On the Depth of Combinatorial Optimization Problems

This paper studies the notion of “depth” of a combinatorial optimization problem, which has already been touched on by several other authors (cf. Weber or Hajek). This notion can be defined as follows: Consider a discrete optimization problem min{r(.u) 1 x E x). Usually hill climbing algorithms work as follows: First, a neighbourhood structure has to be specified on X. Then, starting from an arbitrary solution, say x0, one constructs a sequence _~,,x,,x~,... (hopefully leading to an optimum) such that x, is a neighbour of x,_, for all i. For x0 E ,I’, the depth d(x,) is defined to be the smallest d 2 0 such that there exists a sequence s~~,x,,...,x,, with c(x,) < c(q,) and all intermediate solutions x, having objective value c(x,) 5 c(x”) + d. The depth of the problem is defined to be the maximum depth of a solution xg E X. This notion plays an important role in the theory of Simulated Annealing. Here we want to show that the depth is of some interest in its own right. We will prove some upper bounds and investigate the computational complexity of the depth function for some selected examples.