Modifying Colourings Between Time-Steps to Tackle Changes in Dynamic Random Graphs

Hardy, Bradley; Lewis, Rhyd; Thompson, Jonathan M.

doi:10.1007/978-3-319-30698-8_13

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…Moving away from the edge dynamic GCP, some of our previous work has introduced and explored the effects of modification operators for the vertex dynamic GCP without future change information (see Hardy et al 2016). We plan to extend this work, as we have done here, to explore the situation where information regarding the likelihood of future changes is provided.…”

Section: Discussionmentioning

confidence: 99%

Tackling the edge dynamic graph colouring problem with and without future adjacency information

2017

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Many real world operational research problems, such as frequency assignment and exam timetabling, can be reformulated as graph colouring problems (GCPs). Most algorithms for the GCP operate under the assumption that its constraints are fixed, allowing us to model the problem using a static graph. However, in many real-world cases this does not hold and it is more appropriate to model problems with constraints that change over time using an edge dynamic graph. Although exploring methods for colouring dynamic graphs has been identified as an area of interest with many real-world applications, to date, very little literature exists regarding such methods. In this paper we present several heuristic methods for modifying a feasible colouring at time-step t into an initial, but not necessarily feasible, colouring for a "similar" graph at time-step t + 1. We will discuss two cases; (1) where changes occur at random, and (2) where probabilistic information about future changes is provided. Experimental results are also presented and the benefits of applying these particular modification methods are investigated.

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Section: Discussionmentioning

confidence: 99%

Tackling the edge dynamic graph colouring problem with and without future adjacency information

2017

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“…In light of the computational intractability of graph coloring, previous work on dynamic graph coloring is devoted mostly to heuristics and experimental results [43,51,59,29,28,48,53]. From the theoretical standpoint, it is natural to consider the combinatorial aspects of dynamic coloring or to study restricted families of graphs; to the best of our knowledge, the only work on this pioneering front is that of Barba et al from WADS'17 [5] and Bhattacharya et al from SODA'18 [12].…”

Section: Dynamic Graph Coloringmentioning

confidence: 99%

Improved Dynamic Graph Coloring

Solomon

Wein

2020

ACM Trans. Algorithms

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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.

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“…In light of the computational intractability of graph coloring, previous work on dynamic graph coloring is devoted mostly to heuristics and experimental results [65,74,82,46,45,71,76]. From the theoretical standpoint, it is natural to consider the combinatorial aspects of dynamic coloring or to study restricted families of graphs; to the best of our knowledge, the only work on this pioneering front is that of Barba et al from WADS'17 [9] and Bhattacharya et al from SODA'18 [19].…”

Section: Dynamic Graph Coloringmentioning

confidence: 99%

Improved Dynamic Graph Coloring

Solomon¹,

Wein²

2019

Preprint

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This paper 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− for any > 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.Towards understanding the combinatorial aspects of this problem, one may assume a blackbox 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 ) (resp., O(Cβ))-coloring while recoloring O(β) (resp., O(βn 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 two-fold:• 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 Õ( C β log 2 n)-coloring with O(β) recolorings per update, where the Õ notation supresses 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 it also 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 above 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.

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Modifying Colourings Between Time-Steps to Tackle Changes in Dynamic Random Graphs

Cited by 3 publications

References 16 publications

Tackling the edge dynamic graph colouring problem with and without future adjacency information

Tackling the edge dynamic graph colouring problem with and without future adjacency information

Improved Dynamic Graph Coloring

Improved Dynamic Graph Coloring

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