LINEAR TIME APPROXIMATION ALGORITHM FOR MULTICOLORING LATTICE GRAPHS WITH DIAGONALS

Abstract

Let P be a subset of 2-dimensional integer lattice points P={1, 2, ..., m}×{1, 2, ..., n}⊆Z^2. We consider the graph G_P with vertex set P satisfying that two vertices in P are adjacent if and only if Euclidean distance between the pair is less than or equal to √<2>. Given a non-negative vertex weight vector ω∈Z^P_+, a multicoloring of (G_P, ω) is an assignment of colors to P such that each vertex υ∈P admits ω(υ) colors and every adjacent pair of two vertices does not share a common color. We show the NP-completeness of the problem to determine the existence of a multicoloring of (G_P, ω) with strictly less than (4/3)ω colors where ω denotes the weight of a maximum weight clique. We also propose an O(mn) time approximation algorithm for multicoloring (G_P, ω). Our algorithm finds a multicoloring with at most (4/3)ω+4 colors Our algorithm based on the property that when n=3, we can find a multicoloring of (G_P, ω) with ω colors easily, since an undirected graph associated with (G_P, ω) becomes a perfect graph.