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

Abstract

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.

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“…To adopt this to our inapproximability proof, we follow the approach by Dinur and Guruswami [9] and Guruswami et al [11] who used it to prove lower bounds for some maximum constraint satisfaction problems and hypergraph colorability.…”

“…We note THEORY OF COMPUTING, Volume 13 (10), 2017, pp. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] that an inapproximability of homogeneous linear equations, albeit with a stronger guarantee on the structure of the system, forms the core of the inapproximability of Min-Bisection [14].…”

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“…Saket [20] showed that unless NP ⊆ DTIME(2 poly log n ), it is not possible to color a 2-colorable 4-uniform hypergraph with poly log n colors. We remark that recently, with the discovery of the short code [5], there has been a sequence of works [7,10,16,21,15] which have considerably improved the status of the approximate coloring question. In particular, we know that it is quasi-NP-hard to color a 2-colorable 8-uniform hypergraph with 2 (log n) c colors for some constant c ∈ (0, 1).…”

$2^{(\log N)^{1/10-o(1)}}$ Hardness for Hypergraph Coloring

“…Recently, in [17], Guruswami, Harsha, Håstad, Srinivasan and Varma proved the first super-polylogarithmic hardness result for hypergraph coloring, showing hardness for coloring 2-colorable 8-uniform hypergraphs with 2 2 Ω( √ log log n) colors. Their reduction uses the Low-Degree-Long-Code proposed in [7], based on techniques for testing Reed-Muller codes developed in [12].…”

“…There has been much focus on hardness of hypergraph coloring recently. In [17], Guruswami, Håstad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with 2 2 Ω( √ log log N ) colors. Their result is obtained by composing standard Label-Cover with an inner-verifier based on Low-Degree-Long-Code, using Reed-Muller code testing results by Dinur and Guruswami [12].…”