Theorem I If F is regular,then holant(G,{fvvF) can be computed in time poly(.2(treewidth(G)) SSM: Pr(a(e)=clA)-Pr(o(e)=clA,OB)<poly(V)exp(-t) compute Pr(g(e)=c|TA)±是 FPTAS in time poly(n) for Holant Theorem(Demaine-Hajiaghayi'04) For apex-minor-free graphs, treewidth of t-ball is O(). Theorem II If g is planar (apex-minor-free),F is regular,then SSM FPTAS for Holant(g,F)0WTIV\(G, F) Theorem II If G is planar (apex-minor-free), F is regular, then SSM FPTAS for SSM: G B e t A ￾ ￾8Z(￾(e) = c | ￾A) ￾ 8Z(￾(e) = c | ￾A, ￾B) ￾ ￾ ￾ XWTa(|V |) M`X(￾t) Theorem (Demaine-Hajiaghayi’04) For apex-minor-free graphs, treewidth of t-ball is O(t). compute in time 8Z(￾(e) = c | ￾A) ± 1 n XWTa (n) FPTAS for Holant If F is regular, then PWTIV\(G, {fv}v￾V ￾ F) can be computed in time Theorem I XWTa(|V |) · 2O(\ZMM_QL\P(G))