Affine optimal k-proper connected edge colorings

Abstract

AbstractWe introduce affine optimalk-proper connected edge colorings as a variation on Fujita’s notion of optimalk-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 $$E_G$$ E G , such a coloring corresponds to a decomposition of $$E_G$$ E G into color classes $$C_1, C_2, \ldots , C_n$$ C 1 , C 2 , … , C n , with associated weights $$w_1, w_2, \ldots , w_n$$ w 1 , w 2 , … , w n , minimizing a specified affine function $${\mathcal {A}}\, {:=}\,\sum _{i=1}^{n} \left( w_i \cdot |C_i|\right)$$ 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 $$\zeta _{{\mathcal {A}}}^k(G)$$ ζ A k ( G ) as the minimum possible value of $${\mathcal {A}}$$ A under a k-proper connectivity requirement. For any fixed number of color classes, we show that computing $$\zeta _{{\mathcal {A}}}^k(G)$$ ζ A k ( G ) is treewidth fixed parameter tractable. However, we also show that determining $$\zeta _{{\mathcal {A}}^{\prime }}^k(G)$$ ζ A ′ k ( G ) with the affine function $${\mathcal {A}}^{\prime } \, {:=}\,0 \cdot |C_1| + |C_2|$$ A ′ : = 0 · | C 1 | + | C 2 | is NP-hard for 2-connected planar graphs in the case where $$k = 1$$ k = 1 , cubic 3-connected planar graphs for $$k = 2$$ k = 2 , and k-connected graphs $$\forall k \ge 3$$ ∀ k ≥ 3 . We also show that no fully polynomial-time randomized approximation scheme can exist for approximating $$\zeta _{{\mathcal {A}}^{\prime }}^k(G)$$ ζ A ′ k ( G ) under any of the aforementioned constraints unless $$NP=RP$$ N P = R P .