TY - GEN
T1 - A Fast Coloring Oracle for Average Case Hypergraphs
AU - Marcussen, Cassandra
AU - Pyne, Edward
AU - Rubinfeld, Ronitt
AU - Shapira, Asaf
AU - Tauber, Shlomo
N1 - Publisher Copyright:
© Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber; licensed under Creative Commons License CC-BY 4.0.
PY - 2025/9/15
Y1 - 2025/9/15
N2 - 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).
AB - 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).
KW - average-case algorithms
KW - graph coloring
KW - local computation algorithms
UR - https://www.scopus.com/pages/publications/105019517203
U2 - 10.4230/LIPIcs.APPROX/RANDOM.2025.61
DO - 10.4230/LIPIcs.APPROX/RANDOM.2025.61
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AN - SCOPUS:105019517203
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2025
A2 - Ene, Alina
A2 - Chattopadhyay, Eshan
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 28th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2025 and the 29th International Conference on Randomization and Computation, RANDOM 2025
Y2 - 11 August 2025 through 13 August 2025
ER -