Affiliation:
1. Department of Computer Science, University of Oxford, Oxford, UK
Abstract
A linearly ordered (LO)
k
-colouring of an
r
-uniform hypergraph assigns an integer from {1, ... ,
k
} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for
r
= 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS’21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results.
First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO
k
-colouring with
\( k=O(\sqrt [3]{n \log \log n / \log n} \)
.
Second, given an
r
-uniform hypergraph that admits an LO 2-colouring, we establish
NP
-hardness of finding an LO
k
-colouring for every constant uniformity
r
≥
k
+2. In fact, we determine relationships between polymorphism minions for all uniformities
r
≥ 3, which reveals a key difference between
r
<
k
+2 and
r
≥
k
+2 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing
NP
-hardness of finding an LO
k
-colouring for LO ℓ-colourable
r
-uniform hypergraphs for 2 ≤ ℓ ≤
k
and
r
≥
k
- ℓ + 4.
Funder
European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
UKRI
Clarendon Fund Scholarship
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Reference41 articles.
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2. Per Austrin, Amey Bhangale, and Aditya Potukuchi. 2020. Improved inapproximability of rainbow coloring. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). 1479–1495. DOI:10.1137/1.9781611975994.90
3. $(2+\varepsilon)$-Sat Is NP-hard
4. Libor Barto, Diego Battistelli, and Kevin M. Berg. 2021. Symmetric promise constraint satisfaction problems: Beyond the boolean case. In Proceedings of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS’21), LIPIcs Vol. 187. 10:1–10:16. DOI:10.4230/LIPIcs.STACS.2021.10
5. Algebraic Approach to Promise Constraint Satisfaction
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