Applying Convex Integer Programming: Sum Multicoloring and Bounded Neighborhood Diversity
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In the past 30 years, results regarding special classes of integer linear (and, more generally, convex) programs ourished. Applications in the field of parameterized complexity were called for and the call has been answered, demonstrating the importance of connecting the two fields. The classical result due to Lenstra states that solving Integer Linear Programming in Fixed dimension is polynomial. Later, Khachiyan and Porkolab has extended this result to optimizing a quasiconvex function over a convex set. While applications of the former result have been known for over 10 years, it seems the latter result has not been applied much in the parameterized setting yet. We give one such application. Specifically, we deal with the Sum Coloring problem and a generalization thereof called Sum-Total Multicoloring, which is similar to the preemptive Sum Multicoloring problem. In Sum Coloring, we are given a graph G = (V,E) and the goal is to find a proper coloring c V→N minimizing ∑_v∈ V c(v). By formulating these problems as convex integer programming in small dimension, we show fixed-parameter tractability results for these problems when parameterized by the neighborhood diversity of G, a parameter generalizing the vertex cover number of G.
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