Rapid mixing of hypergraph independent sets

Abstract

We prove that the mixing time of the Glauber dynamics for sampling independent sets on n‐vertex k‐uniform hypergraphs is Ofalse(nnormallognfalse) when the maximum degree Δ satisfies Δ ≤ c2k/2, improving on the previous bound Bordewich and co‐workers of Δ ≤ k − 2. This result brings the algorithmic bound to within a constant factor of the hardness bound of Bezakova and co‐workers which showed that it is NP‐hard to approximately count independent sets on hypergraphs when Δ ≥ 5·2k/2.

Uniform sampling through the Lovasz local lemma

“…Simultaneous to our work, Hermon, Sly, and Zhang [24] showed that Markov chains for monotone k-CNF formulas are rapidly mixing, if d ≤ c2 k/2 for a constant c. In another parallel work, Moitra [29] gave a novel algorithm to sample solutions for general k-CNF when d 2 k/60 . We note that neither results are directly comparable to ours and the techniques are very different.…”

“…This indicates that our condition on r in Theorem 25 is necessary for Algorithm 6. In contrast, Hermon et al [24] show that on a linear hypergraph (including the hypertree), the Markov chain mixes rapidly for degrees higher than the general bound. It is unclear how to combine the advantages from these two approaches.…”

Uniform Sampling Through the Lovász Local Lemma

“…Simultaneous to our work, Hermon et al [24] showed that Markov chains for monotone k-CNF formulas are rapidly mixing, if d ≤ c2 k /2 for a constant c. In another parallel work, Moitra [29] gave a novel algorithm to sample solutions for general k-CNF when d 2 k /60 . We note that neither results are directly comparable to ours, and the techniques are very different.…”

“…Unfortunately, our algorithm can be exponentially slow when the intersection s is not large enough. In sharp contrast, as shown by Hermon et al [24], Markov chains mix rapidly for d ≤ c2 k /k 2 when s = 1.…”

“…Both of these two samplers are approximate, whereas ours is exact. Moreover, ours does not require monotonicity (unlike Hermon et al [24]) and allows larger d than Moitra [29] but at the cost of an extra intersection lower bound. Unfortunately, our algorithm can be exponentially slow when the intersection s is not large enough.…”