58 research outputs found
Realizability of Polytopes as a Low Rank Matrix Completion Problem
This article gives necessary and sufficient conditions for a relation to be
the containment relation between the facets and vertices of a polytope. Also
given here, are a set of matrices parameterizing the linear moduli space and
another set parameterizing the projective moduli space of a combinatorial
polytope
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
Complexity of the positive semidefinite matrix completion problem with a rank constraint.
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is NP-hard for any fixed integer k ≥ 2. Equivalently, for k ≥ 2, it is NP-hard to test membership in the rank constrained elliptope Ek(G), i.e., the set of all partial matrices with off-diagonal entries specified at the edges of G, that can be completed to a positive semidefinite matrix of rank at most k. Additionally, we show that deciding membership in the convex hull of Ek(G) is also NP-hard for any fixed integer k ≥ 2
Selected Open Problems in Discrete Geometry and Optimization
A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers
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