On the chordality of polynomial sets in triangular decomposition in top-down style

Transcription

1 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

2 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

3 Chordal graph Perfect elimination ordering / chordal graph G = (V, E) a graph with V = {x 1,..., x n }: An ordering x i1 < x i2 < < x in 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

4 Triangular set and decomposition Triangular set in K[x 1,..., x n ] with x 1 < < x n T 1 (x 1,..., x s1 ) T 2 (x 1,..., x s1,..., x s2 ) T 3 (x 1,..., x s1,..., x s2,..., x s3 ). T r (x 1,..., x s1,..., x s2,..., x s3,...,..., x sr ) 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

5 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?)

6 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.

7 Associated graphs of polynomial sets K[x 1,..., x 5 ] P= {x 2 + x 1, x 3 + x 1, x x 2, x x 3, x 5 + x 2, x 5 + x 3 + x 2 } Q = {x 2 + x 1, x 3 + x 1, x 3, x x 2, x x 3, x 5 + x 2 } Figure: Associated graphs G(P) (chordal) and G(Q) (not chordal)

8 Chordal graphs in Gaussian elimination Tutorial by Chandrasekaran: Guassian elimination w.r.t. a perfect elimination ordering = no new fill-ins = sparse Gaussian elimination: 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)

9 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

10 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

11 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

12 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

13 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 }, where P (k) = P (k) R (k) for k = 1,..., n 1, we have G( P) G(P) P = {P (1), P (2),..., P (n) } : G(P) chordal P = { P (1), P(2),..., T } : G( P) = = s.t. P (1) R (1), P (2) R (2),..., supp(t ) supp(p (n) ) In particular, supp(t ) = supp(p (n) ) = G( P) = G(P)

14 mapping f i f i : 2 K[xi]\K[xi 1] (K[x i ] \ K[x i 1]) 2 K[xi 1] Some notations 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 The above theorem becomes red i (P) := red i (red i+1 ( (red n (P)) )) G(red n (P)) G(P), and the equality holds if supp(t n ) = supp(p (n) ).

15 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).

16 Counter example for successive inclusions supp(t i ) supp(red i+1 (P) (i) ): then in general we will NOT have Example G(red 1 (P)) G(red n 1 (P)) G(red n (P)) G(P) P = {x 2 + x 1, x 3 + x 1, x x 2, x 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 x 2, x x 3, x 5 + x 2 } T 4 = prem(x x 3, x x 2 ) = x 2 x 4 + x 3, R 4 = {prem(x 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 }.

17 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)

18 Wang s method: binary decomposition tree [Wang 93]: Wang s method, simply-structured algorithm for triangular decomposition in top-down style T = ini(t )x s k + tail(t ) P := P \ P (i) {T } {prem(p, T ) : P P}, Q := Q {ini(t )}, P := P \ {T } {ini(t ), tail(t )}, Q := Q,

19 Wang s method: left child Proposition: Wang s method applied to F K[x 1,..., x n ], chordal F K[x 1,..., x n ] chordal, x 1 < < x n perfect elimination ordering: (P, Q, i) arbitrary node in the binary decomposition tree such that G(P) G(F), T P with minimal degree in x i. Denote Then G(P ) G(F). P = P \ P (i) {T } {prem(p, T ) : P P (i) }. G(P ) G(F) on the conditions that G(F) is chordal and G(P) G(F)

20 Wang s method: right child Proposition (P, Q, i) arbitrary node in the binary decomposition tree, T P (i) with minimal degree in x i. Denote P = P \ {T } {ini(t ), tail(t )}. Then G(P ) G(P ). In particular, supp(tail(t )) = supp(t ) = G(P ) = G(P). G(P ) G(P ) under no conditions

21 Wang s method: any node Theorem: Wang s method applied to F K[x 1,..., x n ], chordal F K[x 1,..., x n ] chordal, x 1 < < x n perfect elimination ordering: For any node (P, Q, i) in the binary decomposition tree, G(P) G(F) Corollary: Wang s method applied to F K[x 1,..., x n ], chordal F K[x 1,..., x n ] chordal, x 1 < < x n perfect elimination ordering: For any triangular set T computed by Wang s method, G(T ) G(F) Problems solved

22 Variable sparsity of polynomial sets Variable sparsity G(F) = (V, E) associated graph of F = {F 1,..., F r } K[x 1,..., x n ]. Define the variable sparsity s v (F) of F as ( ) 2 s v (F) = E /, V denominator: edge number of a complete graph of V vertexes Sparse triangular decomposition Sparse Gaussian elimination = sparse triangular decomposition in topdown style: multivariate generalization, on-going work sparse Gröbner bases [Faugère, Spaenlehauer, Svartz 2014] sparse FGLM algorithms [Faugère, Mou 2011, 2017]

23 Complexity analysis for triangular decomposition in top-down style Chordal completion For a graph G, another graph G is called a chordal completion of G if G is chordal with G as its subgraph. The treewidth of a graph G is defined to be the minimum of the sizes of the largest cliques in all the possible chordal completions of G. many NP-complete problems related to graphs can be solved efficiently for graphs of bounded treewidth [Arnborg, Proskurowski 1989] Complexities for computing Gröbner bases for polynomial sets with small treewidth [Cifuents and Parrilo 2016] Reminding you of the inclusion of graphs for Wang s method The input chordal associated graph: upper bound Complexities for triangular decomposition: first for polynomial sets with chordal graphs / small treewidth

24 Future works Chordality in regular decomposition in top-down style: the most popular triangular decomposition = A dynamic multi-branch decomposition tree More other graph structures? perfect, linear, directed graphs... = Suggestion welcome! Thanks!

25 A refined algorithm for regular decomposition Input: a polynomial set F K[x] Output: a variable ordering x and a regular decomposition Φ of F with respect to x 1 Compute the variable sparsity s v of F 2 If s v is smaller than some sparsity threshold s 0 (F is sparse), then 1 If G(F) is chordal, then compute its perfect elimination ordering x 1 2 Else compute its chordal completion G(F) 2 and a perfect elimination ordering x of G(F) 3 Compute the regular decomposition of F with respect to x with a top-down algorithm 3 1 [Rose, Tarjan, and Lueker 1976] 2 [Bodlaender and Koster 2008] 3 Say, [Wang 2000]

26 Sparse triangular decomposition A sparse polynomial system arising from the lattice reachability problem [Cifuentes and Parrilo 2017], [Diaconis, Eisenbud, Sturmfels 1998] F i := {x k x k+3 x k+1 x k+2 : k = 1, 2,..., i }, i Z >0 Figure: Associated graph of F i

27 Sparse triangular decomposition Comparisons of timings for computing regular decomposition of one class of chordal and variable sparse polynomials [Cifuentes and Parrilo 2017] F i := {x k x k+3 x k+1 x k+2 : k = 1, 2,..., i }, i Z >0 Table: Regular decomposition with RegSer in Epsilon: top-down n s v t p t r t r t r /t p Table: Regular decomposition with RegularChains in Maple: not top-down n s v t p t r t r t r /t p

28 Sparse triangular decomposition Comparisons of timings for computing regular decomposition of one class of chordal and variable sparse polynomials [Cifuentes and Parrilo 2017] F i := {x k x k+3 x k+1 x k+2 : k = 1, 2,..., i }, i Z >0 Table: Regular decomposition with RegSer in Epsilon: top-down n s v t p t r t r t r /t p Table: Regular decomposition with RegularChains in Maple: not top-down n s v t p t r t r t r /t p