Sampling Lovasz Local Lemma [Moi'19,GLLZ'19,FGYZ'20,FHY'20,JPV'20,JPV'21] ·Given local lemma type conditions,,efficiently sample a uniform o∈∑ ·k-CNF formula ·Constructive::d≤2k Perfect uniform ·Approx.uniform:d≤20.175k [UPV'21] d≤20.175k ·Sampling hardness:d≥2k/2 [BGG+'19] Simple algorithm ·Hypergraph coloring Simple analysis Perfect uniform ·Constructive:△sqgk ·Approx.uniform:△sgk/3 △≤qk3 [UPV'21] Expected running 。General atomic CSPs time:0(n logn) ·Constructive:epA≤l Perfect uniform ·Approx.uniform:p0.142A≤1 [UPV'20] p0.175As1 Sampling Lovász Local Lemma [Moi’19, GLLZ’19, FGYZ’20, FHY’20, JPV’20, JPV’21] • Given local lemma type conditions, efficiently sample a uniform 𝜎 ∈ Σ • 𝑘-CNF formula • Constructive: 𝑑 ≲ 2 𝑘 • Approx. uniform: 𝑑 ≲ 2 0.175𝑘 [JPV’21] • Sampling hardness: 𝑑 ≳ 2 𝑘/2 [BGG+’19] • Hypergraph coloring • Constructive: Δ ≲ 𝑞 𝑘 • Approx. uniform: Δ ≲ 𝑞 𝑘/3 [JPV’21] • General atomic CSPs • Constructive: 𝑒𝑝Δ ≤ 1 • Approx. uniform: 𝑝 0.142Δ ≲ 1 [JPV’20] Perfect uniform 𝑑 ≲ 2 0.175𝑘 Perfect uniform Δ ≲ 𝑞 𝑘/3 Perfect uniform 𝑝 0.175Δ ≲ 1 Simple algorithm Simple analysis Expected running time: 𝑂(𝑛 log𝑛)