Heuristic analysis, linear programming and branch and bound

Abstract

We consider two questions arising in the analysis of heuristic algorithms.

• (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.

“The methods used for designing such (heuristic) algorithms tend to the rather problem specific, although a few guiding principles have been identified and can provide a useful starting point”. M.R. Garey and D.S. Johnson: Computers and Intractibility [11, Ch. 6, p. 122].

This research was supported by a Senior Visiting Research Fellowship from the Science Research Council, while the author was on leave from CORE, Université Catholique de Louvain at Louvain-la-Neuve, Belgium.