66 research outputs found
Sparse multivariate polynomial interpolation in the basis of Schubert polynomials
Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in
the study of cohomology rings of flag manifolds in 1980's. These polynomials
generalize Schur polynomials, and form a linear basis of multivariate
polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials,
which generalize skew Schur polynomials, and expand in the Schubert basis with
the generalized Littlewood-Richardson coefficients.
In this paper we initiate the study of these two families of polynomials from
the perspective of computational complexity theory. We first observe that skew
Schubert polynomials, and therefore Schubert polynomials, are in \CountP
(when evaluating on non-negative integral inputs) and \VNP.
Our main result is a deterministic algorithm that computes the expansion of a
polynomial of degree in in the basis of Schubert
polynomials, assuming an oracle computing Schubert polynomials. This algorithm
runs in time polynomial in , , and the bit size of the expansion. This
generalizes, and derandomizes, the sparse interpolation algorithm of symmetric
polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied
Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general
enough to accommodate any linear basis satisfying certain natural properties.
Applications of the above results include a new algorithm that computes the
generalized Littlewood-Richardson coefficients.Comment: 20 pages; some typos correcte
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers
We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and Qiao et al. (STACS 2011), in this paper we design a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.
The mathematical tools required include representation theory,
Wedderburn\u27s theorem on semisimple algebras, and M.E. Harris\u27s 1980
work on p\u27-automorphisms of abelian p-groups. We use tools from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest
Tur\'an and Ramsey problems for alternating multilinear maps
Guided by the connections between hypergraphs and exterior algebras, we study
Tur\'an and Ramsey type problems for alternating multilinear maps. This study
lies at the intersection of combinatorics, group theory, and algebraic
geometry, and has origins in the works of Lov\'asz (Proc. Sixth British
Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and
Feldman and Propp (Adv. Math., 1992).
Our main result is a Ramsey theorem for alternating bilinear maps. Given , , and an alternating bilinear map with , we show that there exists either a dimension-
subspace such that , or a dimension- subspace
such that . This result has natural
group-theoretic (for finite -groups) and geometric (for Grassmannians)
implications, and leads to new Ramsey-type questions for varieties of groups
and Grassmannians.Comment: 20 pages. v3: rewrite introductio
Deterministic Black-Box Identity Testing -Ordered Algebraic Branching Programs
In this paper we study algebraic branching programs (ABPs) with restrictions
on the order and the number of reads of variables in the program. Given a
permutation of variables, for a -ordered ABP (-OABP), for
any directed path from source to sink, a variable can appear at most once
on , and the order in which variables appear on must respect . An
ABP is said to be of read , if any variable appears at most times in
. Our main result pertains to the identity testing problem. Over any field
and in the black-box model, i.e. given only query access to the polynomial,
we have the following result: read -OABP computable polynomials can be
tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}].
Our next set of results investigates the computational limitations of OABPs.
It is shown that any OABP computing the determinant or permanent requires size
and read . We give a multilinear polynomial
in variables over some specifically selected field , such that
any OABP computing must read some variable at least times. We show
that the elementary symmetric polynomial of degree in variables can be
computed by a size read OABP, but not by a read OABP, for
any . Finally, we give an example of a polynomial and two
variables orders , such that can be computed by a read-once
-OABP, but where any -OABP computing must read some variable at
least $2^n
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