Chordal Graphs in Triangular Decomposition
in Top-Down Style111This work was partially supported by the National Natural Science Foundation of China (NSFC 11401018 and 11771034)
In this paper, we first prove that when the associated graph of a polynomial set is chordal, a particular triangular set computed by a general algorithm in top-down style for computing the triangular decomposition of this polynomial set has an associated graph as a subgraph of this chordal graph. Then for Wang’s method and a subresultant-based algorithm for triangular decomposition in top-down style and for a subresultant-based algorithm for regular decomposition in top-down style, we prove that all the polynomial sets appearing in the process of triangular decomposition with any of these algorithms have associated graphs as subgraphs of this chordal graph. These theoretical results can be viewed as non-trivial polynomial generalization of existing ones for sparse Gaussian elimination, inspired by which we further propose an algorithm for sparse triangular decomposition in top-down style by making use of the chordal structure of the polynomial set. The effectiveness of the proposed algorithm for triangular decomposition, when the polynomial set is chordal and sparse with respect to the variables, is demonstrated by preliminary experimental results.
Key words: Triangular decomposition, chordal graph, top-down style, regular decomposition, sparsity
In this paper we establish some underlying connections between graph theory and symbolic computation by studying the changes of associated graphs of polynomial sets in the process of decomposing an arbitrary polynomial set with a chordal associated graph into triangular sets with algorithms in top-down style. The study in this paper is directly inspired by the pioneering work of Cifuentes and Parrilo. In  they showed for the first time the connections between chordal graphs and triangular sets when they introduced the concept of chordal networks of polynomial sets and proposed an algorithm for constructing chordal networks based on computation of triangular decomposition. In particular, they found experimentally that for polynomial sets with chordal associated graphs, the algorithms for triangular decomposition due to Wang (e.g., his algorithm for regular decomposition in ) become more efficient. In this paper, with clarification of the changes of associated graphs of polynomial sets in triangular decomposition in top-down style, we are able to provide a theoretical explanation for their experimental observation (see Remark 33). It is worth mentioning that Cifuentes and Parrilo also studied the connections between chordal graphs and Gröbner bases in , but they found that the chordal structures of polynomial sets are destroyed in the process of computing Gröbner bases .
Chordal graphs have been applied to many scientific and engineering problems like existence of perfect phylogeny in reconstruction of evolutionary trees . Two of these applications are of particular interest to us and are closely related to the study in this paper: sparse Gaussian elimination and sparse sums-of-squares decomposition. For the former problem, it is shown that the Cholesky factorization of a symmetric positive definite matrix does not introduce new fill-ins if the associated graph of the matrix is chordal, and on the basis of this observation algorithms for sparse Gaussian elimination have been proposed by using the property that the sparsity of the matrix can be kept if the associated graph of the matrix is chordal [31, 32, 23]. For the latter, structured sparsity arising from polynomial optimization problems is studied and utilized by using the chordal structures, resulting in sparse algorithms for sums-of-squares decomposition of multivariate polynomials [34, 35, 43, 40].
The underlying ideas of the study in this paper are similar to those in the two successful applications of chordal graphs above: we show that the chordality of associated graphs of polynomial sets is preserved in a few algorithms for triangular decomposition in top-down style, as it is in the Cholesky factorization of symmetric matrices, and we propose a sparse algorithm for triangular decomposition in top-down style based on the chordal structure in a simiar way to what have been done for sparse Gaussian elimination and sparse sums-of-squares decomposition.
Like the Gröbner basis which has been greatly developed in its theory, methods, implementations, and applications [8, 16, 17, 18, 14], the triangular set is another powerful algebraic tool in the study on and computation of polynomials symbolically, especially for elimination theory and polynomial system solving [41, 20, 26, 36, 2, 39, 11], with diverse applications [42, 10]. The process of decomposing a polynomial set into finitely many triangular sets or systems (probably with additional properties like being regular or normal, etc.) with associated zero and ideal relationships is called triangular decomposition of the polynomial set. Triangular decomposition of polynomial sets can be regarded as polynomial generalization of Gaussian elimination for solving linear equations.
The top-down strategy in triangular decomposition means that the variables appearing in the input polynomial set are handled in a strictly decreasing order, and it is a common strategy in the design and implementations of algorithms for triangular decomposition. In particular, most algorithms for triangular decomposition due to Wang are in top-down style [36, 37, 38]. Algorithms for triangular decomposition in top-down style with refinement in the Boolean settings and over finite fields have also been proposed and applied to cryptoanalysis [10, 21, 24]. The fact that elimination in it is performed in a strictly decreasing order makes triangular decomposition in top-down style the closest among all kinds of triangular decomposition to Gaussian elimination, in which the elimination of entries in different columns of the matrix is also performed in a strict order.
In this paper the chordal structures of polynomial sets appearing in the algorithms for triangular decomposition in top-down style are studied. The main contributions of this paper include: 1) Under the conditions that the input polynomial set is chordal and a perfect elimination ordering is used as the variable ordering, we study the influence of general reduction in triangular decomposition in top-down style on the associated graphs of polynomial sets and prove that one particular triangular set computed by algorithms for triangular decomposition in top-down style has an associated graph as a subgraph of the input chordal graph (in Section 3). 2) Under the same conditions, we show (in Section 4) that in the process of triangular decomposition with Wang’s algorithm, any polynomial set (and thus any of the computed triangular sets) has an associated graph as a subgraph of the input chordal graph. 3) The same results are proved for subresultant-based algorithms for triangular decomposition and regular decomposition in top-down style (in Sections 5 and 6 respectively). 4) The variable sparsity of polynomial sets is defined with their associated graphs, and an effective refinement by using the variable sparsity and chordality of input polynomial sets is proposed to speedup triangular decomposition in top-down style (in Section 7). This paper is an extension of , and the contributions 3) and 4) listed above are new.
With triangular decomposition in top-down style viewed as polynomial generalization of Gaussian elimination, the contributions listed above are indeed polynomial generalizations of the roles chordal structures play in Gaussian elimination and of algorithms for sparse Gaussian elimination. As one may expect, these polynomial generalizations are highly non-trivial because of the complicated process of triangular decomposition due to various splitting strategies involved in specific algorithms. Furthermore, these contributions reveal theoretical properties of triangular decomposition in top-down style from the view point of graph theory, and we hope this paper can stimulate more study on triangular decomposition by using concepts and methods from graph theory.
Let be a field, and be the multivariate polynomial ring over in the variables . For the sake of simplicity, we write as , as for some integer , and as .
2.1 Associated graph and chordal graph
For a polynomial , define the (variable) support of , denoted by , to be the set of variables in which effectively appear in . For a polynomial set , its support .
Let be a polynomial set in . Then the associated graph of , denoted by , is an undirected graph with the vertex set and the edge set .
The associated graphs of
are shown in Figure 1.
Let be a graph with . Then an ordering of the vertices is called a perfect elimination ordering of if for each , the restriction of on the following set
is a clique. A graph is said to be chordal if there exists a perfect elimination ordering of it.
An equivalent condition for a graph to be chordal is the following: for any cycle contained in of four or more vertices, there is an edge connecting two vertices in . The edge in this case is called a chord of . A chordal graph is also called a triangulated one. For an arbitrary graph , another graph is called a chordal completion of if is chordal and is its subgraph.
From the algorithmic point of view, there exist effective algorithms for testing whether an arbitrary graph is chordal (in case of a chordal graph, a perfect elimination ordering will also be returned)  and for finding a chordal completion of an arbitrary graph , though the problem of finding the minimal chordal completion is NP-hard .
A polynomial set is said to be chordal if its associated graph is chordal.
2.2 Triangular set and triangular decomposition
Throughout this subsection the variables are ordered as . For an arbitrary polynomial , the greatest variable appearing in is called its leading variable, denoted by . Let . Write with , , and . Then the polynomials and are called the initial and tail of and denoted by and respectively, and is called the leading degree of and denoted by . For two polynomial sets , the set of common zeros of in is denoted by , and , where is the algebraic closure of .
An ordered set of non-constant polynomials is called a triangular set if . A pair with is called a triangular system if is a triangular set, and for each and any , we have .
Given a triangular set , the saturated ideal of is . In particular, for an integer , forms a (truncated) triangular set in , and we denote . For an arbitrary polynomial set , we denote for an integer and denote .
A triangular set is said to be regular or called a regular set if for each , the canonical image of in is neither zero nor a zero-divisor. A triangular system is called a regular system if for each , the following conditions hold: (a) either or ; (b) for any and , we have .
The definitions above of regular set and regular system are algebraic (in the language of ideals) and geometric (in the language of zeros) respectively. The connections between regular sets and regular systems have been clarified in [38, 39].
Let be a polynomial set. Then a finite number of triangular sets (triangular systems respectively) are called a triangular decomposition of if the zero relationship holds, where ( holds respectively). In particular, a triangular decomposition is called a regular decomposition if each of its triangular sets or systems is regular.
When no ambiguity occurs, the process for computing the triangular decomposition of a polynomial set is also called triangular decomposition of . As one may find from Definitions 6 and 8, triangular systems are generalization of triangular sets. For a triangular system , is a triangular set which represents the equations , while is a polynomial set which represents the inequations .
There exist many algorithms for decomposing polynomial sets into triangular sets or systems with different properties. One of the main strategies for designing such algorithms for triangular decomposition is to carry out reduction on polynomials containing the greatest (unprocessed) variable until there is only one such polynomial left, at the same time producing new polynomials whose leading variables are strictly smaller than the currently processed variable.
For an arbitrary polynomial set , the smallest integer such that or for each is called the level of and denoted by . Obviously a polynomial set containing no constant forms a triangular set if .
Let be a polynomial set in and be a set of pairs of polynomial sets, initialized with . Then an algorithm for computing triangular decomposition of is said to be in top-down style if for each polynomial set with , this algorithm handles the polynomials in and to produce finitely many polynomials sets and such that the following conditions hold:
for each , and for ;
there exists some integer such that or , and the other are put into for later computation.
In this paper we are interested mainly in algorithms for triangular decomposition in top-down style. Note that the above definition, compared with the corresponding one in , imposes additional conditions on the polynomial sets representing inequations, for the authors find that it is difficult to study the polynomial sets alone when the interactions between and occur in certain algorithms (see Section 6 for more details).
2.3 Pseudo division and subresultant regular subchain
Two commonly used algebraic operations on multivariate polynomials to perform reduction in algorithms for triangular decomposition are pseudo division and computation of the resultant of two polynomials. The algorithms for triangular decomposition in top-down style studied in this paper rely heavily on these two algebraic operations.
For any two polynomials , there exist polynomials and an integer such that and . Furthermore, if is fixed, then and are unique. The process above of computing and from and is called the pseudo division of with respect to , and the polynomials and here are called the pseudo quotient and pseudo remainder of with respect to and denoted by and respectively.
Suppose further that . Write and with . Denote by the sylvester matrix of and with respect to . Then the determinant is called the Sylvester resultant of and with respect to .
For two integers , define to be the submatrix of obtained by deleting the last rows of ’s coefficients, the last rows of ’s coefficients, and the last columns except the -th one. Then the polynomial is called the th subresultant of and with respect to . In particular, the th subresultant is said to be regular if .
Let be two polynomials such that , and be the th resultant of and with respect to for , where when and otherwise. Then the sequence is called the subresultant chain of and with respect to . Furthermore, let be the regular subresultants in with . Then the sequence is called the subresultant regular subchain of and with respect to .
There exist strong connections between the subresultant chain and the greatest common divisor of two polynomials. The reader is referred to [28, Chap. 7] for more details on this.
3 General triangular decomposition in top-down style
In this section, the graph structures of polynomial sets in general algorithms for triangular decomposition in top-down style are studied when the input polynomial set is chordal. We start this section with the connections between the associated graphs of a triangular set reduced from a chordal polynomial set and the chordal associated graph.
Let be a chordal polynomial set with as one perfect elimination ordering of . For , let be a polynomial such that and ( is set null if ). Then is a triangular set, and . In particular, if for , then .
It is straightforward that is a triangular set because if for .
For any edge , there exists an integer such that . Then , and thus and . Since is chordal with as a perfect elimination ordering and , , we know that by Definition 3. This proves the inclusion .
In the case when for , next we show the inclusion , which implies the equality . For any , there exists an integer and a polynomial such that with . Since , we know that and thus . ∎
The following theorem relates the associated graph of a chordal polynomial set and that of the polynomial set after reduction with respect to one variable.
Let be a chordal polynomial set such that and is one perfect elimination ordering of . Let be a polynomial such that and , and be a polynomial set such that . Then for the polynomial set , where for , we have . In particular, if , then .
To prove the inclusion , it suffices to show that for each edge , we have . For an arbitrary edge , there exists a polynomial and an integer such that and .
If , then , and by we have . This implies that and by the chordality of we have .
Else if , then by there are two cases for accordingly: when , clearly ; when , we have , and thus , and the chordality implies .
In particular, if , then by for and we have . This proves the equality . ∎
Next we introduce some notations to formulate the reduction process in Theorem 12. Denote the power set of a set by . For an integer , let be a mapping
such that and , where is understood as . For a polynomial set and a fixed integer , suppose that for some as stated above. Now define the result of reduction with respect to as the polynomial set by defining all its subsets for as follows.
for simplicity, and the polynomial set is the result of successive reduction with respect to . Following the above terminologies, the conclusions of Theorem 12 can be reformulated as: , and the equality holds if .
Indeed, the reduction process above is commonly used in algorithms for triangular decomposition in top-down style, and the mapping in (2) is abstraction of specific reductions used in different kinds of algorithms for triangular decomposition . For example, one specific kind of such reduction is performed by using pseudo divisions, and in this case in (2) consists of pseudo remainders which do not contain .
Let be a chordal polynomial set with as one perfect elimination ordering of . For each , suppose that for some as in (2) and , where is understood as . Then .
Repeated use of Theorem 12 implies
and the conclusion follows. ∎
Proposition 14 holds because after every reduction remains the same as the chordal graph , and thus the hypotheses of Theorem 12 remain satisfied. If we weaken the condition in Proposition 14 to , then in general we will not have
as shown by the following example (though the last inclusion always holds because is chordal).
Let us continue with Example 13 with and , where . Take
The associated graph is shown below. Note that but .
Despite of this example where successive inclusions of the associated graphs in the reduction chain does not hold, it can be proved that for each , is a subgraph of the original chordal graph .
Let be a chordal polynomial set with as one perfect elimination ordering of and be as defined in (4) for . Then for each and any two variables and , if there exists an integer such that , then .
We induce on the integer . In the case , from the proof of Theorem 12 one can easily find that the conclusion holds . Now suppose that the conclusion holds for , and next we prove that it also holds for , namely for any and , if there exists such that , then .
Since , by (3) we consider the following three cases of .
(1) If , then , and thus . By the inductive assumption we have .
(2) If , then , and thus by the inductive assumption we have .
(3) If , then there exists a polynomial set such that and .
(3.1) If , then , and by the inductive assumption we know that .
(3.2) If , then . Next we consider the following three cases. (3.2.1) : with the same argument as in (a) we know that . (3.2.2) : by the induction assumption we know that . (3.2.3) and : Since , by the induction assumption we have ; since , by the induction assumption we have . Then by the chordality of , and imply that .
This ends the proof. ∎
Let be a chordal polynomial set with as one perfect elimination ordering of and be as defined in (4) for . Then for each , .
By the construction of , we know that all the vertices of are also vertices of . For each edge , there exists an integer and a polynomial such that and . Then by Lemma 16, we know that , and thus . ∎
Let be a chordal polynomial set with as one perfect elimination ordering of and be as defined in (4) for . If does not contain any nonzero constant, then forms a triangular set such that .
Corollary 18 tells us that under the conditions that the input polynomial set is chordal and the variable ordering is one perfect elimination ordering, the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style with reduction satisfying the conditions (2) and (3) is a subgraph of the associated graph of the input polynomial set. In fact, this triangular set is usually the “main branch” in the triangular decomposition in the sense that other branches are obtained by adding additional constrains in the splitting in the process of triangular decomposition.
Note that in the case when the input polynomial set is not chordal, a process of chordal completion can be carried out on to generate a chordal graph (in the worst case this chordal completion results in a complete graph which is trivially chordal). After this chordal completion the conditions of Corollary 18 will be satisfied.
The chordality of any triangular set other than the specific one above in a triangular decomposition computed by an algorithm in top-down style is dependent on the splitting strategy in the algorithm. In the following sections, we study several specific algorithms for triangular decomposition in top-down style and prove that the associated graphs of all the polynomial sets in the decomposition process of these algorithms are subgraphs of the associated graph of a chordal input polynomial set.
4 Wang’s method for triangular decomposition in top-down style
A simply-structured algorithm was proposed by Wang for triangular decomposition in top-down style in 1993 , which is referred to as Wang’s method in the literature (see. e.g., ). Next the chordality of polynomial sets in the decomposition process of Wang’s method is studied.
4.1 Wang’s method revisited
For the self-containness of this paper, Wang’s method for triangular decomposition is outlined in Algorithm 1 below. In this algorithm and those to follow, the data structure is used to represent two polynomial sets and such that or for . For a set consisting of tuples in the form , denote . The subroutine returns an element from a set and then removes it from .
The decomposition process in Wang’s method (Algorithm 1) applied to can be viewed as a binary tree with its root as . The nodes of this binary tree are all the tuples picked from , and each node has two child nodes and , where