Dynamic Algorithms for Graph Coloring

Bhattacharya, Sayan; Chakrabarty, Deeparnab; Henzinger, Monika; Nanongkai, Danupon

doi:10.1137/1.9781611975031.1

Cited by 44 publications

(66 citation statements)

References 26 publications

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“…In such a dynamic setting, recomputing the solution from scratch after every update can be prohibitively time consuming, and it is natural to seek dynamic algorithms that provide faster updates. In the last few decades, efficient dynamic algorithms have been discovered for many combinatorial optimization problems, particularly in graphs such as shortest paths [Fre85,DI04,ACK17a,IKLS17], connectivity [HK99, HDLT01, WN17, ACK17b], maximal independent set and coloring [BCHN18,AOSS18,OSSW18]. For many of these problems, maintaining exact solutions is prohibitively expensive under various complexity conjectures [AW14, KPP16,AWY18,HKNS15a], and thus the best approximation bounds are sought.…”

Section: Introductionmentioning

confidence: 99%

Dynamic set cover: improved algorithms and lower bounds

Abboud¹,

Addanki

Grandoni

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We give new upper and lower bounds for the dynamic set cover problem. First, we give a (1 + ε)f -approximation for fully dynamic set cover in O(f 2 log n/ε 5 ) (amortized) update time, for any ǫ > 0, where f is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to O(f 2 /ε 5 ), while still obtaining an (1 + ε)f -approximation. These are the first algorithms that obtain an approximation factor linear in f for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of O(f 2 ) in the dynamic setting.To complement our upper bounds, we also show that a linear dependence of the update time on f is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of O(f 1−ε ) must have an approximation factor that is Ω(n δ ) for some constant δ > 0 under the Strong Exponential Time Hypothesis.

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Section: Introductionmentioning

confidence: 99%

Dynamic set cover: improved algorithms and lower bounds

Abboud¹,

Addanki

Grandoni

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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show abstract

“…This is evident by the polynomial gap between the update time of best known deterministic algorithms compared to randomized ones for many dynamic problems. For example, a maximal matching can be maintained in a fully dynamic graph with O(1) update time via a randomized algorithm [43], assuming a nonadaptive oblivious adversary, while the best known deterministic algorithm for this problem requires Θ( √ m) update time [37] (see [13] for a similar situation for (∆ + 1)-coloring of vertices of a graph).…”

Section: Problem Statement and Our Resultsmentioning

confidence: 99%

“…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

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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“…In the context of problem specific approaches, algorithms have been designed to update solutions after a dynamic change has happened. 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].…”

Section: Introductionmentioning

confidence: 99%

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

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 for Graph Coloring

Cited by 44 publications

References 26 publications

Dynamic set cover: improved algorithms and lower bounds

Dynamic set cover: improved algorithms and lower bounds

Fully dynamic maximal independent set with sublinear update time

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

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