Optimality Clue for Graph Coloring Problem

In this paper, we present a new approach which qualifies or not a solution found by a heuristic as a potential optimal solution. Our approach is based on the following observation: for a minimization problem, the number of admissible solutions decreases with the value of the objective function. For the Graph Coloring Problem (GCP), we confirm this observation and present a new way to prove optimality. This proof is based on the counting of the number of different k-colorings and the number of independent sets of a given graph G. Exact solutions counting problems are difficult problems (\#P-complete). However, we show that, using only randomized heuristics, it is possible to define an estimation of the upper bound of the number of k-colorings. This estimate has been calibrated on a large benchmark of graph instances for which the exact number of optimal k-colorings is known. Our approach, called optimality clue, build a sample of k-colorings of a given graph by running many times one randomized heuristic on the same graph instance. We use the evolutionary algorithm HEAD [Moalic et Gondran, 2018], which is one of the most efficient heuristic for GCP. Optimality clue matches with the standard definition of optimality on a wide number of instances of DIMACS and RBCII benchmarks where the optimality is known. Then, we show the clue of optimality for another set of graph instances. Optimality Metaheuristics Near-optimal

Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget

In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, the updates consist of insertions only, and the final instance consists of n intervals, then we can achieve an amortized recourse budget of $\u1d4aa({k⁷ log n})$ while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bartłomiej Bosek et al., 2020] in terms of both k and n. We complement this result by showing the lower bound of $Ω(n)$ on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As an additional application of our techniques we include a new combinatorial result on coloring unit circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs $\u1d49c$. We show that if there is a set $\u1d49c'$ of non-intersecting unit arcs of size $L²-1$ such that $\u1d49c ∪ \u1d49c'$ does not contain L+1 arcs intersecting in one point, then it is possible to color $\u1d49c$ with L colors. This complements the work on circular arc coloring [Belkale and Chandran, 2009; Tucker, 1975; Valencia-Pabon, 2003], which specifies sufficient conditions needed to color $\u1d49c$ with L+1 colors or more

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum