Improved Dynamic Graph Coloring

Abstract

This article studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n 1-ε for any ε > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring or alternatively, study restricted families of graphs. Toward understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C -coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS ’17, Barba et al. devised two complementary algorithms: for any β > 0, the first (respectively, second) maintains an O(Cβn 1/β ) (respectively, O(Cβ) -coloring while recoloring O(β) (respectively, O(β n 1/β )) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for β = O(1): any algorithm that maintains a C -coloring of an n -vertex dynamic forest must recolor Ω (n 2 C(C-1)) vertices per update, for any constant C ≥ 2. Our contribution is twofold: • We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: for any β > 0, we get a Ô (Cβlog 2 n)-coloring with O(β) recolorings per update, where the Ô notation suppresses polyloglog(n) factors. In particular, for β = O(1), we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound but also it unveils a rather surprising phenomenon: the trade-off between the number of colors and recolorings is highly non-symmetric. • For uniformly sparse graphs, we use low out-degree orientations to strengthen the preceding result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest. From this data structure, we get a deterministic algorithm for graphs of arboricity ɑ that maintains an O(ɑ log 2 n)-coloring in amortized O(1) time.

Time Complexity Analysis of Randomized Search Heuristics for the Dynamic Graph Coloring Problem

“…For c = 2 , this gives a lower bound of (|V|) . By a result in [17] (which improves upon [39]), one can maintain an O(log n)-coloring of a planar graph with amortized polylogarithmic update time. There is a line of research on dynamically maintaining a ( + 1)-coloring of a graph with maximum degree at most [4,5,16] and the current best algorithm has O(1) update time [5,16].…”

“…Dynamic algorithms have been proposed to maintain proper coloring for graphs with maximum degree at most , 1 with the goal of using as few colors as possible while keeping the (amortized) update time small [3,4]. There exist algorithms that aim to perform as few (amortized) vertex recolorings as possible in order to maintain a proper coloring in a dynamic graph [2,39]. There have also been studies of k-list coloring in a dynamic graph such that each update corresponds to adding one vertex (together with the incident edges) to the graph (e.g.…”

Runtime analysis of randomized search heuristics for dynamic graph coloring

“…For c = 2, this gives a lower bound of Ω(|V |). By a result in [37] (which improves upon [23]), one can maintain an O(log n)-coloring of a planar graph with amortized polylogarithmic update time. There is a line of research on dynamically maintaining a (∆ + 1)-coloring of a graph with maximum degree at most ∆ [21,38,39] and the current best algorithm has O(1) update time [38,39].…”

“…Dynamic algorithms have been proposed to maintain proper coloring for graphs with maximum degree at most ∆, 1 with the goal of using as few colors as possible while keeping the (amortized) update time small [20,21]. There exist algorithms that aim to perform as few (amortized) vertex recolorings as possible in order to maintain a proper coloring in a dynamic graph [22,23]. There have also been studies of k-list coloring in a dynamic graph such that each update corresponds to adding one vertex (together with the incident edges) to the graph (e.g.…”

Explicit and Implicit Dynamic Coloring of Graphs with Bounded Arboricity

“…Later, Solomon and Wein [30] provided a dynamic O(α max log 2 n)-coloring algorithm with poly(log log n) update time. Note that the number of colors used by [30] depends on maximum arboricity α max over all graphs G i . Hence, we ask the following question.…”