Backgrounds Problems Top-down Wang Applications On the chordality of polynomial sets in triangular decomposition in top-down style Chenqi Mou joint work with Yang Bai LMIB–School of Mathematics and Systems Science Beihang University, China ISSAC 2018, New York, USA June 19, 2018
Backgrounds Problems Top-down Wang Applications Chordal graph G = ( V, E ) chordal ⇐ ⇒ for any cycle C contained in G of four or more vertexes, there is an edge e ∈ E \ C connects two vertexes in C . Figure: An illustrative chordal graph
Backgrounds Problems Top-down Wang Applications Chordal graph Perfect elimination ordering / chordal graph G = ( V, E ) a graph with V = { x 1 , . . . , x n } : An ordering x i 1 < x i 2 < · · · < x i n of the vertexes is called a perfect elimination ordering of G if for each j = i 1 , . . . , i n , the restriction of G on X j = { x j } ∪ { x k : x k < x j and ( x k , x j ) ∈ E } is a clique. A graph G is said to be chordal if there exists a perfect elimination ordering of it. Figure: Chordal VS non-chordal graphs
Backgrounds Problems Top-down Wang Applications Triangular set and decomposition Triangular set in K [ x 1 , . . . , x n ] with x 1 < · · · < x n T 1 ( x 1 , . . . , x s 1 ) T 2 ( x 1 , . . . , x s 1 , . . . , x s 2 ) T 3 ( x 1 , . . . , x s 1 , . . . , x s 2 , . . . , x s 3 ) . . . T r ( x 1 , . . . , x s 1 , . . . , x s 2 , . . . , x s 3 , . . . , . . . , x s r ) Triangular decomposition Polynomial set F ⊂ K [ x 1 , . . . , x n ] ⇓ Triangular sets T 1 , . . . , T t s.t. Z ( F ) = � t i =1 Z ( T i / ini( T i )) � Solving F = 0 = ⇒ solving all T i = 0 � Multivariate generalization of Gaussian elimination
Backgrounds Problems Top-down Wang Applications Inspired by the pioneering works of D. Cifuentes P.A. Parrilo (from MIT) [Cifuentes and Parrilo 2017] : Connections between triangular sets and chordal graphs Experimental observations: algorithms for computing triangular sets due to Wang become more efficient when the input polynomial set is chordal ( = ⇒ Why?)
Backgrounds Problems Top-down Wang Applications Associated graphs of polynomial sets supp( F ) for F ⊂ K [ x 1 , . . . , x n ] : the set of variables which appear in F Associated graphs F ⊂ K [ x 1 , . . . , x n ] , associated graph G ( F ) of F is an undirected graph: (a) vertexes of G ( F ) : the variables in supp( F ) (b) edge ( x i , x j ) in G ( F ) : if there exists one polynomial F ∈ F with x i , x j ∈ supp( F ) Chordal polynomial set A polynomial set F ⊂ K [ x 1 , . . . , x n ] is said to be chordal if G ( F ) is chordal.
Backgrounds Problems Top-down Wang Applications Associated graphs of polynomial sets K [ x 1 , . . . , x 5 ] P = { x 2 + x 1 , x 3 + x 1 , x 2 4 + x 2 , x 3 4 + x 3 , x 5 + x 2 , x 5 + x 3 + x 2 } Q = { x 2 + x 1 , x 3 + x 1 , x 3 , x 2 4 + x 2 , x 3 4 + x 3 , x 5 + x 2 } Figure: Associated graphs G ( P ) (chordal) and G ( Q ) (not chordal)
Backgrounds Problems Top-down Wang Applications Chordal graphs in Gaussian elimination Tutorial by Chandrasekaran: Guassian elimination w.r.t. a perfect elimination ordering = ⇒ no new fill-ins = ⇒ sparse Gaussian elimina- tion: sparse + chordal [Parter 61, Rose 70, Gilbert 94] Matrix with a chordal associated graph = ⇒ Matrix in echolon-form with a subgraph (credits to J. Gilbert)
Backgrounds Problems Top-down Wang Applications Triangular decomposition in top-down style The variables are handled in a strictly decreasing order: x n , x n − 1 , . . . , x 1 widely used strategy [Wang 1993, 1998, 2000], [Chai, Gao, Yuan 2008] the closest to Gaussian elimination algorithms due to Wang are mostly in top-down style (!!): Efficient when chordal ⇓ Chordal graphs in Gaussian elimination ⇓ Triangular decomposition in top-down style closest to Gaussian elimination ⇓ Algorithms due to Wang are in top-down style
Backgrounds Problems Top-down Wang Applications Triangular decomposition in top-down style The variables are handled in a strictly decreasing order: x n , x n − 1 , . . . , x 1 widely used strategy [Wang 1993, 1998, 2000], [Chai, Gao, Yuan 2008] the closest to Gaussian elimination algorithms due to Wang are mostly in top-down style (!!): Efficient when chordal ⇓ Chordal graphs in Gaussian elimination ⇓ Triangular decomposition in top-down style closest to Gaussian elimination ⇓ Algorithms due to Wang are in top-down style
Backgrounds Problems Top-down Wang Applications Triangular decomposition in top-down style The variables are handled in a strictly decreasing order: x n , x n − 1 , . . . , x 1 widely used strategy [Wang 1993, 1998, 2000], [Chai, Gao, Yuan 2008] the closest to Gaussian elimination algorithms due to Wang are mostly in top-down style (!!): Efficient when chordal ⇓ Chordal graphs in Gaussian elimination ⇓ Triangular decomposition in top-down style closest to Gaussian elimination ⇓ Algorithms due to Wang are in top-down style
Backgrounds Problems Top-down Wang Applications Problems For a chordal polynomial set: Changes of graph structures of the polynomial sets in the process of triangular decomposition in top-down style Relationships (like inclusion) between associated graphs of computed triangular sets and the input polynomial set Chordal graphs in Gaussian elimination = ⇒ Chordal graphs in triangular decomposition in top-down style: multivariate generalization
Backgrounds Problems Top-down Wang Applications Reduction w.r.t. one variable in triangular decomposition P ⊂ K [ x 1 , . . . , x n ] : P ( i ) = { P ∈ P : lv( P ) = x i } Theorem P ⊂ K [ x 1 , . . . , x n ] chordal, x 1 < · · · <x n perfect elimination ordering: Let T ∈ K [ x 1 , . . . , x n ] with lv( T ) = x n and supp( T ) ⊂ supp( P ( n ) ) , and R ⊂ K [ x 1 , . . . , x n ] with supp( R ) ⊂ supp( P ( n ) ) \ { x n } . Then for P = { ˜ ˜ P (1) , . . . , ˜ P ( n − 1) , T } , P ( k ) = P ( k ) ∪ R ( k ) for k = 1 , . . . , n − 1 , we have G ( ˜ where ˜ P ) ⊂ G ( P ) P = {P (1) , P (2) , P ( n ) } : . . . , G ( P ) chordal ⇓ ⇓ ⇓ ⊃ P = { ˜ ˜ P (1) , P (2) , ˜ G ( ˜ . . . , T } : P ) = s.t. = P (1) ∪ R (1) , P (2) ∪ R (2) , . . . , supp( T ) ⊂ supp( P ( n ) ) ⇒ G ( ˜ � In particular, supp( T ) = supp( P ( n ) ) = P ) = G ( P )
Backgrounds Problems Top-down Wang Applications Some notations mapping f i f i : 2 K [ x i ] \ K [ x i − 1 ] → ( K [ x i ] \ K [ x i − 1 ]) × 2 K [ x i − 1 ] P �→ ( T, R ) s.t supp( T ) ⊂ supp( P ) and supp( R ) ⊂ supp( P ) (where K [ x 0 ] = K ). P ⊂ K [ x 1 , . . . , x n ] and a fixed integer i (1 ≤ i ≤ n ) , suppose that ( T i , R i ) = f i ( P ( i ) ) for some f i . For j = 1 , . . . , n , define P ( j ) , if j > i red i ( P ( j ) ) := { T i } , if j = i P ( j ) ∪ R ( j ) i , if j < i and red i ( P ) := ∪ n j =1 red i ( P ( j ) ) . In particular, write red i ( P ) := red i (red i +1 ( · · · (red n ( P )) · · · )) The above theorem becomes G (red n ( P )) ⊂ G ( P ) , and the equality holds if supp( T n ) = supp( P ( n ) ) .
Backgrounds Problems Top-down Wang Applications Reduction w.r.t. all variables in triangular decomposition P = {P (1) , P (2) , P ( n − 1) , P ( n ) } : . . . , G ( P ) chordal ⇓ ⇓ ⇓ ⇓ ⊃ red n ( P ) = { ˜ P (1) , P (2) , ˜ P ( n − 1) , ˜ . . . , T n } : G (red n ( P )) ⇓ ⇓ ⇓ ⇓ ?? red n − 1 ( P ) = { ˜ ˜ P (1) , ˜ P (2) , ˜ . . . , T n − 1 , T n } : G (red n − 1 ( P )) . . . ?? red 1 ( P ) = { T 1 , T 2 , . . . , T n − 1 , T n } : G (red 1 ( P )) Proposition P ⊂ K [ x 1 , . . . , x n ] chordal, x 1 < · · · < x n perfect elimination ordering: For each i (1 ≤ i ≤ n ) , suppose that ( T i , R i ) = f i (red i +1 ( P ) ( i ) ) for some f i and supp( T i ) = supp(red i +1 ( P ) ( i ) ) . Then G (red 1 ( P )) = · · · = G (red n − 1 ( P )) = G (red n ( P )) = G ( P ) .
Backgrounds Problems Top-down Wang Applications Counter example for successive inclusions supp( T i ) ⊂ supp(red i +1 ( P ) ( i ) ) : then in general we will NOT have G (red 1 ( P )) ⊂ · · · ⊂ G (red n − 1 ( P )) ⊂ G (red n ( P )) ⊂ G ( P ) Example P = { x 2 + x 1 , x 3 + x 1 , x 2 4 + x 2 , x 3 4 + x 3 , x 5 + x 2 , x 5 + x 3 + x 2 } Q = red 5 ( P ) = { x 2 + x 1 , x 3 + x 1 , x 3 , x 2 4 + x 2 , x 3 4 + x 3 , x 5 + x 2 } ⇓ T 4 = prem( x 3 4 + x 3 , x 2 4 + x 2 ) = − x 2 x 4 + x 3 , R 4 = { prem( x 2 4 + x 2 , − x 2 x 4 + x 3 ) } = { x 2 3 − x 3 2 } , ⇓ Q ′ := red 4 ( P ) = { x 2 + x 1 , x 3 + x 1 , x 2 3 − x 3 2 , x 3 , − x 2 x 4 + x 3 , x 5 + x 2 } .
Backgrounds Problems Top-down Wang Applications Subgraphs of the input chordal graph Theorem P ⊂ K [ x 1 , . . . , x n ] chordal, x 1 < · · · < x n perfect elimination ordering: For each i = n, . . . , 1 , G (red i ( P )) ⊂ G ( P ) . Corollary P ⊂ K [ x 1 , . . . , x n ] chordal, x 1 < · · · < x n perfect elimination ordering: If T := red 1 ( P ) does not contain any nonzero constant, then T forms a triangular set such that G ( T ) ⊂ G ( P ) . T above: the main component in the triangular decomposition Valid for ANY algorithms for triangular decomposition in top-down style Problem: what about the other triangular sets? (splitting strategies)
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