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A Fast Coloring Oracle for Average Case Hypergraphs

Authors Cassandra Marcussen, Edward Pyne , Ronitt Rubinfeld, Asaf Shapira, Shlomo Tauber

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  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • average-case algorithms
  • local computation algorithms
  • graph coloring

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Abstract

Hypergraph 2-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen 2-colorable 3-uniform hypergraph. Lee, Molla, and Nagle recently extended this to k-uniform hypergraphs for all k ≥ 3. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants.
Our first result in this paper is a new simple and elementary deterministic 2-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only O(n).
Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a 2-coloring of an average 2-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex v, assigns color red/blue to v while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal 2-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time O(1).

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Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber. A Fast Coloring Oracle for Average Case Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.61

Author Details

Cassandra Marcussen
  • Harvard University, Cambridge, MA, USA
Edward Pyne
  • MIT, Cambridge, MA, USA
Ronitt Rubinfeld
  • MIT, Cambridge, MA, USA
Asaf Shapira
  • Tel Aviv University, Israel
Shlomo Tauber
  • Tel Aviv University, Israel

Funding

  • Marcussen, Cassandra: Supported in part by an NDSEG fellowship, and by NSF Award 2152413 and a Simons Investigator Award to Madhu Sudan.
  • Pyne, Edward: Supported by a Jane Street Graduate Research Fellowship and NSF awards CCF-2310818 and CCF-2127597.
  • Rubinfeld, Ronitt: Supported by NSF awards DMS-2022448 and CCF-2310818.
  • Shapira, Asaf: Supported in part by ERC Consolidator Grant 863438.
  • Tauber, Shlomo: Supported in part by ERC Consolidator Grant 863438.

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