On Minimizing the Number of ADMs - Tight Bounds for an Algorithm Without Preprocessing

Minimizing the number of electronic switches in optical networks is a main research topic in recent studies. In such networks we assign colors to a given set of lightpaths. Thus the lightpaths are partitioned into cycles and paths. The switching cost is minimized when the number of paths is minimized. The problem of minimizing the switching cost is NP-hard, and approximation algorithms have been suggested for it. Many of these algoritms have a preprocessing stage, in which they first find cycles. The basic algorithm eliminates cycles of size at most l, and is known to have a performance guarantee of $OPT+\frac{1}{2}(1+\epsilon)N$, where OPT is the cost of an optimal solution, N is the number of lightpaths, and $0 \leq \epsilon \leq \frac{1}{l+2}$, for any given odd l. Without preprocessing phase (i.e. l=1), this reduces to $OPT + \frac{2}{3} N$. We develop a new technique for the analysis of the upper bound and prove a tight bound of $OPT+ \frac{3}{5}N$ for the performance of this algorithm.