142,338 research outputs found
Composition problems for braids: Membership, Identity and Freeness
In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, , have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-complete for braids with only three strands. The membership problem is decidable in NP for , but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least five strands, but decidability of these problems for remains open. Finally we show that the freeness problem for semigroups of braids from is also decidable in NP. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms
On the Expansion of Group-Based Lifts
A -lift of an -vertex base graph is a graph on
vertices, where each vertex of is replaced by vertices
and each edge in is replaced by a matching
representing a bijection so that the edges of are of the form
. Lifts have been studied as a means to efficiently
construct expanders. In this work, we study lifts obtained from groups and
group actions. We derive the spectrum of such lifts via the representation
theory principles of the underlying group. Our main results are:
(1) There is a constant such that for every , there
does not exist an abelian -lift of any -vertex -regular base graph
with being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix
at most in magnitude). This can be viewed as an analogue of the
well-known no-expansion result for abelian Cayley graphs.
(2) A uniform random lift in a cyclic group of order of any -vertex
-regular base graph , with the nontrivial eigenvalues of the adjacency
matrix of bounded by in magnitude, has the new nontrivial
eigenvalues also bounded by in magnitude with probability
. In particular, there is a constant such that for
every , there exists a lift of every Ramanujan graph in
a cyclic group of order with being almost Ramanujan. We use this to
design a quasi-polynomial time algorithm to construct almost Ramanujan
expanders deterministically.
The existence of expanding lifts in cyclic groups of order
can be viewed as a lower bound on the order of the largest abelian group
that produces expanding lifts. Our results show that the lower bound matches
the upper bound for (upto in the exponent)
Homomorphic encryption and some black box attacks
This paper is a compressed summary of some principal definitions and concepts
in the approach to the black box algebra being developed by the authors. We
suggest that black box algebra could be useful in cryptanalysis of homomorphic
encryption schemes, and that homomorphic encryption is an area of research
where cryptography and black box algebra may benefit from exchange of ideas
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
In this paper we show that certain special cases of the hidden subgroup
problem can be solved in polynomial time by a quantum algorithm. These special
cases involve finding hidden normal subgroups of solvable groups and
permutation groups, finding hidden subgroups of groups with small commutator
subgroup and of groups admitting an elementary Abelian normal 2-subgroup of
small index or with cyclic factor group.Comment: 10 page
An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem
In this paper we consider the problem of testing whether two finite groups
are isomorphic. Whereas the case where both groups are abelian is well
understood and can be solved efficiently, very little is known about the
complexity of isomorphism testing for nonabelian groups. Le Gall has
constructed an efficient classical algorithm for a class of groups
corresponding to one of the most natural ways of constructing nonabelian groups
from abelian groups: the groups that are extensions of an abelian group by
a cyclic group with the order of coprime with . More precisely,
the running time of that algorithm is almost linear in the order of the input
groups. In this paper we present a quantum algorithm solving the same problem
in time polynomial in the logarithm of the order of the input groups. This
algorithm works in the black-box setting and is the first quantum algorithm
solving instances of the nonabelian group isomorphism problem exponentially
faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the
Proceedings of the 27th International Symposium on Theoretical Aspects of
Computer Science (STACS 2010
Fast Monte Carlo Algorithms for Permutation Groups
AbstractWe introduce new, elementary Monte Carlo methods to speed up and greatly simplify the manipulation of permutation groups (given by a list of generators). The methods are of a combinatorial character, using only elementary group theory. The key idea is that under certain conditions, "random subproducts" of the generators successfully emulate truly random elements of a group. We achieve a nearly optimal O(n3 logcn) asymptotic running time for membership testing, where n is the size of the permutation domain. This is an improvement of two orders of magnitude compared to known elementary algorithms and one order of magnitude compared to algorithms which depend on heavy use of group theory. An even greater asymptotic speedup is achieved for normal closures, a key ingredient in group-theoretic computation, now constructible in Monte Carlo time O(n2 logcn), i.e., essentially linear time (as a function of the input length). Some of the new techniques are sufficiently general to allow polynomial-time implementations in the very general model of "black box groups" (group operations are performed by an oracle). In particular, the normal closure algorithm has a number of applications to matrix-group computation. It should be stressed that our randomized algorithms are not heuristic: the probability of error is guaranteed not to exceed a bound ϵ > 0, prescribed by the user. The cost of this requirement is a factor of |log ϵ| in the running time
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
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