This paper combines local search and approximation into “approximation of local optima”, i.e., an attempt to escape hardness of exact local optimization by trying to find solutions which are approximately as good as the worst local optimum. The complexity of several well known optimization problems under this approach is investigated. Our main tool is a special reduction called the “strong PLS-reduction” which preserves cost and local search structure in a very strict sense. Completeness for the class of NP-optimization/local search problems under this reduction allows to deduce that both approximation of global and of local optima cannot be achieved efficiently (unless P = NP resp. P = PLS). We show that the (weighted) problems Min 4-DNF, MaxHopfield, Min/Max 0-1-Programming, Min Independent Dominating Set, and Min Traveling Salesman are all complete under the new reduction.
Moreover the unweighted Min 3-DNF problem is shown to be complete for the class of NP-optimization problems with polynomially bounded cost functions under an approximation preserving reduction. This implies that the logically defined class Min ∑0, the minimization analogue of max SNP, does not capture any (low) approximation degree.