Hypergraph Decompositions and Toric Ideals Elizabeth Gross and Kaie - PowerPoint PPT Presentation

Hypergraph Decompositions and Toric Ideals Elizabeth Gross and Kaie Kubjas June 9, 2015

Toric ideal of a hypergraph ◮ H = ( E , V ) hypergraph ◮ I H is kernel of the monomial map K [ p e : e ∈ E ] → K [ q v : v ∈ V ] � p e �→ q v v ∈ e ◮ the toric ideal is parametrized by the vertex-edge incidence matrix  1 0 1 0 0  1 0 0 1 0     1 0 0 0 1     0 1 1 0 0     0 1 0 1 0   0 1 0 0 1

Main problem Sonja Petrovi´ c and Despina Stasi (2014), Toric algebra of hypergraphs, J. Algebraic Combin. , 39 , 187-208: Question Given a hypergraph H that is obtained by identifying vertices from two smaller hypergraphs H 1 and H 2 , is it possible to obtain generating set of I H from the generating set of I H 1 and I H 2 ? x

Hierarchical models d2 d1 d4 d3 grade by the values on the intersection d2 d2 x d1 d4 d3 d3

Toric fiber products ◮ K [ x ] = K [ x i j : i ∈ [ r ] , j ∈ [ s i ]] , K [ y ] = K [ y i k : i ∈ [ r ] , k ∈ [ t i ]] ◮ multigraded by k ) = a i ∈ Z d deg( x i j ) = deg( y i ◮ I ⊆ K [ x ] , J ⊆ K [ y ] homogeneous wrt the multigrading ◮ K [ z ] = K [ z i jk : i ∈ [ r ] , j ∈ [ s i ] , k ∈ [ t i ]] ◮ φ I , J : K [ z ] → K [ x ] / I ⊗ K K [ y ] / J , z i jk �→ x i j ⊗ y i k ◮ toric fiber product: I × A J = ker( φ I , J ) ◮ Sullivant 2007; Engstr¨ om, Kahle, Sullivant 2014; Kahle, Rauh 2014; Rauh, Sullivant 2014+

Possibility for toric fiber products? ◮ One idea: ◮ V = V 1 ∪ V 2 ◮ H i is the subhypergraph induced by V i ◮ grade edges in H i by their incidence vectors for V 1 ∩ V 2 ◮ However, there are problems: ◮ the ideals are not homogeneous ◮ knowing the incidence vectors for the intersection V 1 ∩ V 2 is not enough information ◮ Multigrading has to remember which edges can be glued

Modified construction ◮ H = ( V , E ) hypergraphs ◮ V = V 1 ∪ V 2 ◮ H i = ( V i , E i ) is the subhypergraph induced by V i ◮ E i = { e ∩ V i : e ∈ E , e ∩ V i � = ∅} ◮ we view E 1 and E 2 as multisets ->

Modified construction ◮ K [ H i ] = K [ x e : e ∈ E , e ∩ V i � = ∅ ] ◮ multigraded by deg( x e ) = deg( y e ) = u e if e ∩ V 1 � = ∅ and e ∩ V 2 � = ∅ , deg( x e ) = deg( y e ) = 0 otherwise ◮ K [ H ] = K [ z e : e ∈ E ] , ◮ ring homomorphism φ I H 1 , I H 2 : K [ H ] → K [ H 1 ] / I H 1 ⊗ K [ H 2 ] / I H 2 defined by z e �→ x e ⊗ y e if e ∩ V 1 � = ∅ and e ∩ V 2 � = ∅ , z e �→ x e ⊗ 1 if e ⊆ V 1 \ V 2 , z e �→ 1 ⊗ y e if e ⊆ V 2 \ V 1 ◮ I H = ker( φ I H 1 , I H 2 )

Graver basis Definition A Graver basis of a toric ideal I consists of all the binomials p + − p − ∈ I such that there is no other binomial q + − q − ∈ I such that q + divides p + and q − divides p − . i : i ∈ I } , Gr ( I H 2 ) = { y v + Gr ( I H 1 ) = { x u + j − y v − i − x u − : j ∈ J } j i ) = (deg( x u + i ) , deg( x u − ◮ ( α + i , α − i )) for all i ∈ I j ) = (deg( y v + j ) , deg( y v − ◮ ( β + j , β − j )) for all j ∈ J

Graver basis L = { ( a 1 , . . . , a I , b 1 , . . . , b J ) ∈ Z I + J : a i α sign( a i ) b j β sign( b j ) � � = , i j a i α − sign( a i ) b j β − sign( b j ) � � = } i j Define a partial order on R I + J by x � x ′ ⇔ sign( x i ) = sign( x ′ i ) and | x i | ≤ | x ′ i | for i = 1 , . . . , I + J Let S be equal to the set of minimal elements in L wrt the partial order.

Gluing ◮ E ′ = { e ∈ E : e ∩ V 1 � = ∅ , e ∩ V 2 � = ∅} e ⊆ V 1 \ V 2 x a + e ∈ E ′ x c + e ⊆ V 1 \ V 2 x a − e ∈ E ′ x c − ◮ f = � � − � � e e e e e e e e e ⊆ V 2 \ V 1 y b + e ∈ E ′ y c + e ⊆ V 1 \ V 2 y b − e ∈ E ′ y c − ◮ g = � � − � � e e e e e e e e e ⊆ V 1 \ V 2 z a + e ⊆ V 2 \ V 1 z b + e ∈ E ′ z c + ◮ glue( f , g ) = � e � e � e − e e e e ⊆ V 1 \ V 2 z a − e ⊆ V 2 \ V 1 z b − e ∈ E ′ z c − � � � e e e e e e ◮ we say f and g are compatible ◮ F 1 ⊆ I H 1 and F 2 ⊆ I H 2 consist of binomials ◮ Glue( F 1 , F 2 ) = { glue( f , g ) : f ∈ F 1 , g ∈ F 2 compatible }

Graver basis Theorem The Graver basis of I H is given by sign ( ai ) − sign ( ai ) ) a i − � � ( x u ( x u ) a i , { glue ( f , g ) ∈ k [ H ] : f = i i i ∈ I i ∈ I sign ( bj ) − sign ( bj ) ) b j − � ( y v � ( y v ) b j g = j j j ∈ J j ∈ J for ( a 1 , . . . , a I , b 1 , . . . , b J ) ∈ S } .

Graver basis

The compatible projection property Definition Let F 1 ⊂ I 1 and F 2 ⊂ I 2 . The pair F 1 and F 2 has compatible projection property if for all compatible pairs x u + − x u − ∈ I H 1 and y v + − y v − ∈ I H 2 there exist x u + i − x u − i , monomial multiples of elements of F 1 , i = 1 , . . . , m , and y v + j − y v − j , monomial multiples of elements of F 2 , j = 1 , . . . , n , such that 1. x u + − x u − = � x u + and y v + − y v − = � y v + j − y v − i − x u − j , i 2. if i 1 < i 2 < . . . < i k are indices where deg( x u + i ) − deg( x u − i ) � = 0 and j 1 < j 2 < . . . < j l are indices where deg( y v + j ) − deg( y v − j ) � = 0, then k = l and deg( x u + ih ) = deg( y v + ih ) − deg( x u − jh ) − deg( y v − jh ) for all h ∈ [ k ].

Markov basis deg Theorem Let F 1 ⊂ I H 1 and F 2 ⊂ I H 2 be Markov bases. Then Glue ( F 1 , F 2 ) is a Markov basis of I H if and only if F 1 and F 2 satisfy the compatible projection property.

Monomial sunflower Consider the monomial sunflower. We can construct larger monomial sunflowers by taking an even number of copies of H and identifying all copies of the vertex v 1 . We consider 128 copies of the sunflower. If we split it into two, then computing a Markov basis using Macaulay2 interface for 4ti2 is 10 times faster compared to computing a Markov basis for the original hypergraph. v 1