The Hardness of 3-Uniform Hypergraph Coloring

Dinur, Irit; Regev, Oded; Smyth, Clifford

doi:10.1007/s00493-005-0032-4

Cited by 87 publications

(118 citation statements)

References 22 publications

(37 reference statements)

Supporting

Mentioning

115

Contrasting

Order By: Relevance

“…For proving hardness results for learning the intersection of two halfspaces, we make critical use of recent hardness results due to Dinur et al [11] on the hardness of coloring 2-colorable, 3-uniform hypergraphs. We give a reduction from -coloring k-colorable, 3-uniform hypergraphs to properly learning intersections of k halfspaces by halfspaces.…”

Section: Amplifying Hardness Results For Approximate Graph Coloringmentioning

confidence: 99%

See 1 more Smart Citation

The complexity of properly learning simple concept classes

Alekhnovich¹,

Braverman

Feldman

et al. 2008

Journal of Computer and System Sciences

View full text Add to dashboard Cite

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes:• We show that unless NP = RP, there is no polynomial-time PAC learning algorithm for DNF formulas where the hypothesis is an OR-of-thresholds. Note that as special cases, we show that neither DNF nor OR-of-thresholds are properly learnable unless NP = RP. Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NP-hard to learn the intersection of 2 halfspaces by the intersection of k halfspaces for any constant k 0. Previous work held for the case when k = .• Assuming that NP DTIME(2 n ) for a certain constant < 1 we show that it is not possible to learn size s decision trees by size s k decision trees for any k 0. Previous hardness results for learning decision trees held for k 2. • We present the first non-trivial upper bounds on properly learning DNF formulas. More specifically, we show how to learn size s DNF by DNF in time 2Õ ( √ n log s) .The hardness results for DNF formulas and intersections of halfspaces are obtained via specialized graph products for amplifying the hardness of approximating the chromatic number as well as applying recent work on the hardness of approximate hypergraph coloring. The hardness results for decision trees, as well as the new upper bounds, are obtained by developing a connection between automatizability in proof complexity and learnability, which may have other applications.

show abstract

Section: Amplifying Hardness Results For Approximate Graph Coloringmentioning

confidence: 99%

“…Recently, several researchers [11,14,20] have shown that it is hard to color uniform hypergraphs, i.e., hypergraphs where each hyperedge is of equal size.…”

Section: Hardness Results For Smaller Concept Classesmentioning

confidence: 99%

The complexity of properly learning simple concept classes

Alekhnovich¹,

Braverman

Feldman

et al. 2008

Journal of Computer and System Sciences

View full text Add to dashboard Cite

show abstract

“…Prior to this result, the best inapproximability result for O(1)-colorable 3-uniform hypergraphs were as follows: Khot [11] showed that it is quasi-NP-hard to color a 3-colorable 3-uniform hypergraphs with (log log N ) 1/9 colors and Dinur, Regev and Smyth [7] showed that it is quasi-NP-hard to color a 2-colorable 3-uniform hypergraphs with (log log N ) 1/3 colors (observe that 2 O(log log N/ log log log N ) is exponentially larger than (log log N ) Ω(1) ). For 2-colorable 3-uniform hypergraphs, the result of Dinur et.…”

Section: Theorem 13 (3-colorable 3-uniform Hypergraphs) Assuming Np /mentioning

confidence: 99%

“…al. [7] only rules out colorability by (log log N ) Ω(1) , while a recent result due to Khot and Saket [13] shows that it is hard to find a δN -sized independent set in a given N -vertex 2-colorable 3-uniform hypergraph assuming the d-to-1 games conjecture. Our improved inapproximability result is obtained by adapting Khot's proof to the low-degree long code using the new noise function over F 3 .…”

Section: Theorem 13 (3-colorable 3-uniform Hypergraphs) Assuming Np /mentioning

confidence: 99%

See 1 more Smart Citation

Super-Polylogarithmic Hypergraph Coloring Hardness via Low-Degree Long Codes

Guruswami¹,

Håstad²,

Harsha³

et al. 2017

SIAM J. Comput.

View full text Add to dashboard Cite

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the "short code" of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.In particular, we prove quasi-NP-hardness of the following problems on n-vertex hypergraphs:• Coloring a 2-colorable 8-uniform hypergraph with 2 2 Ω( √ log log n) colors.• Coloring a 4-colorable 4-uniform hypergraph with 22 Ω( √ log log n) colors.• Coloring a 3-colorable 3-uniform hypergraph with (log n) Ω(1/ log log log n) colors.In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, (log n)colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the lowdegree long code was a multipartite structural restriction in the PCP construction of DinurGuruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a "query doubling" method exploiting additional properties of the 8-query test. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form. The latter step involves extending the key algebraic ingredient of Dinur-Guruswami concerning testing binary Reed-Muller codes to the ternary alphabet.

show abstract

Inapproximability of Combinatorial Optimization Problems

Trevisan

2013

Paradigms of Combinatorial Optimization

View full text Add to dashboard Cite

show abstract

The Hardness of 3-Uniform Hypergraph Coloring

Cited by 87 publications

References 22 publications

The complexity of properly learning simple concept classes

The complexity of properly learning simple concept classes

Super-Polylogarithmic Hypergraph Coloring Hardness via Low-Degree Long Codes

Inapproximability of Combinatorial Optimization Problems

Contact Info

Product

Resources

About