Competitive Vertex Recoloring

Abstract Motivated by placement of jobs in physical machines, we introduce and analyze the problem of online recoloring, or online disengagement. In this problem, we are given a set of $n$ weighted vertices and a $k$-coloring of the vertices (vertices represent jobs, and colors represent physical machines). Edges, representing conflicts between jobs, are inserted in an online fashion. After every edge insertion, the algorithm must output a proper $k$-coloring of the vertices. The cost of a recoloring is the sum of weights of vertices whose color changed. Our aim is to minimize the competitive ratio of the algorithm, i.e., the ratio between the cost paid by the online algorithm and the cost paid by an optimal, offline algorithm. We consider a couple of polynomially-solvable coloring variants. Specifically, for 2-coloring bipartite graphs we present an $O(\log n)$-competitive deterministic algorithm and an $\Omega(\log n)$ lower bound on the competitive ratio of randomized algorithms. For $(\Delta+1)$-coloring, where $\Delta$ is the maximal node degree, we present tight bounds of $\Theta(\Delta)$ and $\Theta(\log\Delta)$ on the competitive ratios of deterministic and randomized algorithms, respectively (where $\Delta$ denotes the maximum degree). We also consider the fully dynamic case which allows edge deletions as well as insertions. All our algorithms are applicable to the case where vertices are weighted and the cost of recoloring a vertex is its weight. All our lower bounds hold even in the unweighted case.