Combinatorial Optimization II pp 121-134 | Cite as

# Heuristic analysis, linear programming and branch and bound

## Abstract

- (i)
Is there a general procedure involved when analysing a particular problem heuristic?

- (ii)
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.

## Key words

Algorithm Analysis Benders’ Algorithm Bin Packing Branch and Bound Duality Gaps Dynamic Programming (Euclidean) Travelling Salesman Heuristic Longest Hamiltonian Tour Matching Heuristic (Mimimum Length) Eulerian Tours (Multidimensional) Knapsack Optimising Problems Uncapacitated*k*-location

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J.F. Benders, “Partitioning procedures for solving mixed-variables programming problems”,
*Numerische Mathematik*4 (1962) 238–252.MATHCrossRefMathSciNetGoogle Scholar - [2]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 - [3]V. Chvatal, “The covering problem”, in:
*Lecture notes on heuristics*(McGill University, 1978).Google Scholar - [4]N. Christofides, “Worst case analysis of a new heuristic for the travelling salesman problem”, GSIA report No. 388, Carnegie-Mellon University (1976).Google Scholar
- [5]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.MATHCrossRefMathSciNetGoogle Scholar - [6]J. Edmonds and E.L. Johnson, “Matching, Euler tours and the chinese postman”,
*Mathematical Programming*5 (1973) 88–124.MATHCrossRefMathSciNetGoogle Scholar - [7]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.MATHCrossRefMathSciNetGoogle Scholar - [8]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
- [9]D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,
*Mathematical Programming*1 (1971) 168–194.MATHCrossRefMathSciNetGoogle Scholar - [10]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.MATHMathSciNetGoogle Scholar - [11]M.R. Garey and D.S. Johnson,
*Computers and intractibility*(W.H. Freeman, San Francisco, CA, 1979).Google Scholar - [12]P.C. Gilmore and R.E. Gomory, “A linear programming approach to the cutting stock problem”,
*Operations Research*9 (1961) 849–859.MATHCrossRefMathSciNetGoogle Scholar - [13]R.G. Jeroslow, “Cutting plane theory: algebraic methods”,
*Discrete Mathematics*23 (1978) 121–150.MATHCrossRefMathSciNetGoogle Scholar - [14]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 - [15]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
- [16]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.MATHMathSciNetGoogle Scholar - [17]L.A. Wolsey, “Integer programming duality: price functions and sensitivity analysis”, Mimeo, London School of Economics (1978).Google Scholar