| Autor: |
Björklund, Andreas, Curticapean, Radu, Husfeldt, Thore, Kaski, Petteri, Pratt, Kevin |
| Rok vydání: |
2024 |
| Předmět: |
|
| Druh dokumentu: |
Working Paper |
| Popis: |
In this paper we further explore the recently discovered connection by Bj\"{o}rklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an $n$-vertex graph can be computed deterministically in $O(1.99982^n)$ time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the $2^n\operatorname{poly}(n)$ time algorithm for chromatic number by Bj\"{o}rklund, Husfeldt, and Koivisto [SICOMP 2009]. Our technique is a combination of earlier algorithms for detecting $k$-colorings for small $k$ and enumerating $k$-colorable subgraphs, with an extension and derandomisation of Pratt's tensor-based algorithm for balanced three-way partitioning to the unbalanced case. |
| Databáze: |
arXiv |
| Externí odkaz: |
|
