On the complexity of the flow coloring problem

Abstract

a b s t r a c tMotivated by bandwidth allocation under interference constraints in radio networks, we define and investigate an optimization problem that combines the classical flow and edge coloring problems in graphs. Let G = (V , E) be a graph with a demand function b : V → Z + and a gateway gis a set with one flow for each source node. Every flow φ defines a multigraph G φ with vertex set V and all edges in the paths in φ. A distance-d edge coloring of a flow φ is an edge coloring of G φ such that two edges with the same color are at distance at least d in G. The distance-d flow coloring problem (FCP d ) is the problem of obtaining a flow φ on G with a distance-d edge coloring where the number of used colors is minimum. For any fixed d ≥ 3, we prove that FCP d is NP-hard even on a bipartite graph with just one source node. For d = 2, we also prove NP-hardness on a bipartite graph with multiples sources. For d = 1, we show that the problem is polynomial in 3-connected graphs and bipartite graphs. Finally, we show that a list version of the problem is inapproximable in polynomial time by a factor of O(log n) even on n-vertex paths, for any d ≥ 1.

Round weighting problem and gathering in radio networks with symmetrical interference

“…In this article, we are considering a symmetrical interference model that has also been studied in [16,21]. The authors in [16] proved that RWP remains NP-hard even on a bipartite graph with one source, for any d 3 fixed. For d = 2, they also proved NP-hardness on a bipartite graph with multiples sources.…”