Parameterized Algorithms for Load Coloring Problem

One way to state the Load Coloring Problem (LCP) is as follows. Let G = (V,E) be graph and let f : V → {red,blue} be a 2-coloring. An edge e ∈ E is called red (blue) if both end-vertices of e are red (blue). For a 2-coloring f , let r′ f and b′f be the number of red and blue edges and let μf (G) = min{r ′ f , b ′ f}. Let μ(G) be the maximum of μf (G) over all 2-colorings. We introduce the parameterized problem k-LCP of deciding whether μ(G) ≥ k, where k is the parameter. We prove that this problem admits a kernel with at most 7k. Ahuja et al. (2007) proved that one can find an optimal 2-coloring on trees in polynomial time. We generalize this by showing that an optimal 2-coloring on graphs with tree decomposition of width t can be found in time O(2). We also show that either G is a Yes-instance of k-LCP or the treewidth of G is at most 2k. Thus, k-LCP can be solved in time O(4).