We consider two questions arising in the analysis of heuristic algorithms.
Is there a general procedure involved when analysing a particular problem heuristic?
How can heuristic procedures be incorporated into optimising algorithms such as branch and bound?
In answer to (i) we present one possible procedure, and discuss the cutting stock and travelling salesman problems from this point of view. Noting that the analysis of a heuristic is often based on a linear programming relaxation, we then show how certain heuristics can be integrated into enumeration schemes to produce branch and bound algorithms whose worst case behaviour steadily improves as the enumeration develops. We take the multidimensional knapsack problem, the uncapacitated K-location problem, and the travelling salesman problem as examples.
A.K. Chandra, D.S. Hirchberg and C.K. Wong, “Approximate algorithms for some generalised knapsack problems”, Theoretical Computer Science 3 (1976) 293–304.CrossRefMathSciNetGoogle Scholar
V. Chvatal, “The covering problem”, in: Lecture notes on heuristics (McGill University, 1978).Google Scholar
N. Christofides, “Worst case analysis of a new heuristic for the travelling salesman problem”, GSIA report No. 388, Carnegie-Mellon University (1976).Google Scholar
G. Cornuejols, M.L. Fisher and G.L. Nemhauser, “Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms”, Management Science 23 (1977) 789–810.zbMATHCrossRefMathSciNetGoogle Scholar
M.L. Fisher, G.L. Nemhauser and L.A. Wolsey, “An analysis of approximations for finding a maximum weight Hamiltonian circuit”, Operations Research 27 (1979) 799–809.zbMATHCrossRefMathSciNetGoogle Scholar
A.M. Frieze, “Worst case analysis of algorithms for travelling salesman problems”, Technical report, Department of Computer Science and Statistics, Queen Mary College, London (1978).Google Scholar
M.R. Garey and D.S. Johnson, “Strong NP-completeness results: motivations, examples and implications”, Journal of the Association of Computing Machinery 25 (1978) 499–508.zbMATHMathSciNetGoogle Scholar
M.R. Garey and D.S. Johnson, Computers and intractibility (W.H. Freeman, San Francisco, CA, 1979).Google Scholar
D.S. Johnson, A. Demers, J.D. Ullman, M.R. Garey and R.L. Graham, “Worst case performance bounds for simple one-dimensional packing algorithms”, Society for Industrial and Applied Mathematics Journal on Computing 3 (1974) 299–325.MathSciNetGoogle Scholar
G.L. Nemhauser and L.A. Wolsey, “Maximizing submodular set functions: formulations, algorithms and applications”, CORE D.P 7832, University of Louvain-la-Neuve, Belgium (1978).Google Scholar
D.J. Rosenkrantz, R.E. Stearns and P.M. Lewis, “An analysis of several heuristics for the travelling salesman problem”, Society for Industrial and Applied Mathematics Journal on Computing 6 (1977) 563–581.zbMATHMathSciNetGoogle Scholar
L.A. Wolsey, “Integer programming duality: price functions and sensitivity analysis”, Mimeo, London School of Economics (1978).Google Scholar