### Abstract

This work studies the hardness of finding independent sets in hypergraphs which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1 - ε)-almost 2-colorable if removing an ε fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph 2-colorable. In particular we prove the following results. • For an arbitrarily small constant γ > 0, there is a constant ξ > 0, such that, given a 4-uniform hypergraph on n vertices which is (1 - ε)-almost 2- colorable for ε = 2^{-(logn) ξ} it is quasi-NP-hard^{1} to find an independent set of n/ (2 ^{(logn)1-γ}) vertices. • For any constants ε δ > 0, given as input a 3- uniform hypergraph on n vertices which is (1 - ε)-aImost 2-colorable, it is NP-hard to find an independent set of δn vertices. • Assuming the d-to-1 Games Conjecture the following holds. For any constant δ > 0. given a 2- colorable 3-uniform hypergraph on n vertices, it is NP-hard to find an independent set of δn vertices. The hardness result on independent set in almost 2- colorable 3-uniform hypergraphs was earlier known only assuming the Unique Games Conjecture. In this work we prove the result unconditionally, combining Fourier analytic techniques with the Multi-Layered PCP of [11]. For independent sets in 2-colorable 3-uniform hy- pergaphs we prove the first strong hardness result, albeit assuming the d-to-1 Games Conjecture. Our reduction uses the d-to-1 Game as a starting point to construct a Multi-Layered PCP with the smoothness property. We use analytical techniques based on the Invariance Principle of Mossel [36], The smoothness property is crucially exploited in a manner similar to recent work of Hastad [20] and Wenner [45], Our result on almost 2-colorable 4-uniform hypergraphs gives the first nearly polynomial hardness factor for independent set in hypergraphs which are (almost) colorable with constantly many colors. It partially bridges the gap between the previous best lower bound of poly(log n) and the algorithmic upper bounds of n^{Ω(1)}. This also exhibits a bottleneck to improving the algorithmic techniques for hypergraph coloring.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 1607-1625 |

Number of pages | 19 |

ISBN (Print) | 9781611973389 |

State | Published - 2014 |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: Jan 5 2014 → Jan 7 2014 |

### Other

Other | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
---|---|

Country | United States |

City | Portland, OR |

Period | 1/5/14 → 1/7/14 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014*(pp. 1607-1625). Association for Computing Machinery.

**Hardness of finding independent sets in 2-colorable and almost 2-colorable hypergraphs.** / Khot, Subhash; Saket, Rishi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014.*Association for Computing Machinery, pp. 1607-1625, 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, OR, United States, 1/5/14.

}

TY - GEN

T1 - Hardness of finding independent sets in 2-colorable and almost 2-colorable hypergraphs

AU - Khot, Subhash

AU - Saket, Rishi

PY - 2014

Y1 - 2014

N2 - This work studies the hardness of finding independent sets in hypergraphs which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1 - ε)-almost 2-colorable if removing an ε fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph 2-colorable. In particular we prove the following results. • For an arbitrarily small constant γ > 0, there is a constant ξ > 0, such that, given a 4-uniform hypergraph on n vertices which is (1 - ε)-almost 2- colorable for ε = 2-(logn) ξ it is quasi-NP-hard1 to find an independent set of n/ (2 (logn)1-γ) vertices. • For any constants ε δ > 0, given as input a 3- uniform hypergraph on n vertices which is (1 - ε)-aImost 2-colorable, it is NP-hard to find an independent set of δn vertices. • Assuming the d-to-1 Games Conjecture the following holds. For any constant δ > 0. given a 2- colorable 3-uniform hypergraph on n vertices, it is NP-hard to find an independent set of δn vertices. The hardness result on independent set in almost 2- colorable 3-uniform hypergraphs was earlier known only assuming the Unique Games Conjecture. In this work we prove the result unconditionally, combining Fourier analytic techniques with the Multi-Layered PCP of [11]. For independent sets in 2-colorable 3-uniform hy- pergaphs we prove the first strong hardness result, albeit assuming the d-to-1 Games Conjecture. Our reduction uses the d-to-1 Game as a starting point to construct a Multi-Layered PCP with the smoothness property. We use analytical techniques based on the Invariance Principle of Mossel [36], The smoothness property is crucially exploited in a manner similar to recent work of Hastad [20] and Wenner [45], Our result on almost 2-colorable 4-uniform hypergraphs gives the first nearly polynomial hardness factor for independent set in hypergraphs which are (almost) colorable with constantly many colors. It partially bridges the gap between the previous best lower bound of poly(log n) and the algorithmic upper bounds of nΩ(1). This also exhibits a bottleneck to improving the algorithmic techniques for hypergraph coloring.

AB - This work studies the hardness of finding independent sets in hypergraphs which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1 - ε)-almost 2-colorable if removing an ε fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph 2-colorable. In particular we prove the following results. • For an arbitrarily small constant γ > 0, there is a constant ξ > 0, such that, given a 4-uniform hypergraph on n vertices which is (1 - ε)-almost 2- colorable for ε = 2-(logn) ξ it is quasi-NP-hard1 to find an independent set of n/ (2 (logn)1-γ) vertices. • For any constants ε δ > 0, given as input a 3- uniform hypergraph on n vertices which is (1 - ε)-aImost 2-colorable, it is NP-hard to find an independent set of δn vertices. • Assuming the d-to-1 Games Conjecture the following holds. For any constant δ > 0. given a 2- colorable 3-uniform hypergraph on n vertices, it is NP-hard to find an independent set of δn vertices. The hardness result on independent set in almost 2- colorable 3-uniform hypergraphs was earlier known only assuming the Unique Games Conjecture. In this work we prove the result unconditionally, combining Fourier analytic techniques with the Multi-Layered PCP of [11]. For independent sets in 2-colorable 3-uniform hy- pergaphs we prove the first strong hardness result, albeit assuming the d-to-1 Games Conjecture. Our reduction uses the d-to-1 Game as a starting point to construct a Multi-Layered PCP with the smoothness property. We use analytical techniques based on the Invariance Principle of Mossel [36], The smoothness property is crucially exploited in a manner similar to recent work of Hastad [20] and Wenner [45], Our result on almost 2-colorable 4-uniform hypergraphs gives the first nearly polynomial hardness factor for independent set in hypergraphs which are (almost) colorable with constantly many colors. It partially bridges the gap between the previous best lower bound of poly(log n) and the algorithmic upper bounds of nΩ(1). This also exhibits a bottleneck to improving the algorithmic techniques for hypergraph coloring.

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UR - http://www.scopus.com/inward/citedby.url?scp=84902095947&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84902095947

SN - 9781611973389

SP - 1607

EP - 1625

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery

ER -