Chordal Networks of Polynomial Ideals

SIAM Journal on Discrete Mathematics, 2016

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction, and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry-in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gröbner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gröbner bases algorithms in many cases. The reason is because all computations are done on "smaller" rings of size equal to the treewidth of the graph (instead of the total number of variables). In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization, and differential equations.

the electronic journal of …, 2010

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

Theory of Computing Systems, 2021

Let F[X] be the polynomial ring over the variables X = {x 1 , x 2 ,. .. , x n }. An ideal I = p 1 (x 1),. .. , p n (x n) generated by univariate polynomials {p i (x i)} n i=1 is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results. Let f (X) ∈ F[ 1 ,. .. , r ] be a (low rank) polynomial given by an arithmetic circuit where i : 1 ≤ i ≤ r are linear forms, and I = p 1 (x 1),. .. , p n (x n) be a univariate ideal. Given α ∈ F n , the (unique) remainder f (X) (mod I) can be evaluated at α in deterministic time d O(r) • poly(n), where d = max{deg(f), deg(p 1). .. , deg(p n)}. This yields a randomized n O(r) algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n O(r) algorithm for evaluating the permanent of a n × n matrix of rank r, over any field 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

Discrete Mathematics, 1995

Let K be a field; let ~mK" be a finite set and let 3(N)mK[xl ..... x,] be the ideal of ~. A purely combinatorial algorithm to obtain a linear basis of the quotient algebra K Ix1 ..... x,]/3(~) is given. Such a basis is represented by an n-dimensional Ferrers diagram of monomials which is minimal with respect to the inverse lexicographical order ~<i.l.-It is also shown how this algorithm can be extended to the case in which ~ is an algebraic multiset. A few applications are stated (among them, how to determine a reduced Grfbner basis of 3(,~) with respect to %i.1. without using Buchberger's algorithm).

2002

Following ideas from (Hei83, DFGS91, MT97) and applying the tech- niques proposed in (May89, KM96, K¨ uh98), we present a deterministic al- gorithm for computing the dimension of a polynomial ideal requiring poly- nomial working space.

Journal of Symbolic Computation, 2003

We describe a (finite) algorithm to determine the set of characteristics of a system of polynomial equations with integer coefficients by using the theory of Gröbner bases. This gives us a proof that the set of characteristics must be either finite and not containing zero, or containing zero and cofinite. Another, algebraic, proof of this is given in the appendix. These results carry over to systems of polynomial equations over a principal ideal domain and also yields an algorithm for finding the characteristic set of a matroid.

Journal of Symbolic Computation, 2009

The Buchberger-Möller algorithm is a well-known efficient tool for computing the vanishing ideal of a finite set of points. If the coordinates of the points are (imprecise) measured data, the resulting Gröbner basis is numerically unstable. In this paper we introduce a numerically stable Approximate Vanishing Ideal (AVI) Algorithm which computes a set of polynomials that almost vanish at the given points and almost form a border basis. Moreover, we provide a modification of this algorithm which produces a Macaulay basis of an approximate vanishing ideal. We also generalize the Border Basis Algorithm ([Kehrein, A., Kreuzer, M., 2006. Computing border bases. J. Pure Appl. Algebra 205, 279-295]) to the approximate setting and study the approximate membership problem for zero-dimensional polynomial ideals. The algorithms are then applied to actual industrial problems.

Journal of Complexity, 2006

Let F1, F2, . . . , Ft be multivariate polynomials (with complex coefficients) in the variables z 1 , z 2 , . . . , z n . The common zero locus of these polynomials, V (F 1 , F 2 , . . . , F t ) = {p ∈ C n |F i (p) = 0 for 1 ≤ i ≤ t}, determines an algebraic set. This algebraic set decomposes into a union of simpler, irreducible components. The set of polynomials imposes on each component a positive integer known as the multiplicity of the component. Multiplicity plays an important role in many practical applications. It determines roughly "how many times the component should be counted in a computation." Unfortunately, many numerical methods have difficulty in problems where the multiplicity of a component is greater than one. The main goal of this paper is to present an algorithm for determining the multiplicity of a component of an algebraic set. The method sidesteps the numerical stability issues which have obstructed other approaches by incorporating a combined numerical-symbolic technique.

2007

We present an algorithm for computing Gröbner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Möller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Möller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Gröbner bases to the problem of network reconstruction in molecular biology.

Journal of Symbolic Computation, 2012

We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.