The traveling salesman problem: An update of research

During the course of the last few years, attacks on the traveling salesman problem have resulted in a variety of often innovative and rather powerful computational procedures. In this article, we present a review of these results for problems defined on weighted and unweighted graphs. Some account of computational behavior for exact algorithms is provided; however, the primary coverage deals with the strategy of particular procedures. In addition, we include some aspects of nonexact algorithms with major interest confined to the establishment of worst-case bounds.

[1]  Kenneth Steiglitz,et al.  On the Complexity of Local Search for the Traveling Salesman Problem , 1977, SIAM J. Comput..

[2]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[3]  Mandell Bellmore,et al.  Pathology of Traveling-Salesman Subtour-Elimination Algorithms , 1971, Oper. Res..

[4]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[5]  S. M. Roberts,et al.  Systematic generation of Hamiltonian circuits , 1966, CACM.

[6]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[7]  Richard M. Karp,et al.  Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane , 1977, Math. Oper. Res..

[8]  C. Nash-Williams,et al.  Hamiltonian circuits in graphs and digraphs , 1969 .

[9]  Martin Grötschel,et al.  On the symmetric travelling salesman problem I: Inequalities , 1979, Math. Program..

[10]  Thomas L. Magnanti,et al.  Deterministic network optimization: A bibliography , 1977, Networks.

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  Richard M. Karp,et al.  A Patching Algorithm for the Nonsymmetric Traveling-Salesman Problem , 1979, SIAM J. Comput..

[13]  P. Miliotis,et al.  Using cutting planes to solve the symmetric Travelling Salesman problem , 1978, Math. Program..

[14]  Bruce L. Golden,et al.  A statistical approach to the tsp , 1977, Networks.

[15]  Richard M. Karp,et al.  An algorithm to solve the m × n assignment problem in expected time O(mn log n) , 1980, Networks.

[16]  Alex Karel Obruca Spanning Tree Manipulation and the Travelling Salesman Problem , 1968, Comput. J..

[17]  M. Meyniel Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe oriente , 1973 .

[18]  Vasek Chvátal,et al.  Edmonds polytopes and weakly hamiltonian graphs , 1973, Math. Program..

[19]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[20]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[21]  G. L. Thompson,et al.  A Heuristic Approach to Solving Travelling Salesman Problems , 1964 .

[22]  Nicos Christofides,et al.  Algorithms for Large-scale Travelling Salesman Problems , 1972 .

[23]  G. Dirac Some Theorems on Abstract Graphs , 1952 .

[24]  Robert S. Garfinkel,et al.  Technical Note - On Partitioning the Feasible Set in a Branch-and-Bound Algorithm for the Asymmetric Traveling-Salesman Problem , 1973, Oper. Res..

[25]  G. L. Nemhauser,et al.  Tight bounds for christofides' traveling salesman heuristic , 1978, Math. Program..

[26]  Egon Balas,et al.  A restricted Lagrangean approach to the traveling salesman problem , 1981, Math. Program..

[27]  L. Lesniak-Foster,et al.  Some recent results in hamiltonian graphs , 1977 .

[28]  Eugene L. Lawler,et al.  A solvable case of the traveling salesman problem , 1971, Math. Program..

[29]  Nicos Christofides,et al.  Technical Note - Bounds for the Travelling-Salesman Problem , 1972, Oper. Res..

[30]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[31]  Jamie J. Goode,et al.  The traveling salesman problem: A duality approach , 1977, Math. Program..

[32]  V. Chvátal On Hamilton's ideals , 1972 .

[33]  M. R. Rao,et al.  The travelling salesman problem and a class of polyhedra of diameter two , 1974, Math. Program..

[34]  Christos H. Papadimitriou,et al.  The adjacency relation on the traveling salesman polytope is NP-Complete , 1978, Math. Program..

[35]  Nicos Christofides,et al.  Strong sufficient conditions for the existence of Hamiltonian circuits in undirected graphs , 1981, J. Comb. Theory, Ser. B.

[36]  Jan Karel Lenstra,et al.  Some Simple Applications of the Travelling Salesman Problem , 1975 .

[37]  Martin Grötschel,et al.  On the symmetric travelling salesman problem II: Lifting theorems and facets , 1979, Math. Program..

[38]  G. Croes A Method for Solving Traveling-Salesman Problems , 1958 .

[39]  Patrick D. Krolak,et al.  A man-machine approach toward solving the traveling salesman problem , 1970, DAC '70.

[40]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[41]  Selmer M. Johnson,et al.  On a Linear-Programming, Combinatorial Approach to the Traveling-Salesman Problem , 1959 .

[42]  P. Miliotis,et al.  Integer programming approaches to the travelling salesman problem , 1976, Math. Program..

[43]  Victor Klee,et al.  Combinatorial Optimization: What is the State of the Art , 1980, Math. Oper. Res..

[44]  Gerald L. Thompson,et al.  Computational Performance of Three Subtour Elimination Algorithms for Solving Asymmetric Traveling Salesman Problems. , 1977 .

[45]  John Adrian Bondy,et al.  A method in graph theory , 1976, Discret. Math..

[46]  M. M. Flood The Traveling-Salesman Problem , 1956 .

[47]  Rainer E. Burkard,et al.  Travelling Salesman and Assignment Problems: A Survey , 1979 .

[48]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[49]  O. Ore Note on Hamilton Circuits , 1960 .

[50]  A. Bagchi,et al.  Neighborhood Search Algorithms for Guaranteeing Optimal Traveling Salesman Tours Must Be Inefficient , 1976, J. Comput. Syst. Sci..

[51]  Jakob Krarup,et al.  Improvements of the Held—Karp algorithm for the symmetric traveling-salesman problem , 1974, Math. Program..

[52]  Nicos Christofides,et al.  The Shortest Hamiltonian Chain of a Graph , 1970 .

[53]  T. H. C. Smith A LIFO implicit enumeration algorithm for the asymmetric travelling salesman problem using a one-arborescence relaxation , 1980 .

[54]  George L. Nemhauser,et al.  The Traveling Salesman Problem: A Survey , 1968, Oper. Res..

[55]  Shen Lin Computer solutions of the traveling salesman problem , 1965 .