Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms , directed homomorphisms , and injective directed homomorphisms , and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: • The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. • All the polynomial families are hard under p -projections.

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Succinct Algebraic Branching Programs Characterizing Non-uniform Complexity Classes

Fundamentals of Computation Theory - Lecture Notes in Computer Science ◽

10.1007/978-3-642-22953-4_18 ◽

2011 ◽

pp. 205-216 ◽

Cited By ~ 3

Author(s):

Guillaume Malod

Keyword(s):

Complexity Classes ◽

Branching Programs ◽

Algebraic Branching Programs

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Average-case linear matrix factorization and reconstruction of low width algebraic branching programs

Computational Complexity ◽

10.1007/s00037-019-00189-0 ◽

2019 ◽

Vol 28 (4) ◽

pp. 749-828 ◽

Cited By ~ 1

Author(s):

Neeraj Kayal ◽

Vineet Nair ◽

Chandan Saha

Keyword(s):

Matrix Factorization ◽

Linear Matrix ◽

Branching Programs ◽

Average Case ◽

Algebraic Branching Programs

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The Computational Complexity of Plethysm Coefficients

Computational Complexity ◽

10.1007/s00037-020-00198-4 ◽

2020 ◽

Vol 29 (2) ◽

Author(s):

Nick Fischer ◽

Christian Ikenmeyer

Keyword(s):

Matrix Multiplication ◽

Complexity Classes ◽

Algebraic Complexity ◽

Test Bed ◽

Discrete Tomography ◽

Geometric Complexity Theory ◽

Special Cases ◽

The Matrix ◽

Kronecker Coefficients ◽

Coordinate Rings

AbstractIn two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity.In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.

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