Affine optimal k-proper connected edge colorings

We introduce affine optimal k-proper connected edge colorings as a variation on Fujita’s notion of optimal k-proper connected colorings (Fujita in Optim Lett 14(6):1371–1380, 2020. https://doi.org/10.1007/s11590-019-01442-9) with applications to the frequency assignment problem. Here, for a simple undirected graph G with edge set \(E_G\), such a coloring corresponds to a decomposition of \(E_G\) into color classes \(C_1, C_2, \ldots , C_n\), with associated weights \(w_1, w_2, \ldots , w_n\), minimizing a specified affine function \({\mathcal {A}}\, {:=}\,\sum _{i=1}^{n} \left( w_i \cdot |C_i|\right)\), while also ensuring the existence of k vertex disjoint proper paths (i.e., simple paths with no two adjacent edges in the same color class) between all pairs of vertices. In this context, we define \(\zeta _{{\mathcal {A}}}^k(G)\) as the minimum possible value of \({\mathcal {A}}\) under a k-proper connectivity requirement. For any fixed number of color classes, we show that computing \(\zeta _{{\mathcal {A}}}^k(G)\) is treewidth fixed parameter tractable. However, we also show that determining \(\zeta _{{\mathcal {A}}^{\prime }}^k(G)\) with the affine function \({\mathcal {A}}^{\prime } \, {:=}\,0 \cdot |C_1| + |C_2|\) is NP-hard for 2-connected planar graphs in the case where \(k = 1\), cubic 3-connected planar graphs for \(k = 2\), and k-connected graphs \(\forall k \ge 3\). We also show that no fully polynomial-time randomized approximation scheme can exist for approximating \(\zeta _{{\mathcal {A}}^{\prime }}^k(G)\) under any of the aforementioned constraints unless \(NP=RP\).