Linearly ordered colourings of hypergraphs

Conference item

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(√{nlog log n}/log n), where n is the number of vertices of the input hypergraph. This is established by building on ideas from algorithms designed for approximate graph colourings. Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO 3-colouring for every constant uniformity r ≥ 5. In fact, we determine the precise relationship of polymorphism minions for all uniformities r ≥ 3, which reveals a key difference between r = 3,4 and r ≥ 5 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+1)-colouring for LO k-colourable r-uniform hypergraphs for k ≥ 2 and r ≥ 5.

Authors: Nakajima, T-V; Zivny, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Leibniz International Proceedings in Informatics, LIPIcs
• Journal: Proceedings of the 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
• Volume: 229
• Article number: 128
• Publication date: 2022-06-28
• Acceptance date: 2022-04-10
• Event title: International Colloquium on Automata, Languages and Programming (ICALP22)
• Event location: Paris, France
• Event start date: 2022-07-04
• Event end date: 2022-07-08
• DOI: 10.4230/LIPIcs.ICALP.2022.128
• ISSN: 1868-8969
• Language: English
• Keywords: FFR