## 5 Summary  
In this paper, we investigated the problem of minimizing hardware cost in optical networks in the scenario in which at most \(g\) (\(g\) is the grooming factor) lightpaths can share the same color on any network edge. It is worth noticing that, while most previous work takes into account only the cost due to electronic components (i.e., ADMs), we focused on the combined cost due to ADMs and optical components (OADMs). In particular, we considered the cost function of \(f(\alpha)=\alpha|OADMs|+(1-\alpha)|ADMs|\), where \(0\leq\alpha\leq 1\), and we have studied the fundamental topologies of ring and chain networks.  
We showed that finding an optimal coloring, i.e., a coloring minimizing the considered cost function, is NP-complete even on chains and when the grooming factor is \(g=2\), for any value of \(\alpha>0\). We then presented a general technique that, given an \(r\)-approximation algorithm working on particular instances of our problem (i.e., instances in which all requests share a same edge of the network), builds a new algorithm for general instances having an approximation ratio \(O(r\log n)\). We exploited this technique in order to obtain an \(O(\sqrt{g}\log n)\)-approximation algorithms for our problem.  
It is worth noticing that our approximation ratio of \(2\sqrt{g}\left(\left\lceil\log n\right\rceil+1\right)\) is better than the trivial one (which has a \(2g\) factor) in case \(g>(\left\lceil\log n\right\rceil+1)^{2}\). Moreover, our algorithm has the following interesting property: the returned solution simultaneously achieves the claimed approximation ratio with respect to every fixed value of \(\alpha\).  
There are several interesting left open problems:  
* Strengthening the \(NP\)-completeness result, either towards the direction of proving an \(APX\)-hardness result or towards an impossibility of a sub-exponential algorithm under the exponential time hypothesis (\(ETH\), introduced in [33]). In this respect, it is worth noting that the current reduction cannot be exploited in order to obtain these extensions. In particular, with respect to the former direction, the considered 11 problem is a decision problem and it is worth investigating whether a related optimization problem could be exploited in order to prove the \(APX\)-hardness. On the other hand, with respect to the latter direction, even if 11 were to be proven to admit no sub-exponential algorithm under the ETH, our reduction builds an instance of 11 of quadratic size with respect to the one of 11 and therefore would not be able to provide a similar result for 11.
* Improving the achieved approximation ratio for the considered topologies of ring and chain networks.  
* Extending the algorithm and the analysis to other network topologies.
* Considering the online version of the problem in which lightpath requests are not given in advance but arrive over time.  
**Author Contributions:** Formal analysis, M.F., G.M., L.M., M.S. and S.Z.; Writing--original draft, G.M., L.M. and M.S.; Writing--review & editing, M.F., G.M., L.M., M.S. and S.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.