Dynamic Graph Coloring

Barba, Luis; Cardinal, Jean; Korman, Matias; Langerman, Stefan; Renssen, André van; Roeloffzen, Marcel; Verdonschot, Sander

doi:10.1007/s00453-018-0473-y

“…An important performance measure of a dynamic algorithm is its adjustment complexity (sometimes called recourse) that counts the number of vertices (or edges) that need to be inserted to or deleted from the maintained solution after each update (see, e.g. [3,13,17,24]). For many natural graph problems such as maintaining a maximal matching, constant worst-case adjustment complexity can be trivially achieved since one edge update cannot ever necessitate more than a constant number of changes in the maintained solution.…”

Section: Problem Statement and Our Resultsmentioning

confidence: 99%

Fully dynamic maximal independent set with sublinear update time

Assadi

¹

,

Onak²,

Schieber³

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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A maximal independent set (MIS) can be maintained in an evolving m-edge graph by simply recomputing it from scratch in O(m) time after each update. But can it be maintained in time sublinear in m in fully dynamic graphs?We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time O(min{∆, m 3/4 }), where ∆ is a fixed bound on the maximum degree in the graph and m is the (dynamically changing) number of edges.We further present a distributed implementation of our algorithm with O(min{∆, m 3/4 }) amortized message complexity, and O(1) amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a non-adaptive oblivious adversary.

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“…For dynamically changing graphs, a number of dynamic graph coloring algorithms have been proposed. A general lower bound limits their efficiency: for any dynamic algorithm A that maintains a c-coloring of a graph, there exists a dynamic forest such that A must recolor at least (|V| 2∕c(c−1) ) vertices per update on average, for any c ≥ 2 [2]. For c = 2 , this gives a lower bound of (|V|) .…”

Section: Related Work On (Dynamic) Graph Coloring In the Context Of Problem Specific Approachesmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Note that deleting edges from the graph can never worsen the current coloring, hence we focus on adding edges only. 2 The assumption that the graph is updated by adding edges is natural in many practical scenarios. For example, the web graph is explored gradually by a crawler that adds edges as they are discovered; the citation networks (in which nodes are research papers and edges indicate the citations between two papers) and collaboration networks (in which nodes are scientific researchers and edges correspond to collaborations) grow by adding edges.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Bossek

¹

,

Neumann

²

,

Pan

³

et al. 2021

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We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1) Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Runtime analysis of randomized search heuristics for dynamic graph coloring

Bossek

¹

,

Neumann

²

,

Pan

³

et al. 2019

Proceedings of the Genetic and Evolutionary Computation Conference

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We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1) Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient.

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Dynamic Graph Coloring

Cited by 16 publications

References 34 publications

Fully dynamic maximal independent set with sublinear update time

Fully dynamic maximal independent set with sublinear update time

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

Runtime analysis of randomized search heuristics for dynamic graph coloring

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