A Strongly Subcubic Combinatorial Algorithm for Triangle Detection with Applications
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909131058708480 |
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| author | Dumitrescu, Adrian |
| author_facet | Dumitrescu, Adrian |
| contents | We revisit the algorithmic problem of finding a triangle in a graph: We give a randomized combinatorial algorithm for triangle detection in a given $n$-vertex graph with $m$ edges running in $O(n^{7/3})$ time, or alternatively in $O(m^{4/3})$ time. This may come as a surprise since it invalidates several conjectures in the literature. In particular,
- the $O(n^{7/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection based on matrix multiplication running in $O(n^ω) = O(n^{2.372})$ time, due to Itai and Rodeh (1978).
- the $O(m^{4/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection in sparse graphs based on matrix multiplication running in $O(m^{2ω/(ω+1)})= O(m^{1.407})$ time due to Alon, Yuster, and Zwick (1997).
- the $O(n^{7/3})$ time algorithm for triangle detection leads to a $O(n^{25/9} \log{n})$ time combinatorial algorithm for $n \times n$ Boolean matrix multiplication, by a reduction of V. V. Williams and R.~R.~Williams (2018).This invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014).
- the $O(m^{4/3})$ runtime invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014) that any combinatorial algorithm for triangle detection requires $m^{3/2 - o(1)}$ time.
- as a direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the $k$-clique problem, surpassing an almost $40$ years old algorithm of Ne{š}et{ř}il and Poljak (1985). This result strongly disproves the combinatorial $k$-clique conjecture.
- as another direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the \textsc{Max-Cut} problem, surpassing an almost $20$ years old algorithm of R.~R.~Williams (2005). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Strongly Subcubic Combinatorial Algorithm for Triangle Detection with Applications Dumitrescu, Adrian Data Structures and Algorithms Combinatorics We revisit the algorithmic problem of finding a triangle in a graph: We give a randomized combinatorial algorithm for triangle detection in a given $n$-vertex graph with $m$ edges running in $O(n^{7/3})$ time, or alternatively in $O(m^{4/3})$ time. This may come as a surprise since it invalidates several conjectures in the literature. In particular, - the $O(n^{7/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection based on matrix multiplication running in $O(n^ω) = O(n^{2.372})$ time, due to Itai and Rodeh (1978). - the $O(m^{4/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection in sparse graphs based on matrix multiplication running in $O(m^{2ω/(ω+1)})= O(m^{1.407})$ time due to Alon, Yuster, and Zwick (1997). - the $O(n^{7/3})$ time algorithm for triangle detection leads to a $O(n^{25/9} \log{n})$ time combinatorial algorithm for $n \times n$ Boolean matrix multiplication, by a reduction of V. V. Williams and R.~R.~Williams (2018).This invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014). - the $O(m^{4/3})$ runtime invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014) that any combinatorial algorithm for triangle detection requires $m^{3/2 - o(1)}$ time. - as a direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the $k$-clique problem, surpassing an almost $40$ years old algorithm of Ne{š}et{ř}il and Poljak (1985). This result strongly disproves the combinatorial $k$-clique conjecture. - as another direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the \textsc{Max-Cut} problem, surpassing an almost $20$ years old algorithm of R.~R.~Williams (2005). |
| title | A Strongly Subcubic Combinatorial Algorithm for Triangle Detection with Applications |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2403.01085 |