Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917619708198912 |
|---|---|
| author | Dalirrooyfard, Mina Mathialagan, Surya Williams, Virginia Vassilevska Xu, Yinzhan |
| author_facet | Dalirrooyfard, Mina Mathialagan, Surya Williams, Virginia Vassilevska Xu, Yinzhan |
| contents | We study finding and listing $k$-cliques in a graph, for constant $k\geq 3$, a fundamental problem of both theoretical and practical importance.
Our main contribution is a new output-sensitive algorithm for listing $k$-cliques in graphs, for arbitrary $k\geq 3$, coupled with lower bounds based on standard fine-grained assumptions, showing that our algorithm's running time is tight. Previously, the only known conditionally optimal output-sensitive algorithms were for the case of $3$-cliques by Björklund, Pagh, Vassilevska W. and Zwick [ICALP'14].
Typical inputs to subgraph isomorphism or listing problems are measured by the number of nodes $n$ or the number of edges $m$. Our framework is very general in that it gives $k$-clique listing algorithms whose running times are measured in terms of the number of $\ell$-cliques $Δ_\ell$ in the graph for any $1\leq \ell<k$. This generalizes the typical parameterization in terms of $n$ (the number of $1$-cliques) and $m$ (the number of $2$-cliques).
If the matrix multiplication exponent $ω$ is $2$, and if the size of the output, $Δ_k$, is sufficiently large, then for every $\ell<k$, the running time of our algorithm for listing $k$-cliques is $$\tilde{O}\left(Δ_\ell^{\frac{2}{\ell (k - \ell)}}Δ_k^{1-\frac{2}{k(k-\ell)}}\right).$$ For sufficiently large $Δ_k$, we prove that this runtime is in fact {\em optimal} for all $1 \leq \ell < k$ under the Exact $k$-Clique hypothesis.
In the special cases of $k = 4$ and $5$, our algorithm in terms of $n$ is conditionally optimal for all values of $Δ_k$ if $ω= 2$. Moreover, our framework is powerful enough to provide an improvement upon the 19-year old runtimes for $4$ and $5$-clique detection in $m$-edge graphs, as a function of $m$ [Eisenbrand and Grandoni, TCS'04]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15871 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques Dalirrooyfard, Mina Mathialagan, Surya Williams, Virginia Vassilevska Xu, Yinzhan Data Structures and Algorithms We study finding and listing $k$-cliques in a graph, for constant $k\geq 3$, a fundamental problem of both theoretical and practical importance. Our main contribution is a new output-sensitive algorithm for listing $k$-cliques in graphs, for arbitrary $k\geq 3$, coupled with lower bounds based on standard fine-grained assumptions, showing that our algorithm's running time is tight. Previously, the only known conditionally optimal output-sensitive algorithms were for the case of $3$-cliques by Björklund, Pagh, Vassilevska W. and Zwick [ICALP'14]. Typical inputs to subgraph isomorphism or listing problems are measured by the number of nodes $n$ or the number of edges $m$. Our framework is very general in that it gives $k$-clique listing algorithms whose running times are measured in terms of the number of $\ell$-cliques $Δ_\ell$ in the graph for any $1\leq \ell<k$. This generalizes the typical parameterization in terms of $n$ (the number of $1$-cliques) and $m$ (the number of $2$-cliques). If the matrix multiplication exponent $ω$ is $2$, and if the size of the output, $Δ_k$, is sufficiently large, then for every $\ell<k$, the running time of our algorithm for listing $k$-cliques is $$\tilde{O}\left(Δ_\ell^{\frac{2}{\ell (k - \ell)}}Δ_k^{1-\frac{2}{k(k-\ell)}}\right).$$ For sufficiently large $Δ_k$, we prove that this runtime is in fact {\em optimal} for all $1 \leq \ell < k$ under the Exact $k$-Clique hypothesis. In the special cases of $k = 4$ and $5$, our algorithm in terms of $n$ is conditionally optimal for all values of $Δ_k$ if $ω= 2$. Moreover, our framework is powerful enough to provide an improvement upon the 19-year old runtimes for $4$ and $5$-clique detection in $m$-edge graphs, as a function of $m$ [Eisenbrand and Grandoni, TCS'04]. |
| title | Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2307.15871 |