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Optimisation combinatoire: graphes et programmation linéaire and programmation discréte
144
Book Reviews
topic since it is primarily their research which is the state of the art in polyhedral theory for the TSP. Chapter 8 is a highly specialized mathematical chapter. The presentation is clear and there are a number of research problems for the reader who is interested in challenging research. The solution methodology presented here can also be useful for the closely related VRP as G. Laporte and Y. Norbert illustrated in " C o m b inequalities for the VRP", Publication # 305, Centre de recherche sur les transports, University of Montreal, 1983. In Chapter 9, "Polyhedral computations", the authors describe the computational implications following the theory outlined in their previous chapter. Of the two chapters, the second is easier to follow. The ellipsoid method for linear programs is nicely sketched out here. The computational experiments are very impressive and provide a clear conclusion for the two chapters. Chapter 10, pp. 361-401, is devoted to "Branch and bound methods" and is written by E. Balas and P. Toth. The chapter provides a well-written thorough overview of branch and bound methodology as applied to the TSP. It describes the state of the art for different solution procedures for an exact solution to the TSP. The relaxation approaches are well-sketched as are the bounding rules. If there is a fault, then it is with the presentation of the computational results which cannot be directly compared. Also, in my opinion the comparison of the average performance of the algorithms in terms of the best fit warrants only a paragraph. Chapter 11, pp. 403-429, is a graph theoretical chapter titled "Hamiltonian cycles" written by V. Chvatal. It is an important chapter which provides yet an additional dimension of viewing the problem of the traveling salesman. It complements the rest of the book very nicely; while, as the author writes, "This is a biased e x p o s i t i o n . . . " , it is one which adds to the knowledge and understanding of what it means to solve: a TSP. The chapter provides all the known results on the Hamiltonian and non-Hamiltonian properties of undirected graphs. The chapter includes no discussion of directed graphs, but covers random graphs, planar graphs and powers of graphs. It also contains a good number of exercises and research problems. The book concludes with Chapter 12, pp. 431-448, on "Vehicle routing" by N. Christofides. When the naive outsider asks " W h a t is the travel-
ing salesman problem?", the most frequently used illustration is that of a vehicle route. The author does not have to elaborate on the close tie between vehicle routing and the TSP, so he does not. This chapter is a short and well-presented overview of the VRP. In contrast to the earlier chapters, the material in this chapter has immediate practical implication for an O.R. professional connected with industry, especially the distribution industry. The chapter contains the formulations of the VRP and the state of the art in solution techniques for the problem. For the purists the exact methods are sketched out, and for the practitioners the most successful heuristics are presented. Only the generic VRP is considered, and the presentation is very concise. In summary, I believe this book is a first-rate addition to the library of a serious researcher and student of combinatorial optimization. It could be used in its entirety as the basis for a doctoral student seminar or chapters 3, 5, 6, 7, 9, 10 could be used as a core in a graduate course on computational aspects of combinatorial optimization algorithms. I myself would welcome an opportunity to use this stimulating text for teaching either of those courses. The book encompasses thirty years of research on the TSP. The topic is very well covered, and only minor aspects of the problem were omitted. There are still many interesting avenues of research open to researchers today, and ten or twenty years from now the same title might be used to tell a new story with the same point of departure.
MOSHE DROR Industrial Engineering and Management Ben-Gurion University of the Negev Beersheva, Israel Michel S A K A R O V I T C H
Optimisation Combinatoire: Graphes et Programmarion Lin~aire, and
Programmation Diser+te Volumes 31 and 32 in: Enseignement des Sciences, Hermann, Paris, 1984, xiii + 249 pages, F164.00, xiii + 269 pages, F164.00 The publication of a volume in French, in the area of graphs and combinatorial optimization is
Book Reviews not so frequent an event that it be forbidden to devote more than a few lines to it. If, to begin with, I had to characterize in a few words the book by Michel Sakarovitch (Hermann, Paris 1984; volume " G " : Graphs and Linear Programming; volume "P": Discrete Programming) 1 would say that it is an introductory volume: - pedagogical and attractive: accurate and concise on abstract issues: - as complete as possible: stimulating and open. (1) Pedagogical. It is true that subjects like graph theory lend themselves pretty well to concretizing abstract concepts. However, few authors before have so systematically used the inductive approach which consists in introducing theoretical notions from simple examples and concrete situations: the shortest path problem from a decanting problem in Chapter G2; the well-known marriage problem in Chapter G6 to illustrate the concept of a bipartite g r a p h . . , and much more examples could be a d d e d - - M . Sakarovitch thus holds his readers (as, before them, his students I imagine) breathless by this frequent recourse to mathematical games. (2) Accurate and concise. Once the theoretical bases have been laid and the concepts have been made more precise, if necessary, by pointing out counter-examples or possible traps (another efficient way of stimulating in-depth learning of the beginner) the style switches to that of a more traditional mathematical exposition, where the rules are those of economy of words and consistency in the deductive reasoning: and, in this part, too, the style is perfect, as will be realized at the reading of many passages: introduction of the structures of flows and tensions (G6, Section 3); linear programming (G8-G8-G10); fundamentals of complexity theory (Chapter P2) and others. (3) As complete as possible. One can say, without any risk of being wrong, that nothing really fundamental in the field has been left aside. One will even find included--a rare feature for an introductory book--mentions to fairly recent developments such as: new finite pivoting rules for the simplex method (derived from recent work in the field of oriented matroids); fundamentals of polyhedral combinatorics in Chapter P7: the use of Lagrangean relaxation methods in combinatorial optimization (Chapter P6). Though inevitably succinct, such developments are anyhow most useful
145
to draw the reader's attention upon techniques potentially conducive to future progress. Concerning completeness, purists will only be entitled to object to the somewhat reduced place devoted to Hamiltonian problems as well as to (Khachian's) ellipsoid algorithm together with its consequences in combinatorial optimization (on the charge of "reduced practical usefulness", see G l l , Section 1). This algorithm (which is not quoted in the bibliography) was however to open the road to a brand new approach (Karmarkar, 1984) combining both theoretical efficiency (polynomiality) and practical usefulness, therefore invalidating the remark concerning the "anomaly of linear programming" (Chapter P2, Section l). However, as far as we are concerned, our only regret will be the relative scarcity of bibliographical references which, without growing immoderately, could have at least included the references to the main algorithms described in the text of the book (e.g. Prim, Kruskal, Dijkstra, Dantzig, Khachian . . . . ). (4) Stimulating and open. The author does not restrict himself to the perfected core of the theory, which, without other perspectives, is nothing but a tombstone: polished, maybe, but turned at the past. On the contrary, M. Sakarovitch shows us an evolving discipline, whose liveliness derives from its very incompletions, as is particularly apparent, for instance, from the discussion of Chapter P2 on complexity theory and its related open problems (P = NP?). As a whole, a volume of high pedagogical value, which can be unreservedly recommended, not only to many students or beginners in our discipline, but also to many teachers as a basic text-book in their courses. M. M I N O U X 69 Rue Barrault Paris, France
Roger HARTLEY Linear and Nonlinear Programming: An introduction to linear methods in mathematical programming
Ellis Horwood. Chichester, 1985, 221 pages, £9.95 The reviewed book deals with methods of solving linear programming problems and some prob-
Book Reviews
topic since it is primarily their research which is the state of the art in polyhedral theory for the TSP. Chapter 8 is a highly specialized mathematical chapter. The presentation is clear and there are a number of research problems for the reader who is interested in challenging research. The solution methodology presented here can also be useful for the closely related VRP as G. Laporte and Y. Norbert illustrated in " C o m b inequalities for the VRP", Publication # 305, Centre de recherche sur les transports, University of Montreal, 1983. In Chapter 9, "Polyhedral computations", the authors describe the computational implications following the theory outlined in their previous chapter. Of the two chapters, the second is easier to follow. The ellipsoid method for linear programs is nicely sketched out here. The computational experiments are very impressive and provide a clear conclusion for the two chapters. Chapter 10, pp. 361-401, is devoted to "Branch and bound methods" and is written by E. Balas and P. Toth. The chapter provides a well-written thorough overview of branch and bound methodology as applied to the TSP. It describes the state of the art for different solution procedures for an exact solution to the TSP. The relaxation approaches are well-sketched as are the bounding rules. If there is a fault, then it is with the presentation of the computational results which cannot be directly compared. Also, in my opinion the comparison of the average performance of the algorithms in terms of the best fit warrants only a paragraph. Chapter 11, pp. 403-429, is a graph theoretical chapter titled "Hamiltonian cycles" written by V. Chvatal. It is an important chapter which provides yet an additional dimension of viewing the problem of the traveling salesman. It complements the rest of the book very nicely; while, as the author writes, "This is a biased e x p o s i t i o n . . . " , it is one which adds to the knowledge and understanding of what it means to solve: a TSP. The chapter provides all the known results on the Hamiltonian and non-Hamiltonian properties of undirected graphs. The chapter includes no discussion of directed graphs, but covers random graphs, planar graphs and powers of graphs. It also contains a good number of exercises and research problems. The book concludes with Chapter 12, pp. 431-448, on "Vehicle routing" by N. Christofides. When the naive outsider asks " W h a t is the travel-
ing salesman problem?", the most frequently used illustration is that of a vehicle route. The author does not have to elaborate on the close tie between vehicle routing and the TSP, so he does not. This chapter is a short and well-presented overview of the VRP. In contrast to the earlier chapters, the material in this chapter has immediate practical implication for an O.R. professional connected with industry, especially the distribution industry. The chapter contains the formulations of the VRP and the state of the art in solution techniques for the problem. For the purists the exact methods are sketched out, and for the practitioners the most successful heuristics are presented. Only the generic VRP is considered, and the presentation is very concise. In summary, I believe this book is a first-rate addition to the library of a serious researcher and student of combinatorial optimization. It could be used in its entirety as the basis for a doctoral student seminar or chapters 3, 5, 6, 7, 9, 10 could be used as a core in a graduate course on computational aspects of combinatorial optimization algorithms. I myself would welcome an opportunity to use this stimulating text for teaching either of those courses. The book encompasses thirty years of research on the TSP. The topic is very well covered, and only minor aspects of the problem were omitted. There are still many interesting avenues of research open to researchers today, and ten or twenty years from now the same title might be used to tell a new story with the same point of departure.
MOSHE DROR Industrial Engineering and Management Ben-Gurion University of the Negev Beersheva, Israel Michel S A K A R O V I T C H
Optimisation Combinatoire: Graphes et Programmarion Lin~aire, and
Programmation Diser+te Volumes 31 and 32 in: Enseignement des Sciences, Hermann, Paris, 1984, xiii + 249 pages, F164.00, xiii + 269 pages, F164.00 The publication of a volume in French, in the area of graphs and combinatorial optimization is
Book Reviews not so frequent an event that it be forbidden to devote more than a few lines to it. If, to begin with, I had to characterize in a few words the book by Michel Sakarovitch (Hermann, Paris 1984; volume " G " : Graphs and Linear Programming; volume "P": Discrete Programming) 1 would say that it is an introductory volume: - pedagogical and attractive: accurate and concise on abstract issues: - as complete as possible: stimulating and open. (1) Pedagogical. It is true that subjects like graph theory lend themselves pretty well to concretizing abstract concepts. However, few authors before have so systematically used the inductive approach which consists in introducing theoretical notions from simple examples and concrete situations: the shortest path problem from a decanting problem in Chapter G2; the well-known marriage problem in Chapter G6 to illustrate the concept of a bipartite g r a p h . . , and much more examples could be a d d e d - - M . Sakarovitch thus holds his readers (as, before them, his students I imagine) breathless by this frequent recourse to mathematical games. (2) Accurate and concise. Once the theoretical bases have been laid and the concepts have been made more precise, if necessary, by pointing out counter-examples or possible traps (another efficient way of stimulating in-depth learning of the beginner) the style switches to that of a more traditional mathematical exposition, where the rules are those of economy of words and consistency in the deductive reasoning: and, in this part, too, the style is perfect, as will be realized at the reading of many passages: introduction of the structures of flows and tensions (G6, Section 3); linear programming (G8-G8-G10); fundamentals of complexity theory (Chapter P2) and others. (3) As complete as possible. One can say, without any risk of being wrong, that nothing really fundamental in the field has been left aside. One will even find included--a rare feature for an introductory book--mentions to fairly recent developments such as: new finite pivoting rules for the simplex method (derived from recent work in the field of oriented matroids); fundamentals of polyhedral combinatorics in Chapter P7: the use of Lagrangean relaxation methods in combinatorial optimization (Chapter P6). Though inevitably succinct, such developments are anyhow most useful
145
to draw the reader's attention upon techniques potentially conducive to future progress. Concerning completeness, purists will only be entitled to object to the somewhat reduced place devoted to Hamiltonian problems as well as to (Khachian's) ellipsoid algorithm together with its consequences in combinatorial optimization (on the charge of "reduced practical usefulness", see G l l , Section 1). This algorithm (which is not quoted in the bibliography) was however to open the road to a brand new approach (Karmarkar, 1984) combining both theoretical efficiency (polynomiality) and practical usefulness, therefore invalidating the remark concerning the "anomaly of linear programming" (Chapter P2, Section l). However, as far as we are concerned, our only regret will be the relative scarcity of bibliographical references which, without growing immoderately, could have at least included the references to the main algorithms described in the text of the book (e.g. Prim, Kruskal, Dijkstra, Dantzig, Khachian . . . . ). (4) Stimulating and open. The author does not restrict himself to the perfected core of the theory, which, without other perspectives, is nothing but a tombstone: polished, maybe, but turned at the past. On the contrary, M. Sakarovitch shows us an evolving discipline, whose liveliness derives from its very incompletions, as is particularly apparent, for instance, from the discussion of Chapter P2 on complexity theory and its related open problems (P = NP?). As a whole, a volume of high pedagogical value, which can be unreservedly recommended, not only to many students or beginners in our discipline, but also to many teachers as a basic text-book in their courses. M. M I N O U X 69 Rue Barrault Paris, France
Roger HARTLEY Linear and Nonlinear Programming: An introduction to linear methods in mathematical programming
Ellis Horwood. Chichester, 1985, 221 pages, £9.95 The reviewed book deals with methods of solving linear programming problems and some prob-