Finding Small Complete Subgraphs Efficiently
(2023) 34th International Workshop on Combinatorial Algorithms, IWOCA 2023 In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 13889 LNCS. p.185-196- Abstract
(I) We revisit the algorithmic problem of finding all triangles in a graph G= (V, E) with n vertices and m edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in O(mα) = O(m3 / 2) time, where α= α(G) is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the... (More)
(I) We revisit the algorithmic problem of finding all triangles in a graph G= (V, E) with n vertices and m edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in O(mα) = O(m3 / 2) time, where α= α(G) is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on m and α in the running time O(αℓ-2· m) of the algorithm of Chiba and Nishizeki for listing all copies of Kℓ, where ℓ≥ 3, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of Kℓ, for small ℓ≥ 4. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every ℓ≥ 7.
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- author
- Dumitrescu, Adrian and Lingas, Andrzej LU
- organization
- publishing date
- 2023
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- keywords
- graph arboricity, rectangular matrix multiplication, subgraph detection/counting, triangle
- host publication
- Combinatorial Algorithms - 34th International Workshop, IWOCA 2023, Proceedings
- series title
- Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
- editor
- Hsieh, Sun-Yuan ; Hung, Ling-Ju and Lee, Chia-Wei
- volume
- 13889 LNCS
- pages
- 12 pages
- publisher
- Springer Science and Business Media B.V.
- conference name
- 34th International Workshop on Combinatorial Algorithms, IWOCA 2023
- conference location
- Tainan, Taiwan
- conference dates
- 2023-06-07 - 2023-06-10
- external identifiers
-
- scopus:85163991645
- ISSN
- 1611-3349
- 0302-9743
- ISBN
- 9783031343469
- DOI
- 10.1007/978-3-031-34347-6_16
- language
- English
- LU publication?
- yes
- id
- 565c876a-63c8-4e92-ac46-06177cd8d6e0
- date added to LUP
- 2023-10-06 15:11:42
- date last changed
- 2024-07-12 09:36:38
@inproceedings{565c876a-63c8-4e92-ac46-06177cd8d6e0, abstract = {{<p>(I) We revisit the algorithmic problem of finding all triangles in a graph G= (V, E) with n vertices and m edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in O(mα) = O(m<sup>3 / 2</sup>) time, where α= α(G) is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on m and α in the running time O(α<sup>ℓ</sup><sup>-</sup><sup>2</sup>· m) of the algorithm of Chiba and Nishizeki for listing all copies of K<sub>ℓ</sub>, where ℓ≥ 3, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of K<sub>ℓ</sub>, for small ℓ≥ 4. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every ℓ≥ 7.</p>}}, author = {{Dumitrescu, Adrian and Lingas, Andrzej}}, booktitle = {{Combinatorial Algorithms - 34th International Workshop, IWOCA 2023, Proceedings}}, editor = {{Hsieh, Sun-Yuan and Hung, Ling-Ju and Lee, Chia-Wei}}, isbn = {{9783031343469}}, issn = {{1611-3349}}, keywords = {{graph arboricity; rectangular matrix multiplication; subgraph detection/counting; triangle}}, language = {{eng}}, pages = {{185--196}}, publisher = {{Springer Science and Business Media B.V.}}, series = {{Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)}}, title = {{Finding Small Complete Subgraphs Efficiently}}, url = {{http://dx.doi.org/10.1007/978-3-031-34347-6_16}}, doi = {{10.1007/978-3-031-34347-6_16}}, volume = {{13889 LNCS}}, year = {{2023}}, }