Combinatorial Resultants in the Algebraic Rigidity Matroid

computational complexity, 2013

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant [Val77], rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all n × n matrices over an infinite field have a rigidity of (n − r) 2 . It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Ω(n).

2013

We study the number of hamiltonian circuits, containing a fixed basis, and the number of hyperplanes, which do not contain a fixed basis in perfect matroid designs. Projective and affine finite geometries are considered as examples of such matroids. We give algorithms to find the hyperplanes and the hamiltonian circuits in such cases.

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.

arXiv (Cornell University), 2021

A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such as graphs, matrices, codes and projective geometries. In this work, we define cyclic matroids as matroids over a ground set of size n whose automorphism group contains an n-cycle. We study the properties of such matroids, with special focus on the minimum size of their basis sets. For this, we broadly employ two different approaches: the multiple basis exchange property, and an orbit-stabilizer method, developed by analyzing the action of the cyclic group of order n on the set of bases. We further present some applications of our theory to algebra and geometry, presenting connections to cyclic projective planes, cyclic codes and k-normal elements.

Czechoslovak Mathematical Journal, 2006

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2008

Algebraic circuits combine operations drawn from an algebraic system. In this chapter we develop algebraic and combinatorial circuits for a variety of generally non-Boolean problems, including multiplication and inversion of matrices, convolution, the discrete Fourier transform, and sorting networks. These problems are used primarily to illustrate concepts developed in later chapters, so that this chapter may be used for reference when studying those chapters. For each of the problems examined here the natural algorithms are straight-line and the graphs are directed and acyclic; that is, they are circuits. Not only are straight-line algorithms the ones typically used for these problems, but in some cases they are the best possible. The quality of the circuits developed here is measured by circuit size, the number of circuit operations, and circuit depth, the length of the longest path between input and output vertices. Circuit size is a measure of the work necessary to execute the c...

Arxiv preprint math/0503050, 2005

Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M . The ideas of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this wider setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial), and have little to do with the special nature of graphic matroids or the field R.

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.

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).

arXiv preprint arXiv:1108.5985, 2011

The resultant is the most fundamental tool in algebraic variable elimination; it captures the solvability of an overconstrained polynomial system and yields efficient algorithms for system solving and the implicitization of parametric (hyper) surfaces, to name our main motivation only. Sparse elimination theory characterizes polynomials by their Newton polytope (convex hull of the exponent vectors of nonzero monomials). The (Newton) polytope of a system&#x27;s resultant can be recovered from the secondary polytope ( ...

Lecture Notes in Computer Science, 2007

According to the present state of the theory of the matroid matching problem, the existence of a good characterization to the size of a maximum matching depends on the behavior of certain substructures, called double circuits. In this paper we prove that if a polymatroid has no double circuits at all, then a partition-type min-max formula characterizes the size of a maximum matching. We provide applications of this result to parity constrained orientations and to a rigidity problem.

2019

This text is a development of a preprint of Ankit Gupta. We present an approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-$4$ circuits with bounded top fanin. This approach is similar to Kayal-Shubhangi approach for depth-$3$ circuits. Kayal and Shubhangi based their algorithm on Sylvester-Gallai-type theorem about linear polynomials. We show how it is possible to generalize this approach to depth-$4$ circuits. However we failed to implement this plan completely. We succeeded to construct a polynomial time deterministic algorithm for depth-$4$ circuits with bounded top fanin and its correctness requires a hypothesis. Also we present a polynomial-time (unconditional) algorithm for some subclass of depth-$4$ circuits with bounded top fanin.

arXiv: Computational Complexity, 2017

Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence oracle.

Lecture Notes in Computer Science, 2010

The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, to yield interesting upper bounds on the number of embeddings. We focus on R 2 and R 3 , where Laman graphs and 1-skeleta of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. We establish the first lower bound in R 3 of about 2.52 n , where n denotes the number of vertices. Moreover, our implementation yields upper bounds for n ≤ 10 in R 2 and R 3 , which reduce the existing gaps, and tight bounds up to n = 7 in R 3 .

Advances in Applied Mathematics, 2002

Let G be a matroid on ground set A. The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra E on A by the ideal I generated by circuit boundaries. The quadratic closure A(G) of A(G) is the quotient of E by the ideal generated by the degree-two component of I. We introduce the notion of nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. These results generalize to the degree r closure of A(G).