Phase T ransition of Hypergraph Matchings Yitong Yin Nanjing University Joint work with: Jinman Zhao ( Nanjing Univ. / U Wisconsin )
hardcore model monomer - dimer model λ undirected λ graph λ λ λ G = ( V, E ) λ activity λ λ λ λ independent sets I configurations: matchings M weight: w ( I ) = λ | I | w ( M ) = λ | M | partition function: Z = Σ M : matchings in G w ( M ) Z = Σ I : independent sets in G w ( I ) Gibbs distribution: μ ( I ) = w ( I ) / Z μ ( M ) = w ( M ) / Z approximate counting: FPTAS/FPRAS for Z sampling: sampling from μ within TV-distance ε in time poly( n , log1/ ε )
Decay of Correlation (Weak Spatial Mixing, WSM) v Pr[ v ∈ I | σ ] hardcore model: ` → ∞ I ∼ μ ( d +1) -regular tree boundary condition σ : fixing leaves at level l to be occupied/unoccupied by I WSM: Pr[ v ∈ I | σ ] does not depend on σ when l → ∞ d d uniqueness threshold: λ c = ( d − 1) ( d +1) • λ ≤ λ c ⇔ WSM holds on ( d +1) -regular tree ⇔ Gibbs measure is unique • [Weitz ‘06]: λ < λ c ⇒ FPTAS for graphs with max-degree ≤ d +1 • [Galanis, Š tefankovi č , Vigoda ‘12; Sly, Sun ‘12]: λ > λ c ⇒ inapproximable unless NP=RP
Decay of Correlation (Weak Spatial Mixing, WSM) Pr[ e ∈ M | σ ] e monomer - dimer model: ` → ∞ M ∼ μ regular tree boundary condition σ : fixing leaf-edges at level l to be occupied/unoccupied by M WSM: Pr[ e ∈ M | σ ] does not depend on σ when l → ∞ • WSM always holds ⇔ Gibbs measure is always unique • [Jerrum, Sinclair ’89]: FPRAS for all graphs • [Bayati, Gamarnik, Katz, Nair, Tetali ’08]: FPTAS for graphs with bounded max-degree
CSP (Constraint Satisfaction Problem) a 1 1 b c 2 a b 4 d 3 c 2 3 degree e f d g = 2 degree 5 6 4 e ≤ d max-degree ≤ d 5 f 6 g matching constraint matchings: variables x i ∈ { 0 , 1 } (at-most- 1 )
CSP (Constraint Satisfaction Problem) a 1 1 b c 2 a b 4 d 3 c 2 3 degree e f d g = 2 degree 5 6 4 e ≤ d max-degree ≤ d 5 f 6 g matching constraint matchings: variables x i ∈ { 0 , 1 } (at-most- 1 ) matching constraint independent sets: variables x i ∈ { 0 , 1 } (at-most- 1 ) partition function: X λ k ~ x k 1 Z = x ∈ { 0 , 1 } n satisfying ~ all constraints
CSP (Constraint Satisfaction Problem) c 1 deg ≤ k +1 deg ≤ d +1 x 1 c 2 x 2 c 3 c 4 x 3 c 5 x 4 c 6 x 5 c 7 Boolean at-most- 1 variables constraints partition function: X λ k ~ x k 1 Z = x ∈ { 0 , 1 } n satisfying ~ all constraints
Hypergraph matching vertex set V hypergraph H = ( V, E ) hyperedge e ∈ E, e ⊂ V a matching is an subset M ⊂ E of disjoint hyperedges partition X λ | M | Z λ ( H ) = v 1 e 1 functions: v 3 v 4 v 2 M : matching of H v 9 v 5 v 7 v 8 e 3 v 6 e 4 λ | M | Gibbs e 2 e 5 µ ( M ) = distribution: Z λ ( H )
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * matchings in hypergraphs of max-degree ≤ k+ 1 and max-edge-size ≤ d +1 matching v 1 incidence graph e 1 v 3 v 4 v 2 primal: v 3 v 9 v 5 v 7 e 1 v 8 e 3 v 1 v 6 v 4 e 4 v 2 v 9 e 2 e 5 e 5 e 2 e 3 v 5 v 3 v 8 v 7 e 1 v 4 v 6 v 1 e 5 e 4 dual: e 3 v 9 CSP defined by v 8 matching ( packing ) constraint e 2 v 5 v 2 * v 7 e 4 independent set v 6 independent sets in hypergraphs of max-degree ≤ d+ 1 and max-edge-size ≤ k +1 independent sets: a subset of non-adjacent vertices (to be distinguished with: vertex subsets containing no hyperedge as subset)
Known results c 1 deg ≤ k +1 deg ≤ d +1 x 1 c 2 independent sets of hypergraphs x 2 c 3 of max-degree ≤ d +1 and max-edge-size ≤ k +1 c 4 x 3 partition function: c 5 x 4 X λ k ~ x k 1 c 6 Z = x 5 c 7 x ∈ { 0 , 1 } n satisfying ~ Boolean at-most- 1 all constraints variables constraints Classification of approximability in terms of λ , d, k ? • k =1: hardcore model • d =1: monomer-dimer model • for λ =1: • [Dudek, Karpinski, Rucinski, Szymanska 2014]: FPTAS when d =2, k ≤ 2 • [Liu and Lu 2015] FPTAS when d =2, k ≤ 3
Our Results c 1 deg ≤ k +1 deg ≤ d +1 x 1 c 2 independent sets of hypergraphs x 2 c 3 of max-degree ≤ d +1 and max-edge-size ≤ k +1 c 4 x 3 partition function: c 5 x 4 X λ k ~ x k 1 c 6 Z = x 5 c 7 x ∈ { 0 , 1 } n satisfying ~ Boolean at-most- 1 all constraints variables constraints • uniqueness threshold for ( k +1) -uniform ( d +1) -regular infinite hypertree: d d λ c ( k, d ) = k ( d − 1) d +1 • λ < λ c : FPTAS • : inapproximable unless NP=RP λ > 2 k +1+( − 1) k λ c ≈ 2 λ c k +1
matchings of hypergraphs of max-degree ( k +1) and max-edge-size ( d +1) λ = 1: independent sets of hypergraphs of max-degree ( d +1) and max-edge-size ( k +1) k 6 uniqueness threshold: 5 uniqueness d d threshold hard λ c = k ( d − 1) ( d +1) 4 easy threshold for hardness: 3 [Liu-Lu 2015] 2 k +1+( − 1) k λ c ≈ 2 λ c k +1 2 [Dudek et al . 2014] d 1 2 3 4 5 6 (4,2): independent sets of 3 -uniform hypergraphs of max-degree 5, the only open case for counting Boolean CSP with max-degree 5. (2,4): matchings of 3-uniform hypergraphs of max-degree 5, exact at the critical threshold: 2 2 d d k ( d − 1) ( d +1) = 4 · 1 5 = 1
Spatial Mixing (Decay of Correlation) weak spatial mixing (WSM): Pr[ v is occupied | σ ∂ R ] ≈ Pr[ v is occupied | τ ∂ R ] error < exp (- t ) strong spatial mixing (SSM): Pr[ v is occupied | σ ∂ R , σ Λ ] ≈ Pr[ v is occupied | τ ∂ R , σ Λ ] by self-reduction: H Pr[ v is occupied | σ Λ ] ∂ R is approximable with additive error ε R in time poly( n , 1/ ε ) t v Λ FPTAS for partition function Z
Hardcore model: random regular v bipartite graph v SAW-tree SSM locally like regular tree arbitrary boundary with parity-preserving symmetry condition for hypergraph: Similar... • on infinite regular tree: Gibbs measure is unique semi-translation invariant (invariant under parity-preserving automorphisms) Gibbs measure is unique • algorithm : Gibbs measure is unique on regular tree Yes. generic WSM on regular tree SSM on trees n SSM on graphs self-avoiding walk (SAW) tree FPTAS for graphs • hardness : a sequence of finite graphs G n (random regular No. bipartite graph) is locally like the infinite regular tree • a sequence of labeled G n locally converges to the infinite regular tree with parity labeling
d d Theorem: λ ≤ λ c ( k, d ) = k ( d − 1) d +1 WSM holds for ( k +1) -uniform ( d +1) -regular hypertree Theorem: on infinite uniform regular hypertree WSM SSM Theorem: on infinite ( k , d ) -hypertree for ( ≤ k , ≤ d ) -hypergraphs SSM SSM with the same rate SSM with exponential rate FPTAS all statements are for hypergraph independent sets
Tree Recursion independent sets of hypertree T : Pr[ v is occupied | σ ] let R T = Pr[ v is unoccupied | σ ] v 1 v tree recursion : e 1 e 3 e 2 e i d 1 Y R T = λ v 2 v 2 v 6 v 3 v 4 v 5 1 + P k i v ij j =1 R T ij i =1 e 3 e 4 e 4 e 4 monomer-dimer model: λ v 3 v 4 v 5 v 5 v 3 v 4 R T = 1 + P k j =1 R T j e 4 e 4 hardcore model: fixed by σ v 5 v 5 d 1 Y R T = λ 1 + R T i i =1
Pr[ v is occupied | σ ] let R T = Pr[ v is unoccupied | σ ] d 1 tree recursion : Y R T = λ 1 + P k i j =1 R T ij i =1 d d Theorem: λ ≤ λ c ( k, d ) = k ( d − 1) d +1 WSM holds for ( k +1) -uniform ( d +1) -regular hypertree monotonicity of the recursion the 2 extremal boundaries at level- l are all occupied / all unoccupied d root 1 the recursion becomes Y R ` = λ 1 + kR ` − 1 i =1 whose convergence is the same as d hardcore model: R 0 1 Y ` = λ 0 1 + R 0 ` � 1 with activity λ 0 = k λ i =1
d d Theorem: λ ≤ λ c ( k, d ) = k ( d − 1) d +1 WSM holds for ( k +1) -uniform ( d +1) -regular hypertree Theorem: on infinite uniform regular hypertree WSM SSM Theorem: on infinite ( k , d ) -hypertree for ( ≤ k , ≤ d ) -hypergraphs SSM SSM with the same rate SSM with exponential rate FPTAS
Self-Avoiding Walk Tree (Weitz 2006) T = T ��� ( G, v ) G =( V , E ) 1 v 1 4 2 3 4 3 2 6 6 4 3 5 σ Λ 5 6 6 5 1 6 6 1 5 6 6 4 5 for hardcore: 4 6 6 P G [ v is occupied | σ Λ ] 4 4 = P T [ v is occupied | σ Λ ] if cycle closing > cycle starting if cycle closing < cycle starting
Hypergraph SAW Tree self-avoiding walk(SAW): ( v 0 , e 1 , v 1 , . . . , e ` , v ` ) T = T SAW ( H , v ) is a simple path in incidence graph and v i 62 [ e i v 1 j<i e 1 e 3 e 2 v 2 e 1 v 3 v 2 v 2 v 6 v 3 v 4 v 5 v 6 e 3 v 1 e 3 e 4 e 4 e 4 v 4 e 4 e 2 v 3 v 4 v 5 v 5 v 3 v 4 v 5 e 4 e 4 v 5 v 5 P H [ v is occupied | σ ] mark any cycle-closing vertex unoccupied if: = P T [ v is occupied | σ ] cycle-closing edge locally < cycle-starting edge and occupied if otherwise
T = T SAW ( H , v ) v 2 e 1 v 1 R T v 3 v 6 e 3 e 1 e 3 e 2 v 1 e i v 4 ` e 4 v 2 v 2 v 6 v 3 v 4 v 5 e 2 R T ij v 5 e 3 e 4 e 4 e 4 Pr[ v is occupied | σ ] let R T = Pr[ v is unoccupied | σ ] v 3 v 4 v 5 v 5 v 3 v 4 arbitrary initial values tree recursion : truncated e 4 e 4 d 1 Y R T = λ 1 + P k i v 5 v 5 j =1 R T ij i =1 Theorem: on infinite ( k +1, d +1) -hypertree for ( ≤ k +1, ≤ d +1) -hypergraphs SSM SSM with the same rate SSM with exponential rate FPTAS
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