8 research outputs found
Dynamic Complexity of Planar 3-connected Graph Isomorphism
Dynamic Complexity (as introduced by Patnaik and Immerman) tries to express
how hard it is to update the solution to a problem when the input is changed
slightly. It considers the changes required to some stored data structure
(possibly a massive database) as small quantities of data (or a tuple) are
inserted or deleted from the database (or a structure over some vocabulary).
The main difference from previous notions of dynamic complexity is that instead
of treating the update quantitatively by finding the the time/space trade-offs,
it tries to consider the update qualitatively, by finding the complexity class
in which the update can be expressed (or made). In this setting, DynFO, or
Dynamic First-Order, is one of the smallest and the most natural complexity
class (since SQL queries can be expressed in First-Order Logic), and contains
those problems whose solutions (or the stored data structure from which the
solution can be found) can be updated in First-Order Logic when the data
structure undergoes small changes.
Etessami considered the problem of isomorphism in the dynamic setting, and
showed that Tree Isomorphism can be decided in DynFO. In this work, we show
that isomorphism of Planar 3-connected graphs can be decided in DynFO+ (which
is DynFO with some polynomial precomputation). We maintain a canonical
description of 3-connected Planar graphs by maintaining a database which is
accessed and modified by First-Order queries when edges are added to or deleted
from the graph. We specifically exploit the ideas of Breadth-First Search and
Canonical Breadth-First Search to prove the results. We also introduce a novel
method for canonizing a 3-connected planar graph in First-Order Logic from
Canonical Breadth-First Search Trees
Tree Tribes and Lower Bounds for Switching Lemmas
We show tight upper and lower bounds for switching lemmas obtained by the
action of random -restrictions on boolean functions that can be expressed as
decision trees in which every vertex is at a distance of at most from some
leaf, also called -clipped decision trees. More specifically, we show the
following:
If a boolean function can be expressed as a -clipped
decision tree, then under the action of a random -restriction , the
probability that the smallest depth decision tree for has depth
greater than is upper bounded by .
For every , there exists a function that can be
expressed as a -clipped decision tree, such that under the action of a
random -restriction , the probability that the smallest depth decision
tree for has depth greater than is lower bounded by
, for and , where are universal
constants
Edge Expansion and Spectral Gap of Nonnegative Matrices
The classic graphical Cheeger inequalities state that if M is an n Γ n symmetric doubly stochastic matrix, then
1βΞ»β(M)/2 β€ Ο(M) β€ β2β
(1βΞ»β(M)) where Ο(M) = min_(Sβ[n],|S|β€n/2)(1|S|β_(iβS,jβS)M_(i,j)) is the edge expansion of M, and Ξ»β(M) is the second largest eigenvalue of M. We study the relationship between Ο(A) and the spectral gap 1 β Re Ξ»β(A) for any doubly stochastic matrix A (not necessarily symmetric), where Ξ»β(A) is a nontrivial eigenvalue of A with maximum real part. Fiedler showed that the upper bound on Ο(A) is unaffected, i.e., Ο(A) β€ β2β
(1βReΞ»β(A)). With regards to the lower bound on Ο(A), there are known constructions with Ο(A) β Ξ(1βReΞ»β(A)/log n) indicating that at least a mild dependence on n is necessary to lower bound Ο(A). In our first result, we provide an exponentially better construction of n Γ n doubly stochastic matrices A_n, for which Ο(An) β€ 1βReΞ»β(A_n)/βn. In fact, all nontrivial eigenvalues of our matrices are 0, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of n), by showing that for any doubly stochastic matrix A, Ο(A) β₯ 1βReΞ»β(A)/35β
n As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of n^Ξ± for Β½ β€ Ξ± β€ 1 in lower bounding Ο by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices R with nonnegative entries, to obtain a two-sided gapped refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such R, there is a nonnegative eigenvalue r such that all eigenvalues of R lie within the closed disk of radius r about 0. Further, if R is irreducible, which means Ο(R) > 0 (for suitably defined Ο), then r is positive and all other eigenvalues lie within the open disk, so (with eigenvalues sorted by real part), Re Ξ»β(R) < r. An extension of Fiedler's result provides an upper bound and our result provides the corresponding lower bound on Ο(R) in terms of r β Re Ξ»β(R), obtaining a two-sided quantitative version of the Perron-Frobenius theorem
Edge Expansion and Spectral Gap of Nonnegative Matrices
The classic graphical Cheeger inequalities state that if is an symmetric doubly stochastic matrix, then
where is the edge expansion of , and
is the second largest eigenvalue of . We study the relationship between
and the spectral gap for any doubly
stochastic matrix (not necessarily symmetric), where is a
nontrivial eigenvalue of with maximum real part. Fiedler showed that the
upper bound on is unaffected, i.e.,
. With regards to the
lower bound on , there are known constructions with
indicating that at least a mild dependence on is necessary to lower bound
.
In our first result, we provide an exponentially better construction of
doubly stochastic matrices , for which
In fact, all
nontrivial eigenvalues of our matrices are , even though the matrices are
highly nonexpanding. We further show that this bound is in the correct range
(up to the exponent of ), by showing that for any doubly stochastic matrix
,
Our second result extends these bounds to general nonnegative matrices ,
obtaining a two-sided quantitative refinement of the Perron-Frobenius theorem
in which the edge expansion (appropriately defined), a quantitative
measure of the irreducibility of , controls the gap between the
Perron-Frobenius eigenvalue and the next-largest real part of any eigenvalue
Behavior of O(log n) Local Commuting Hamiltonians
We study the variant of the k-local hamiltonian problem which is a natural generalization of k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamiltonian H over n qubits, where H is a sum of k-local terms acting non-trivially on O(log n) qubits, and all the k-local terms commute, and show the following -
1. We show that a specific case of O(log n) local commuting hamiltonians over the hypercube is in NP using the Bravyi-Vyalyi Structure theorem.
2. We give a simple proof of a generalized area law for commuting hamiltonians (which seems to be a folklore result) in all dimensions, and deduce the case for O(log n) local commuting hamiltonians.
3. We show that traversing the ground space of O(log n) local commuting hamiltonians is QCMA complete.
The first two behaviours seem to indicate that deciding whether the ground space energy of O(log n)-local commuting hamiltonians is low or high might be in NP or possibly QCMA, though the last behaviour seems to indicate that it may indeed be the case that O(log n)-local commuting hamiltonians are QMA complete. </p
QCMA hardness of ground space connectivity for commuting Hamiltonians
In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian . It was shown in [Gharibian and Sikora, ICALP15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians