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Faster Parameterized Algorithms for Deletion to Split Graphs
TLDR
A systematic study of parameterized complexity of the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split and an efficient fixed-parameter algorithms and polynomial sized kernels are given.
Towards an algebraic natural proofs barrier via polynomial identity testing
TLDR
It is observed that a certain kind of algebraic proof cannot be used to prove lower bounds against VP if and only if what the authors call succinct hitting sets exist for VP.
On the Power of Homogeneous Depth 4 Arithmetic Circuits
TLDR
It is shown that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension nO(1) must have size nΩ(√n) and the results strengthen previous works in two significant ways.
Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields
TLDR
The first polynomial time algorithm for indexing necklaces is given, giving a poly(n, log|Σ|)-time computable bijection between \(\{1, \ldots, |\cal N|\}\) and the set of all necklace of length n over a finite alphabet Σ.
The limits of depth reduction for arithmetic formulas: it's all about the top fan-in
TLDR
The results in particular suggest that the method of improved depth reduction and shifted partial derivatives may not be powerful enough to prove superpolynomial lower bounds for (even homogeneous) arithmetic formulas.
Arithmetic Circuits with Locally Low Algebraic Rank
TLDR
A key technical ingredient of the proofs, which may be of independent interest, is a result which states that over any field of characteristic zero, up to a translation, every polynomial in a set of polynomials can be written as a function of the polynOMials in a transcendence basis of the set.
A quadratic lower bound for homogeneous algebraic branching programs
  • Mrinal Kumar
  • Mathematics, Computer Science
    computational complexity
  • 8 June 2019
TLDR
It is shown that any homogeneous algebraic branching program which computes the polynomial x1n+x2n+⋯+xnn has at least Ω(n2) vertices (and hence edges), which seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound.
Some Closure Results for Polynomial Factorization and Applications
TLDR
It is shown that a superpolynomial lower bound for bounded depth arithmetic circuits, for a family of explicit polynomials of degree poly$(\log n)$ implies deterministic sub-exponential time algorithms for polynomial identity testing (PIT) for bounded Depth arithmetic circuits.
Lower Bounds for Matrix Factorization
TLDR
An approach is outlined for proving improved lower bounds through a certain derandomization problem, and this approach is used to prove asymptotically optimal quadratic lower bounds for natural special cases, which generalize many of the common matrix decompositions.
Faster Parameterized Algorithms for Deletion to Split Graphs
TLDR
The problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split is studied and efficient fixed-parameter algorithms and polynomial sized kernels for the problem are given.
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