Parameterized algorithms for load coloring problem

Abstract

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 µ f (G) = min{rWe 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 t ). 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 k ).

Parameterized and Approximation Algorithms for the Load Coloring Problem

“…Indeed, if G / ∈ (c, k)-LC, any DFS on the connected components of G gives a Tremaux tree with depth bounded by c(k + 1) that we may transform into path decomposition of size bounded by c(k + 1) in polynomial time. Since the O * (2 tw(G) )-time algorithm for 2-Load Coloring from [9] can be generalized to an O * (c tw(G) )-time algorithm for c-Load Coloring, there exists a O * (c ck )-time algorithm for this problem.…”

“…For c = 2, the running time O * (4 k ) (first obtained in [9]) can be improved using the result by Kneis et al [13] that a graph with m edges and n vertices has treewidth at most m/5.769 + O(log n). Thus, by Theorem 3 in polynomial time we can reduce a graph G to a graph G with tw(G ) ≤ 1.0401k + O( √ k).…”

“…Thus, for c = 2 we improve the kernel result of [9]. To show our result, we introduce reduction rules, which are new even for c = 2.…”

“…This problem is NP-complete [1], and Gutin and Jones studied its parameterization by k [9]. They proved that 2-Load Coloring is fixed-parameter tractable (FPT) 1 by obtaining a kernel with at most 7k vertices.…”

“…6. Graphs Following [1,9], we consider graphs without loops or multiple edges. (Actually, our results generalize to graphs with loops and multiple edges, see Sect.…”