On comparing algorithms for the maximum clique problem

Züge, Alexandre Prusch; Carmo, Renato

doi:10.1016/j.dam.2018.01.005

Cited by 4 publications

(5 citation statements)

References 21 publications

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“…Let 𝑝 be a constant, 𝐺 ∼ G(𝑛, 𝑝) and 𝐺 * be obtained from 𝐺 using the construction of Lemma 2 with 𝑁 = |𝑉 (𝐺 * )|. Based on a much stronger result from [Pittel, 1982], [Züge and Carmo, 2018] point out that the number of cliques in 𝐺 is 𝑛 Θ(log 𝑛) with high probability. In contrast, Theorem 3 tells us that the expected number of cliques in 𝐺 * is 𝑁 Θ( 𝑁) .…”

Section: Instances With Exponential Number Of Cliquesmentioning

confidence: 99%

“…The experimental results reported in all these cases seem surprising in the light of the theoretical results mentioned above. This apparent contradiction was investigated in [Züge and Carmo, 2018], which shows that branch and bound algorithms for MC have quasi-polynomial average running time (under G(𝑛, 𝑝)for fixed 𝑝.) This leaves open the question of whether these (and other) algorithms do have subexponential running time in the worst case or not.…”

Section: Introductionmentioning

confidence: 99%

“…We say that an execution of a standard branch and bound algorithm for MC on 𝐺 is contained in a clique search tree 𝑇 of 𝐺 if it considers a set of clique subinstances which induces a connected subgraph of 𝑇 containing the root and every choice of pivot in the algorithm is the same as in 𝑇. Using this notation, the following can be obtained from [Züge and Carmo, 2018].…”

Section: Introductionmentioning

confidence: 99%

“…The part of the analysis in [Züge and Carmo, 2018] which concerns us stems from the fact that almost every graph (under G(𝑛, 𝑝)for fixed 𝑝) has a quasi-polynomial number of cliques. Consequently, any standard branch and bound algorithm for MC will have a quasi-polynomial running time for almost any instance (as long as it takes no more than quasi-polynomial time for each subinstance.)…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Instances for the Maximum Clique Problem with Hardness Guarantees

Campos

Carmo²,

Nogueira

2022

Anais Do VII Encontro De Teoria Da Computação (ETC 2022)

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Branch and bound algorithms which use (an estimate on) the chromatic number as the bounding function are reported among the best known exact algorithms for the maximum clique problem. Their average running time is known to be quasi-polynomial and describing families of instances which induce exponential running time for these algorithms is a non-trivial issue. We describe two such families of graphs. We prove that graphs in the first family induce asymptotically higher running time than the average for algorithms whose running time is Ω(c(G)) where c(G) is the number of cliques in G. We also prove that graphs in the second family induce exponential running time for branch and bound algorithms which use the chromatic number as the bounding function.

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Section: Instances With Exponential Number Of Cliquesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Instances for the Maximum Clique Problem with Hardness Guarantees

Campos

Carmo²,

Nogueira

2022

Anais Do VII Encontro De Teoria Da Computação (ETC 2022)

Self Cite

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show abstract

“…Linear programming and mixed integer programming approaches are applicable and such models are presented in [3,[64][65][66]. There are also approaches based on deterministic methods such as branch and bound [67][68][69][70]. The exact algorithms, expressed as codes in programming languages, are described in [44,[71][72][73].…”

Section: Introductionmentioning

confidence: 99%

The Maximum Clique Problem and Integer Programming Models, Their Modifications, Complexity and Implementation

Seda

2023

Symmetry

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The maximum clique problem is a problem that takes many forms in optimization and related graph theory problems, and also has many applications. Because of its NP-completeness (nondeterministic polynomial time), the question arises of its solvability for larger instances. Instead of the traditional approaches based on the use of approximate or stochastic heuristic methods, we focus here on the use of integer programming models in the GAMS (General Algebraic Modelling System) environment, which is based on exact methods and sophisticated deterministic heuristics incorporated in it. We propose modifications of integer models, derive their time complexities and show their direct use in GAMS. GAMS makes it possible to find optimal solutions to the maximum clique problem for instances with hundreds of vertices and thousands of edges within minutes at most. For extremely large instances, good approximations of the optimum are given in a reasonable amount of time. A great advantage of this approach over all the mentioned algorithms is that even if GAMS does not find the best known solution within the chosen time limit, it displays its value at the end of the calculation as a reachable bound.

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A maximum clique based approximation algorithm for wireless link scheduling under SINR model

Kadivar

Mohammadi

2022

Journal of Computer and System Sciences

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On comparing algorithms for the maximum clique problem

Cited by 4 publications

References 21 publications

Instances for the Maximum Clique Problem with Hardness Guarantees

Instances for the Maximum Clique Problem with Hardness Guarantees

The Maximum Clique Problem and Integer Programming Models, Their Modifications, Complexity and Implementation

A maximum clique based approximation algorithm for wireless link scheduling under SINR model

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