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, … , C_n , with associated weights w_1, w_2, … , w_n , minimizing a specified affine function 𝒜 := ∑ _i=1^n( w_i · |C_i|) , 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 ζ _𝒜^k(G) as the minimum possible value of 𝒜 under a k-proper connectivity requirement. For any fixed number of color classes, we show that computing ζ _𝒜^k(G) is treewidth fixed parameter tractable. However, we also show that determining ζ _𝒜^'^k(G) with the affine function 𝒜^' := 0 · |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 ∀ k ≥ 3 . We also show that no fully polynomial-time randomized approximation scheme can exist for approximating ζ _𝒜^'^k(G) under any of the aforementioned constraints unless NP=RP .