Graph G with m vertices andp=(p1,,Pm)>0,let pr=Ilie Pi Alternating-sign independence polynomial: An alternative form of the polynomial: 9s=4()= ∑(-1)"p 9s=9s()= ∑(-1)sp IEInd,ICS IEInd,SEI Shearer region:S =pE(0,1)m:VS s [m],qi(p)>0} ={∈(0,1)m:∀l∈Ind,qu()>0} 个 Lemma 3 q,≥0廿I∈Ind →9s≥q0 VSE [m] 9s≥0HS∈[m] > q,≥pml.im1eInd q1=p'·9mr+() (Lemma 2) =pl.∑yer+0qY (Lemma 1) ≥pl·q0 (q1=p'·9mr+0>0） ≥p.4ml (Lemma 2) ≥p[m].dml (p'≥pm)
Lemma 3 (𝑞𝐼 = 𝑝 𝐼 ⋅ 𝑞෬ 𝑚 ∖Γ + 𝐼 > 0) = 𝑝 𝐼 ⋅ ∑𝑌⊆Γ+ 𝐼 𝑞𝑌 (Lemma 2) ≥ 𝑝 𝐼 ⋅ 𝑞෬ 𝑚 (Lemma 2) Alternating-sign independence polynomial: 𝑞෬𝑠 = 𝑞෬𝑠 𝑝 Ԧ = 𝐼∈Ind,𝐼⊆𝑆 −1 |𝐼| 𝑝 𝐼 An alternative form of the polynomial: 𝑞𝑆 = 𝑞𝑆 𝑝 Ԧ = 𝐼∈Ind,𝑆⊆𝐼 −1 |𝐼∖𝑆| 𝑝 𝐼 𝑞𝐼 ≥ 0 ∀ 𝐼 ∈ Ind Shearer region: 𝒮 = {𝑝 Ԧ ∈ 0,1 𝑚: ∀𝑆 ⊆ [𝑚], 𝑞෬𝐼 𝑝 Ԧ > 0} = {𝑝 Ԧ ∈ 0,1 𝑚: ∀𝐼 ∈ Ind, 𝑞𝐼 𝑝 Ԧ > 0} 𝑞 ෬ 𝑠 ≥ 𝑞∅ ∀ 𝑆 ∈ [m] 𝑞 ෬ 𝑠 ≥ 0 ∀ 𝑆 ∈ [m] 𝑞𝐼 ≥ 𝑝 𝑚 ⋅ 𝑞 ෬ 𝑚 ∀ 𝐼 ∈ Ind 𝑞𝐼 = 𝑝 𝐼 ⋅ 𝑞 ෬ 𝑚 ∖Γ + 𝐼 (Lemma 1) ≥ 𝑝 𝐼 ⋅ 𝑞∅ ≥ 𝑝 [𝑚] ⋅ 𝑞෬ 𝑚 (𝑝 𝐼 ≥ 𝑝 [𝑚] ) Graph G with m vertices and 𝑝 Ԧ = 𝑝1, ⋯ , 𝑝𝑚 > 0, let 𝑝𝐼 = ∏𝑖∈𝐼 𝑝𝑖
Graph G with m vertices andp=(p1,.,pm)>0,let pr Ilie Pi Alternating-sign independence polynomial: An alternative form of the polynomial: s=i,()=∑(-1)川p 9s=qs=∑(-1)八p IEInd,IES IEInd,SCI Shearer region:S={i∈(0,1)m:S∈[ml,i()>0} ={i∈(0,1)m:I E Ind,q()>0} Tight Lovasz Local Lemma (Shearer 1985): pES Each event sets 41,...,4m where p(Ai)=pi and the dependency graph is Gsatisfies Pr>0
Alternating-sign independence polynomial: 𝑞෬𝑠 = 𝑞෬𝑠 𝑝 Ԧ = 𝐼∈Ind,𝐼⊆𝑆 −1 |𝐼| 𝑝 𝐼 An alternative form of the polynomial: 𝑞𝑆 = 𝑞𝑆 𝑝 Ԧ = 𝐼∈Ind,𝑆⊆𝐼 −1 |𝐼∖𝑆| 𝑝 𝐼 Shearer region: 𝒮 = {𝑝 Ԧ ∈ 0,1 𝑚: ∀𝑆 ⊆ [𝑚], 𝑞෬𝐼 𝑝 Ԧ > 0} = {𝑝 Ԧ ∈ 0,1 𝑚: ∀𝐼 ∈ Ind, 𝑞𝐼 𝑝 Ԧ > 0} Tight Lovász Local Lemma (Shearer 1985): Each event sets A1,…,Am where 𝑝 𝐴𝑖 = 𝑝𝑖 and the dependency graph is G satisfies Pr ٿ=��1 𝑚 𝐴𝑖 > 0 𝑝 Ԧ ∈ 𝒮 Graph G with m vertices and 𝑝 Ԧ = 𝑝1, ⋯ , 𝑝𝑚 > 0, let 𝑝𝐼 = ∏𝑖∈𝐼 𝑝𝑖
Graph G with m vertices and节=(p1,…,pm)>0,letp1=Πie Pi Alternating-sign independence polynomial: An alternative form of the polynomial: 。=i,(例=∑(-10川p 9s=qs()=∑(-1)八sp IEInd,ICS IEInd,SEI Shearer region:S =pE(0,1)m:VS [m],qi(p)>0} ={i∈(0,1)m:1∈Ind,q(p)>0} Tight Lovasz Local Lemma (Shearer 1985): pES Each event sets 41,...,Am where p(Ai)=pi and the dependency graph is Gsatisfies Pr>
Alternating-sign independence polynomial: 𝑞෬𝑠 = 𝑞෬𝑠 𝑝 Ԧ = 𝐼∈Ind,𝐼⊆𝑆 −1 |𝐼| 𝑝 𝐼 An alternative form of the polynomial: 𝑞𝑆 = 𝑞𝑆 𝑝 Ԧ = 𝐼∈Ind,𝑆⊆𝐼 −1 |𝐼∖𝑆| 𝑝 𝐼 Shearer region: 𝒮 = {𝑝 Ԧ ∈ 0,1 𝑚: ∀𝑆 ⊆ [𝑚], 𝑞෬𝐼 𝑝 > 0} = {𝑝 Ԧ ∈ 0,1 𝑚: ∀𝐼 ∈ Ind, 𝑞𝐼 𝑝 > 0} Tight Lovász Local Lemma (Shearer 1985): Each event sets A1,…,Am where 𝑝 𝐴𝑖 = 𝑝𝑖 and the dependency graph is G satisfies Pr ٿ=��1 𝑚 𝐴𝑖 > 0 𝑝 Ԧ ∈ 𝒮 Graph G with m vertices and 𝑝 Ԧ = 𝑝1, ⋯ , 𝑝𝑚 > 0, let 𝑝𝐼 = ∏𝑖∈𝐼 𝑝𝑖