816 research outputs found
Characterization and Lower Bounds for Branching Program Size using Projective Dimension
We study projective dimension, a graph parameter (denoted by pd for a
graph ), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving
lower bounds for pd for bipartite graphs associated with a Boolean
function imply size lower bounds for branching programs computing .
Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy
2000), proving super-linear lower bounds for projective dimension of explicit
families of graphs has remained elusive.
We show that there exist a Boolean function (on bits) for which the
gap between the projective dimension and size of the optimal branching program
computing (denoted by bpsize), is . Motivated by the
argument in (Pudl\'ak, R\"odl 1992), we define two variants of projective
dimension - projective dimension with intersection dimension 1 (denoted by
upd) and bitwise decomposable projective dimension (denoted by
bitpdim).
As our main result, we show that there is an explicit family of graphs on vertices such that the projective dimension is , the
projective dimension with intersection dimension is and the
bitwise decomposable projective dimension is .
We also show that there exist a Boolean function (on bits) for which
the gap between upd and bpsize is . In contrast, we
also show that the bitwise decomposable projective dimension characterizes size
of the branching program up to a polynomial factor. That is, there exists a
constant and for any function , . We also study two other
variants of projective dimension and show that they are exactly equal to
well-studied graph parameters - bipartite clique cover number and bipartite
partition number respectively.Comment: 24 pages, 3 figure
Variety Membership Testing in Algebraic Complexity Theory
In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r
Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness
We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field ?, in the blackbox setting.
Our algorithm works for every field ? with | ? | ? d+1, and uses only d log n + log (1/ ?) + O(d log log n) random bits to achieve a success probability 1 - ? for some ? > 0. In the low degree regime that is d ? n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC\u2714 and Bshouty, ITCS\u2714) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions
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Representations of Finite Groups
The workshop Representations of Finite Groups was organised by Joseph Chuang (London), Meinolf Geck (Stuttgart), Markus Linckelmann (London), and Gabriel Navarro (Valencia). It covered a wide variety of aspects of the representation theory of finite groups and related objects, such as algebraic groups
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
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