of V\ {vl} is visited G. Laporte, LH. Osman, A bibliogaphy of routing problems 231 exactly once, and satisfying a number of side constraints. Here are the most common of these restrictions. (i) Capacity constraints: with each vertex vi is associated a non-negative demand qi and the sum of demands in each route may not exceed Q; (ii) Time constraints: here, a travel time tij is associated with each arc or edge and each city of V\ {Vl } has a non-negative service time tSi; the total duration of each route must not exceed a preset bound L; (iii) Time windows: all vehicles leave the depot at time 0 and customer vi must be visited within a time window [ai, bi]. The VRP plays a central role in distribution management and has for a long time attracted the attention of operations researchers. It was described as the Truck Dispatching Problem in an early paper by Dantzig and Ramser [ 115], but the designation VRP is now more common. The problem is still largely unsolved. Exact methods are rarely applicable to problems in excess of 50 vertices [284,295]. Integer linear programming approaches based on constraint relaxation work relatively well on loosely constrainted problems [297,299], while column generation algorithms are better suited for tightly constrained problems [121,131]. Recently, there has been a surge in the development of local search heuristics for the VRP [201]. For recent surveys of VRP algorithms, see Fisher [176], Laporte [284], and Osman [364]. Time constrained VRPs and vehicle scheduling problems are reviewed in Desrosiers, Dumas, Solomon and Soumis [126]. A wide number of variants to the basic problem exist (see, e.g., Assad [15]), such as multiple depots, heterogeneous vehicles, precedence constraints between customers (including the VRP with backhauls), combined pickup and deliveries (including dial-a-ride problems), multiple visits, split deliveries, full load deliveries, multiple-period routing, etc. In the Fleet Size and Mix VRP, the aim is to determine a least cost vehicle fleet to fit the demand. Location-Routing problems investigate the joint effect of locational and routing decisions. A problem closely related to the VRP is the Vehicle Scheduling Problem: here the problem is to design vehicle schedules so as to optimally cover a set of preset routes, tasks or requests. Although these restrictions and variants are often important in practice, they have not received as much attention as the basic problem. 3.1. GENERALAND SURVEY WORK Achuthan and Caccetta [1], Assad [15], Beasley [33], Bodin [56], Bodin and Golden [58], Bodin, Golden, Assad and Ball [59], Bookbinder and Reece [65], Bott and Ballou [67], Christofides [92], Christofides and Mingozzi [99], Christofides, Mingozzi and Toth [101], Christopher [104], Current and Marsh [110], Dantzig and Ramser [115], Dejax [117], Desrochers, Lenstra and Savelsbergh [123], Desrosiers, Dumas, Solomon and Soumis [126], Eilon, Watson-Gandy and Christofides [152], Fisher [176], Golden and Assad [210,211], Hall [233], Institute of Logistics and Distribution Management [242], Laporte [282,284], Lenstra and Rinnooy Kan [307], Osman [364], Osman and Laporte [365], Ronen [417]. 232 3.2. G. Laporte, LH. Osman, A bibliogaphy of routing problems APPLICATIONSAND SOFTWARE Bechara and Galvao [37], Belardo, Duchessi and Seagle [38], Bodin and Salamone [62], Brown, Ellis, Graves and Ronen [74], Brown and Graves [75], Duchessi, Belardo and Seagle [144], Evans and Norback [162], Golden, Bodin and Goodwin [214], Golden and Wasil [219], Holt and Watts [240], Kim [260], Lukka [318], Lysgaard [320], Mathews and Waters [331], O'Neil and Bommer [358], Pape [375], Pooley [386], Potvin, Lapalme and Rousseau [393], Raghavendra, Krishnakumar, Muralidhar, Sarvanan and Raghavendra [401], Robuste, Daganzo and Sou leyrette [415], Russell and Chalinor [424], Sateesh and Ray [433], Savelsbergh [436], Semet and Taillard [442], Sena [443], Solomon, Chalifour, Desrosiers and Boisvert [453], S0rensen [455], Sutcliffe and Board [463], Taylor [467], van Vliet, Boender, Guus and Rinnooy Kan [485], Waters [489,490], Yano, Chan, Ritcher, Cutler, Murity and McGettigan [498], Zielinski [500]. 3.3. FORMULATIONS AND EXACTALGORITHMS FOR THE CLASSICALVRP WITHOUT TIME WINDOWS Agarwal, Mathur and Salkin [3], Araque, Kudva, Morin and Pekny [13], Brodie and Waters [73], Campos, Corber~in and Mota [80], Christofides, Hadjiconstantinou and Mingozzi [98], Christofides, Mingozzi and Toth [100, 102], Cornu6jols and Harche [108], Desrochers and Laporte [122], Fischetti, Toth and Vigo [173], Fisher [174, 175], Fleuren [184], Gavish and Graves [194], Jansen [249], Kulkarni and Bhave [272], Laporte, Mercure and Nobert [293,294], Laporte and Nobert [295,297], Laporte, Nobert and Desrochers [299], Laporte, Nobert and Taillefer [300,301], Li, SimchiLevi and Desrochers [312], Lucena [317], Miller [335], Mingozzi, Christofides and Hadjiconstantinou [340], Naddef [347]. 3.4. HEURISTICALGORITHMS FOR THE CLASSICALVRP WITHOUT TIME WINDOWS Alfa, Heragu and Chen [5], Altinkemer and Gavish [8,9, 10], Baker [19], Ballou [25], Ballou and Agarwal [26], Beasley [34], Beasley and Christofides [35], Bienstock, Bramel and Simchi-Levi [54], Bowerman, Calamai and Hall [66], Bramel, Coffman, Shor and Simchi-Levi [68], Buxey [78,79], Christofides, Mingozzi and Toth [101], Cullen, Jarvis and Ratliff [109], El Ghaziri [157], Fahrin and Wrede [163], Fisher and Jaikumar [177], Foisy and Potvin [185], Gendreau, Hertz and Laporte [199], Gendreau, Laporte and Potvin [201], Golden and Skiscim [217], Haimovich and Rinnooy Kan [230], Haimovich, Rinnooy Kan and Stougie [231], Halse [234], Hiquebran, Alfa and Shapiro [232], Kadaba, Nygard and Juell [255], Kagaya, Kicuchi and Donnelly [256], Kinderwater and Savelsbergh [261], Kopfer, Pankratz and Erkens [266], Li, Simchi-Levi and Desrochers [312], Lukka [318], Macleod and Moll [321], Mole, Johnson and Wells [345], Nelson, Nygard, Griffin G. Laporte, LH. Osman, A bibliogaphy of routing problems 233 and Shreve [352], Nygard and Juell [357], Osman [362,363], Paessens [372], Park and Koelling [376], Potvin [387], Potvin, Lapalme and Rousseau [392,393,394], Pureza and Franca [400], Renaud, Boctor and Laporte [408], Ryan, Hjorring and Glover [427], Salhi and Rand [428], Savelsbergh [435], Semet and Taillard [442], Scrensen [455], Stewart and Golden [459], Taillard [465], Thompson [473], Thompson and Psaraftis [474], Van Breedam [482], Vigo [486], Wark and Holt [488], Willard[491 ], Wong and Beasley [495], Woolsey [496], Zanakis, Evans and Vazacopoulos [499]. 3.5. THE VRP WITH TIME WINDOWS Ahn and Shin [4], Atkinson [18], Baker and Schaffer [21], Bramet, Li and Simchi-Levi [69], Bramel and Simchi-Levi [70], Chiang and Russell [90], Derigs and Grabenbauer [120], Desrochers, Desrosiers and Solomon[121], Desrochers, Lenstra, Savelsberg and Soumis [124], Desrosiers, Sauv6 and Soumis [130], Desrosiers, Soumis and Desrochers [131], Desrosiers, Soumis, Desrochers and Sauv6 [132], Dumas, Desrosiers and Soumis [146], Ferland and Fortin [165], Fisher [175], Fisher, J6rnsten and Madsen [ 179], Garcia, Potvin and Rousseau [ 191 ], Haouari, Dejax and Desrochers [236], J~irnsten, Madsen and SCrensen [253], Kolen, Rinnooy Kan and Trienekens [264], Kontoradis and Bard [265], Koskosidis and Powell [268], Koskosidis, Powell and Solomon [269], Langevin and Soumis [278], Min [338], Nygard, Greenberg, Bolkan and Swenson [356], Potvin, Dub6 and Robillard [390], Potvin, Kervahut and Rousseau [391], Potvin and Rousseau [395], Psaraftis, Solomon, Magnanti and Kim [399], Russell [422], Savelsberg [434, 437], Sexton and Choi [446], Sol and Savelsbergh [449], Solomon [450, 451 ], Solomon, Baker and Schaffer [452], Solomon and Desrosiers [454], Thangiah [468], Thangiah, Nygard and Juell [469], Thangiah, Osman, Vinayagamoorthy and Sun [470], Thangiah, Sun and Potvin [472], Van der Bruggen, Lenstra and Shuur [483], van Landeghem [484]. 3.6. VRPsWITH PRECEDENCERELATIONS Desrosiers, Dumas and Soumis [127], G61inas, Desrochers, Desrosiers and Solomon [197], Goetschalckx and Jacobs-Blecha [209], Jacobs-Blecha and Goetschalckx [246], Min, Current and Schilling [339], Toth and Vigo [475]. 3.7. VRPsWITH COMBINED PICKUPAND DELIVERIES Casco, Golden and Wasil [84], Daganzo and Hall [ 113], Deif and Bodin [ 116], Derigs and Metz [119], Dumas, Desrosiers and Soumis [146], Fisher, Jiegang and Bao-Xing [ 178], Fisher, Tang and Zhen [ 180], Min [337], Savelsbergh and Sol [438], Sexton and Choi [446], Sol and Savelsbergh [449], Thangiah, Sun and Potvin [472], Van der Bruggen, Lenstra and Shuur [483]. 234 3.8. G. Laporte, LH. Osman, A bibliogaphy of routing problems MULTI-PERIODVRPs Christofides and Beasley [94], Gaudioso and Paletta [193], Russell and Gribbin [425], Tan and Beasley [466]. 3.9. MULTI-DEPOTVRPs Bianco, Mingozzi and Ricciardelli [53], Carpaneto, Dell'Amico, Fischetti and Toth [82], Chao, Golden and Wasil [86], Dell'Amico, Fischetti and Toth [118], Laporte, Nobert and Taillefer [300], Li and Simchi-Levi [311], Mesquita and Paixo [334], Renaud, Laporte and Boctor [409]. 3.10. FLEETSIZE AND MIX PROBLEMS Beaujon and Turnquist [36], Buxey [78], Desrochers and Verhoog [125], Desrosiers, Sauv6 and Soumis [ 130], Etazadi and Beasley [158], Ferland and Michelon [116], Gheysens, Golden and Assad [206,207], Golden, Assad, Levy and Gheysens [213], Larson, Minkoff and Gregory [302], Mole [344], Osman and Salhi [366], Parikh [377], Ronen [416], Salhi and Rand [430], Salhi, Sari and Sadi [431]. 3.11. LOCATION-ROUTINGPROBLEMS Balakrishnan, Ward and Wong [22], Eiselt and Laporte [155], Hansen, Hegedahl, Hjortkjaer and Obel [235], Laporte [281], Laporte and Dejax [285], Laporte, Louveaux and Mercure [289], Laporte, Nobert and Arpin [298], Laporte, Nobert and Taillefer [300], Madsen [322], Nambiar, Gelders and van Wassenhove [350], Salhi and Rand [429], Srivastava [456], Srivastava and Benton [457]. 3.12. DIAL-A-RIDEAND VEHICLESCHEDULINGPROBLEMS Ballou [25], Ballou and Agarwal [26], Bodin and Sexton [63], Cyrus[l 11], Dell'Amico, Fischetti and Toth [118], Desrosiers, Dumas, Solomon and Soumis [126], Desrosiers, Dumas and Soumis [127], Desrosiers, Dumas and Soumis [128], Dumas, Desrosiers and Soumis [147], Hooker and Natraj [241], Ioachim, Desrosiers, Dumas and Solomon [243], Jaw, Odoni, Psaraftis and Wilson [250], Kim [260], Koskosidis and Powell [268], Koskosidis, Powell and Solomon [269], Kubo and Kasugai [270], Nygard and Juell [357], Psaraftis [396,398], Psaraftis, Solomon, Magnanti and Kim [399], Ribeiro and Soumis [410], Roy, Rousseau, Lapalme and Ferland [420,421], Sexton and Bodin [444,445], Skitt and Levary [448], Solomon [450, 451 ], Solomon, Baker and Schaffer [452], Solomon and Desrosiers [454], Swersey and Ballard [464], Thompson and Psaraftis [474]. G. Laporte, LH. Osman, A bibliogaphy of routing problems 235 3.13. OTHER VARIANTS OF DETERMINISTICVRPs Afrati, Cosmadakis, Papadimitriou, Papageorgiou and Papakostantinou [2], Anily and Federgruen [11, 12], Brenninger-Gtithe [72], Busch [77], Chien, Balakrishnan and Wong [91], Desrosiers, Laporte, Sauv6, Soumis and Taillefer [129], Dror and Ball [134], Dror, Ball and Golden [135], Dror, Laporte and Trudeau [137], Dror and Levy [139], Dror and Trudeau [142, 143], Federgruen and Zipkin [164], Ferland and Michelon [166], Fran~a, Gendreau, Laporte and MiJller [186], Frizzell and Giffin [190], Golden, Assad and Dahl [212], Labb6, Laporte and Mercure [373], Lamatsch [275], Malandraki and Daskin [324], Min [373], Nag [349], Pooley [385], Psaraftis [397], Russell and Igo [426], Thangiah, Osman,Vinayagamoorthy and Sun [471]. 3.14. STOCHASTICVRPs Bastian and Rinnooy Kan [31 ], Beasley [32], Bertsimas [46], Bertsimas, Jaillet, and Odoni [39], Bertsimas and Van Ryzin [48,49,50], Bienstock, Bramel and SimchiLevi [54], Dror [133], Dror, Laporte and Louveaux [136], Dror, Laporte and Trudeau [137], Dror and Trudeau[141], Gendreau, Laporte and Srguin [202,203], Lambert, Laporteand Louveaux [276], Laporte and Louveaux [286], Laporte, Louveaux and Mercure [288,289], Srguin [440], Simchi-Levi [447], Stewart and Golden [460]. 4. The Chinese Postman Problem (CPP) Both the TSP and the VRP are vertex routing problems as they impose that each vertex be visited. In some contexts such as snow plowing or garbage collection, it is required to cover all arcs of G. Such problems are called arc routing problems. The unconstrained case is the Chinese Postman Problem (CPP). The CPP can be defined on an undirected graph, on a directed graph, or on a mixed graph. The Windy Postman Problem (WPP) is defined on an undirected graph, but the cost of traversing an edge depends on the direction of travel. In the Hierarchical CPP, V contains a source s and a sink t. The arcs of A not incident to s or t are partitioned into {Al .... ,Ak}, and an order relation < is imposed on the elements of the partition. The problem is to determine a least cost traversal of G starting at s, ending at t, and servicing the arcs of the partition in such a way that if At, < Aq, then all arcs of Ap are serviced before any arc of Aq. However, arcs of Aq may be serviced before some arcs of At,. The origins of the CPP can be traced back to the work of Euler [159, 160], who sought to determine whether there existed a closed walk traversing each of the seven Krnigsberg bridges exactly once. Such a walk is called an Eulerian tour. Hierholzer [238] addressed the question of determining a closed walk when one exists, while Meigu Guan (or Kwan Mei-Ko) [226] investigated the problem of determining the shortest traversal of a connected graph known to contain no Eulerian tour. Both the 236 G. Laporte, LH. Osman, A bibliogaphy of routing problems pure undirected and directed versions of the CPP can be solved in polynomial time, but the mixed case, the Windy CPP, and several variants of the basic CPP are NPhard. The complexity of the Hierarchical CPP depends on the structure of G and on whether the order relation is complete or not. Recent surveys of Eulerian graphs and of the CPP are provided in Fleischner [182, 183] and in Eiselt, Gendreau and Laporte [153]. 4.1. GENERAL AND SURVEY WORK Assad and Golden [16], Bodin and Levy [60], Christofides [93], Edmonds and Johnson [148], Eiselt, Gendreau and Laporte [ 153], Evans and Minieka [161 ], Fleischner [182,183], Guan [228], Liebling [314], Orloff [361], Papadimitriou [373], Pearn, Assad and Golden [382], Su [461], Win [492]. 4.2. APPLICATIONS Assad and Golden [16], Barahona [27, 28], Bodin, Fagin, Welebny and Greenberg [57], Cebry, De Silva and Disilio [85], Eglese and Murdock[151], Malek, Mourad and Pandya [327], Riccio [411], Riccio and Litke [412], Roy and Rousseau [419]. 4.3. THE UNDIRECTED CPP Edmonds and Johnson [148], Euler [159, 160], Guan [277], Hierholzer [238], Itai, Lipton, Papadimitriou and Rodeh [244], Itai and Rodeh [245], Kesel'man [259], Korach and Penn [267]. 4.4. THE DIRECTED CPP Beltrami and Bodin [41], Edmonds and Johnson [148], Guan and Pulleyblank [229], Lin and Zhao [315], Richey and Parker [413], Richey, Parker and Rardin [414], van Aardenne-Ehrenfest and de Bruijn [481]. 4.5. THE MIXED CPP Brucker [76], Christofides, Benavent, Campos, Corber~ and Mota [95], Edmonds and Johnson [148], Kappauf and Koehler [257], Minieka [341], Nobert and Picard [353], Ralphs [402], Win [492]. 4.6. THE WINDY CPP Gendreau, Laporte and Zhao [205], Gr6tschel and Win [223,224], Guan [227], Pearn and Li [383], Win [492,493]. G. Laporte, I.H. Osman, A bibliogaphy of muting problems 4.7. 237 THE HIERARCHICALCPP Alfa and Liu [6], Dror, Stern and Trudeau [140], G61inas [196]. 4.8. OTHER VARIANTS OF THE CPP Dror, Stern and Trudeau [140], Malandraki and Daskin [324]. 5. The Rural Postman Problem (RPP) In some arc routing problems, only a subset R of the arcs or of the edges have to be visited. Such problems are referred to as Rural Postman Problems and can be viewed as a special case of the Capacitated Arc Routing Problem (CARP) defined as follows. In G, each arc or edge (vi, vj) has a non-negative weight qq. One must service in a least cost fashion all arcs or edges with qq > 0, using identical vehicles of capacity Q, in such a way that the sum of weights on the arcs serviced by the same vehicle does not exceed Q. The classical RPP is obtained by setting R = {('Oi, l)j)EA LJ E:qij> 0} and Q = 0o. The Stacker Crane Problem is defined on a mixed graph and consists of traversing all directed arcs in a least cost fashion. Several other variants of these basic arc routing problems have also been investigated. Common applications of RPPs are mail delivery, snow plowing, garbage collection, meter reading, milk delivery, parking meter collection, school bus routing, etc. For a recent survey of these applications and of RPP algorithms, see Eiselt, Gendreau and Laporte [I 54]. 5.1. GENERALAND SURVEYWORK Benavent, Campos, Corbedin and Mota [42], Christofides [93], Eiselt, Gendreau and Laporte [ 154]. 5.2. APPLICATIONS Alprin [7], Ball and Magazine [24], Bodin, Fagin, Welebny and Greenberg [57], Bodin and Levy [61], Chen, Kallsen, Chen and Tseng [89], Eglese [149], Eglese and Li [150], Eglese and Murdock [151], Haslam and Wright [237], Lemieux and Campagna [306], Levy [308], Levy and Bodin [309,310], Li [313], McBride [332], Minnazzato [342], Negreiros [351], Ong, Goh, Poh and Lim [359], Roy and Rousseau [419], Stern and Dror [458], Tucker and Clohan [478], Wunderlich, Collette, Levy and Bodin [497]. 5.3. THE UNDIRECTEDRPP Christofides, Campos, Corber(m and Mota [96], Corber,~rt and Sanchis [105,106], Frederickson [187], Sanchis [432]. G. Laporte, I.H. Osman, A bibliogaphy of routing problems 238 5.4. 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