Counting with Bounded Treewidth

Transcription

1 Counting with Bounded Treewidth Yitong Yin Nanjing University Joint work with Chihao Zhang (SJTU?)

2 Spin Systems graph G=(V,E) A (u,v) vertices are variables (domain [q]) edge are constraints: e=(u,v) E configuration partition function: 2 [q] V X Y Z = A e ( u, v) 2[q] V e=(u,v)2e A e :[q] [q]! C

3 Holant Problem graph G=(V,E) f v edges are variables (domain [q]) vertices are constraints (signatures) 8v 2 V, f v :[q] deg(v)! C configuration [q] E holant = X 2[q] E Y v2v f v E(v) E(v) = (e1,..., ed) incident edges of v q m configurations where m= E

4 graph G=(V,E) 1 a b 2 c 3 d f e [q] V Y w( )= A e ( u, v) e=(u,v)2e incidence graph B(V,E,F) EQ V where F EQ(x 1,...,x d )= a b c d e f E A e ( 1 x 1 = = x d 0 otherwise Z = X 2[q] V w( ) holant = X 2[q] F Y v2v [E f v F (v)

5 graph G=(V,E) q=2 Boolean symmetric f : {0, 1} d! C f = [f0, f1,..., fd] fv where fi = f(x) for Σjxj = i counting matchings: at every vertex v V, fv = [1, 1, 0, 0..., 0] the at-most-one function holant = X Y [1, 1, 0,...,0] E(v) 2{0,1} E v2v

6 graph G=(V,E) q=2 Boolean symmetric f : {0, 1} d! C f = [f0, f1,..., fd] fv where fi = f(x) for Σjxj = i counting perfect matchings: holant = at every vertex v V, fv = [0, 1, 0, 0..., 0] the exact-one function X Y [0, 1, 0, 0,...,0] E(v) 2{0,1} E v2v

7 graph G=(V,E) q=2 Boolean symmetric f : {0, 1} d! C f = [f0, f1,..., fd] fv where fi = f(x) for Σjxj = i counting edge covers: at every vertex v V, fv = [0, 1, 1, 1..., 1] the at-least-one function holant = X Y [0, 1, 1, 1,...,1] E(v) 2{0,1} E v2v

8 graph G=(V,E) q=2 Boolean symmetric f : {0, 1} d! C f = [f0, f1,..., fd] fv where fi = f(x) for Σjxj = i fv = [1, μ, 1, μ, 1, μ,...] subgraphs world ( Jerrum-Sinclair 90 ): Z = X X E µ odd(x) holant = X odd(x) = #odd-degree vertices in X Y 2{0,1} E v2v [1,µ,1,µ,...] E(v)

9 graph G=(V,E) incidence graph B(V,E,F) n [0,...,0,1,0,...,0] dv/2 n dv/2 [0, 1, 0] V F E counting Eulerian orientations: holant = X 2{0,1} F Y v2v Y [0,...,0, 1, 0,...,0] {z } {z } F (v) deg(v)/2 deg(v)/2 e=(e 1,e 2 )2E [ (e 1 ) 6= (e 2 )]

10 Treewidth tw(g): treewidth of graph G measures how much a graph is like a tree: tw(tree) = 1 tw(k k grid) = k tw(k n) = n-1 q-state spin system on G with n vertices: Courcelle s Theorem: f(q, tw)poly(n) time [Courcelle 90] Junction-tree belief propagation: qo(tw) poly(n) time Holant (tensor networks) on G with O(1) max-degree: qo(tw) poly(n) [Markov, Shi 08] [Arad, Landau 10]

11 Treewidth graph G=(V,E) a tree of bags of vertices: 1. Every vertex is in some bag. 2. Every edge is in some bag. 3. If two bags have a same vertex, then all bags in the path between them have that vertex. width: max bag size-1 tree-decomposition treewidth: width of optimal tree decomposition

12 Separator Decomposition V j V = V V i V j V k S i V k V i given a graph G=(V,E) a binary tree T G of n nodes: 1. Every node is a vertex set Vi V. The root is V. Every leaf is. 2. Every node Vi has a separator Si in G[Vi] separating Vi into Vj and Vk. 3. Vj and Vk are children of Vi. width of T G = max i { S i, V i } V i : vertex boundary of V i in G sw(g): width of optimal T G 1. sw(g) = θ(tw(g)) 2. TG can be constructed in 2 O(tw) poly(n) time

13 FPT Algorithm for Spin System V j V i V j V k S i V k V i max { S i, V i } = O(tw(G) ) for q-state spin systems Z = X Y A e ( u, v) 2[q] V e=(u,v)2e dynamic programming: 2 i for : Z(V i, ) = X feasible 2[q] S i table size: q O(tw(G)) n running time: q O(tw(G)) n Z(V j, )Z(V k, )Z(S i i, [ ) also works for bounded-degree Holant

14 Fine-grained Classification classify the computational complexity of Holant(G, F) in terms of graph family G and function family F Holant(G, F) Classify : G = all graphs: 2O(n) time G = graphs with treewidth k: 2O(k) poly(n) time G = planar graphs: PTAS for log-holant (log-partition function) FPTAS for Holant assuming strong spatial mixing

15 Pinning & Peering symmetric f :[q] d C fix any τ [q] d-k Pin (f) :[q] k! C Pinning: σ [q] k Pinτ( f ) = f(τ1, τ2,..., τd-k,,,..., ) (σ) σ1 σ2 σk ( Peering: equivalent relation between τ,τ [q] d-k τ τ if Pinτ( f )= Pinτ ( f ) k f is r-regular if r = max 0applekappled Pin (f) 2 [q] d k not violating f

16 Boolean symmetric f : {0, 1} d! C τ {0,1} i+j τ has i 1 s and j 0 s f = [ ] [f0, f1,..., fi-1, fi,..., fd-j-1, fd-j,..., fd] = Pinτ( f ) where f i = f(x) for Σ j x j = i f is r-regular if r = max 0applekappled Pin (f) 2 [q] d k not violating f 2-state spin systems: EQ=[1,0,...,0,1] is 2-regular, A e is 3-regular matchings & PMs: [1,1,0,...,0] and [0,1,0,...,0] are 2-regular edge covers: [0,1,1,...,1] is 2-regular subgraphs world: [1, μ, 1, μ,...] is 2-regular Eulerian orientations: [0,...,0, 1, 0,..., 0] is (d/2+1)-regular

17 symmetric f : {0, 1, 2} d! C f60 f50 f51 fij = f(x) f40 f41 f42 where x has i 0 s and j 1 s Pinτ( f ) f30 f31 f32 f33 τ =0112 f20 f21 f22 f23 f24 f10 f11 f12 f13 f14 f15 f00 f01 f02 f03 f04 f05 f06 all d-ary symmetric function is at most bounded-degree Holant is O(1)-regular d + q 1 q 1 -regular

18 Theorem For any constant domain size q 2, Holant of any graph G=(V, E) with r-regular symmetric signatures can be computed in time: r O(n) where n= V r O(tw(G)) n + 2 O(tw(G)) Poly(n) extendable to asymmetric signatures (under proper assumption for evaluating asymmetric functions with unbounded arity); implications in approximate counting on planar graphs: PTAS for log-holant FPTAS for holant assuming strong spatial mixing

19 A Simple r O(n) -time Algorithm given a Holant instance: G=(V,E) V = {v 1, v 2,..., v n } f v1 f vi f v2 f vn all f vi :[q] deg(vi)! C are r-regular f is r-regular if r = max 0applekappled Pin (f) 2 [q] d k f v1 f v2 f vi σ Pin 1 (f vi ) f vn dynamic programming: table size: r n n running time: r n q n = r O(n) holant(g, {f v1,...,f vn }) = X feasible 2[q] deg(v n) f vn ( ) holant G \{v n }, {new f v1,...,f vi } where f vi = ( f vi Pin j (f vi ) v i 6 v n v i is j s nbr. of v n

20 Oracles given a Holant instance: G=(V,E) V = {v 1, v 2,..., v n } fix f vi f v1 f vi f v2 f vn all denoted as fi,1,f k i,2,...,f k i,r k Evaluation oracle: Given (i, j, k) and σ [q] k, returns fi,j( k ) f vi :[q] deg(vi)! C are r-regular and arity k: there are r distinct Pin (f vi ):[q] k! C Pinning oracle: Given (i, j, k) and τ [q] k-l, returns j that f l i,j 0 =Pin (f k i,j) For symmetric function f :[q] d! C with constant q, these oracles can be implemented using: Poly(d) preprocessing time Poly(d) space O(1) query time for Holant with finitely many signatures: this can be hard-wired into the algorithm

21 An FPT Algorithm V i V j V k S i V i for Holant problems: holant = X 2[q] E Y f v E(v) f v max { S i, V i } = O(tw(G) ) V j V k v2v each v S i has unbounded degree but only has 3 classes of edges Peering: τ τ if Pin τ ( f )= Pin τ ( f ) Observation: f(ρστ) is determined by the equivalent classes for ρ,σ,τ enumerate equivalent classes of local configurations!

22 An FPT Algorithm V j f v V i V j V k S i V k V i each v S i has unbounded degree but only has 3 classes of edges τ τ if Pin τ ( f )= Pin τ ( f ) f(ρστ) is determined by the equivalent classes for ρ,σ,τ (f)( )= f is r-regular 1 (f) = (f) 0 Peerτ( f ) is r-regular f f( ) Peer (f) Peer (f) Peer (f) locally enumerate equivalent classes ρ,σ,τ

23 An FPT Algorithm fv Vi Vj Vj V Si k Vk each v Si has unbounded degree but only has 3 classes of edges Vi τ τ if Pinτ( f )= Pinτ ( f ) f(ρστ) is determined by the equivalent classes for ρ,σ,τ (f )( ) = 1 0 (f ) = Holant (Vi, boundary signatures Peer (fv ) at v ) X old boundary signatures of Vj (f ) Holant Vj, and Peer v (fv ) at v 2 Si equiv. classes v, v, v old boundary signatures of Vk at every v2si Holant Vk, and Peer v (fv ) at v 2 Si Y old boundary signatures of Vi Holant Si fv ( v, v, v ) and Peer v (fv ) at v 2 Si = v2si

24 An FPT Algorithm V j f v V i V j V k S i V k V i each v S i has unbounded degree but only has 3 classes of edges τ τ if Pin τ ( f )= Pin τ ( f ) f(ρστ) is determined by the equivalent classes for ρ,σ,τ (f)( )= 1 (f) = (f) 0 fix f v1,f v2,...,f vn Peering oracle: Given i, (equivalent class) τ, returns Peer (f ) vi Pinning oracle & Evaluation oracle: for Peer (f vi ) Evaluation oracle: Given i, (equivalent classes) ρ,σ,τ, returns f vi ( ) For symmetric functions, can be implemented using Poly(n) preprocessing time, Poly(n) space, and O(1) query time

25 Theorem For any constant domain size q 2, Holant of any graph G=(V, E) with r-regular signatures can be computed in time: r O(n) where n= V r O(tw(G)) V + 2 O(tw(G)) Poly( V ) assuming Peering, Pinning, & Evaluation oracles. asymmetric f :[q] d C fix any τ [q] d-k and S 2 [d] d k PinS,τ( f ) = f(τ1,..., τ2,..., τ3,...,τd-k,...) ( at positions in S f is r-regular if r = max 0applekappled max S2( [d] d k) Pin S, (f) 2 [q] d k

26 Planar Decomposition Baker s Decomposition (Baker 93): For all k, a planar graph G can be decomposed into subgraphs G 1,..., G k each of treewidth O(k). Jerrum-Goldberg-McQuillan 12: For spin systems on planar graphs, if always holant exp(ω(n)) then PTAS for log-holant.

27 Correlation Decay strong spatial mixing (SSM): B [q] B ( (e) =c A) ( (e) =c A, B) ( V ) ( (t)) B t G SSM: sufficiency of local information for approx. of? ( (e) =c A) efficiency of local computation e A Theorem (Demaine-Hajiaghayi 04) For apex-minor-free graphs, treewidth of t-ball is O(t).

28 Correlation Decay strong spatial mixing (SSM): B [q] B ( (e) =c A) ( (e) =c A, B) ( V ) ( (t)) B t G SSM: sufficiency of local information for approx. of for planar graphs ( (e) =c A) efficiency of local computation e A for self-reducible problems FPTAS for Holant

29 Conclusion A class of Holant problems behave like boundeddegree #CSP: 2 O(n) - and 2 O(tw) Poly(n)-time algorithms. Regularity implies efficient DP algorithms, but: is not robust to holographic transformation; does not cover inversion-based algorithms. Open question: a Holant problem with Booleandomain symmetric signatures (and is easy for decision) with n Ω(tw) lower bound: must have unbounded degree, must be irregular.

30 Thank you!