New algorithms for linear k -matroid intersection and matroid k -parity problems
TLDR
Algorithms for thek-Matroid Intersection Problem and for the Matroidk-Parity Problem when the matroids are represented over the field of rational numbers andk > 2.Abstract:
We present algorithms for the k-Matroid Intersection Problem and for the Matroid k-Pafity Problem when the matroids are represented over the field of rational numbers and k > 2. The computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids. As an application, we describe new polynomially solvable cases of the k-Dimensional Assignment Problem and of the k-Dimensional Matching Problem. The algorithms use some new identities in mululinear algebra including the generalized Binet-Cauchy formula and its analogue for the Pfaffian. These techniques extend known methods developed earlier for k=2.read more
Citations
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Most Tensor Problems Are NP-Hard
TL;DR: In this paper, it was shown that determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm, approximating an eigen value, eigenvector, singular vector, or the spectral norm is NP-hard and computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.
Posted Content
Most tensor problems are NP-hard
TL;DR: It is proved that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.
Journal ArticleDOI
Algebraic Algorithms for Matching and Matroid Problems
TL;DR: New algebraic approaches for two well-known combinatorial problems: nonbipartite matching and matroid intersection are presented and new randomized algorithms that exceed or match the efficiency of existing algorithms are yielded.
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The Grassmann-Berezin calculus and theorems of the matrix-tree type
TL;DR: In this paper, the Grassmann-Berezin calculus is used to prove two generalizations of the matrix-tree theorem, one of which extends the all minors matrices to the massive matrices, where no condition on row or column sums is imposed.
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Hankel hyperdeterminants and Selberg integrals
TL;DR: In this article, the authors investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1 + i2 +... + ik)).
References
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Book
Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
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The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.