Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

Loera, Jesús A. De; Lee, Jon; Malkin, Peter N.; Margulies, Susan S.

doi:10.1145/1390768.1390797

Cited by 23 publications

(42 citation statements)

References 18 publications

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“…Relevant works include Nesterov [141], Parrilo [142], Lasserre [47], Laurent [143], and De Loera et al [144]. The method of Lasserre [145] works for integer polynomial programs when each variable has an explicit lower and upper bound.…”

Section: Extensions To Polynomial Optimizationmentioning

confidence: 99%

Non-convex mixed-integer nonlinear programming: A survey

Burer

Letchford

2012

Surveys in Operations Research and Management Science

379

238

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Section: Extensions To Polynomial Optimizationmentioning

confidence: 99%

Non-convex mixed-integer nonlinear programming: A survey

Burer

Letchford

2012

Surveys in Operations Research and Management Science

379

238

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“…This well-known method, which Alon referred to as the polynomial method [3,4] recently regained strong interest and emphasizes the alternative view of border bases algorithms in their various incarnations as proof systems which successively uncovers hidden information by making it explicit. In [23,Section 2.3] and [21,22,25] infeasibility of certain combinatorial problems, e.g., 3-colorability of graphs is established using Hilbert's Nullstellensatz and the algorithm NulLA is provided to establish infeasibility by using a linear relaxation. The core of the algorithm is identical to the L-stable span procedure used in the border basis algorithm in [35], which intimately links both procedures.…”

Section: Computing Border Basesmentioning

confidence: 99%

A Polyhedral Characterization of Border Bases

Braun¹,

Pokutta²

2016

SIAM J. Discrete Math.

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A. Border bases arise as a canonical generalization of Gröbner bases, using order ideals instead of term orderings. We provide a polyhedral characterization of all order ideals (and hence all border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we also provide an adaptation of the border basis algorithm of Kehrein and Kreuzer [35] to allow for computing border bases for arbitrary order ideals, given implicitly via maximizing a preference on monomials (variable selection problem), independent of term orderings. The algorithm requires the same size of resources as the border basis algorithm except for some minor overhead. We also show that the underlying variable selection problem of finding an order ideal that supports a border basis is NP-hard and that any linear description of the associated convex hull of all order ideals requires a superpolynomial number of inequalities. IIn many different disciplines and real-world applications one is faced with solving systems of polynomial equations. Often this is simply due to a physical or dynamical system having a natural representation as a system of polynomial equations, but equally often it is due to the sheer expressive power of polynomial systems of equations that allow for easy reformulation. To give an example of the latter, an inequality ax ≤ b with a ∈ R n and b ∈ R can be expressed via a single polynomial equation: ax + u 2 = b. A slightly more involved example is that of the feasible region of a binary program {x | Ax ≤ b, x ∈ {0, 1} n }, which can be captured via rewriting each individual inequality as before, and adding quadratic polynomials x 2 i − x i = 0 for each coordinate i = 1, . . . , n of x. As a consequence, there is a huge need to computationally model, understand, manipulate, and extract the solution set of systems of polynomial equations.A key insight in (computational) commutative algebra is that one can choose a smart ordering on the monomials and compute a special set of generators of the ideal generated by the system of equations that makes many operations easy, and provides a structural insight into the system. One such special set of generators is a Gröbner basis. By now, Gröbner bases are fundamental and standard tools in commutative algebra to actually perform important operations on ideals such as intersection, membership test, elimination, projection, and many more. Border bases arise as a natural generalization of Gröbner bases that can be computed for zero-dimensional ideals, i.e., the associated factor space is a finite-dimensional vector space (see Section 2 for details). While this might seem to be a severe restriction, for many applications it is sufficient. Roughly speaking, whenever the solution set is finite, we are dealing with a zero-dimensional ideal. For example, systems of p...

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“…By putting degree truncations on the certificates, we can transform the theoretic statements into effective algorithmic procedures for constructing certificates. The idea of degree truncations in polynomial identities follows the same principles of the degree truncations with various types of Nullstellen-and Positivstellensätze in [4,6,15]. It is instructive to have a look at two simple examples first.…”

Section: Approximations Based On the Real Nullstellensatzmentioning

confidence: 99%

“…Let G 1,3 denote the Grassmannian of lines in 3-space, which is the variety in P 5 C , defined by P 01 P 23 − P 02 P 13 + P 03 P 12 = 0 , which we consider as a subvariety of (C * ) 6 . The three terms in this quadratic equation involve distinct variables and hence correspond to linearly independent exponent vectors.…”

Section: 1])mentioning

confidence: 99%

Approximating amoebas and coamoebas by sums of squares

Theobald

Wolff

2014

Math. Comp.

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Abstract. Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming.Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of non-containment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.

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Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

Cited by 23 publications

References 18 publications

Non-convex mixed-integer nonlinear programming: A survey

Non-convex mixed-integer nonlinear programming: A survey

A Polyhedral Characterization of Border Bases

Approximating amoebas and coamoebas by sums of squares

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