A logarithmic approximation of linearly ordered colourings
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866912404405747712 |
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| author | Håstad, Johan Martinsson, Björn Nakajima, Tamio-Vesa Živný, Stanislav |
| author_facet | Håstad, Johan Martinsson, Björn Nakajima, Tamio-Vesa Živný, Stanislav |
| contents | A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with $O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$ colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with $O^*(\sqrt[5]{n})$ colours [FSTTCS'24]. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A logarithmic approximation of linearly ordered colourings Håstad, Johan Martinsson, Björn Nakajima, Tamio-Vesa Živný, Stanislav Combinatorics Discrete Mathematics Data Structures and Algorithms A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with $O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$ colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with $O^*(\sqrt[5]{n})$ colours [FSTTCS'24]. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement. |
| title | A logarithmic approximation of linearly ordered colourings |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2404.19556 |