16,949 research outputs found
Locally Testable Codes and Cayley Graphs
We give two new characterizations of (\F_2-linear) locally testable
error-correcting codes in terms of Cayley graphs over \F_2^h:
\begin{enumerate} \item A locally testable code is equivalent to a Cayley
graph over \F_2^h whose set of generators is significantly larger than
and has no short linear dependencies, but yields a shortest-path metric that
embeds into with constant distortion. This extends and gives a
converse to a result of Khot and Naor (2006), which showed that codes with
large dual distance imply Cayley graphs that have no low-distortion embeddings
into .
\item A locally testable code is equivalent to a Cayley graph over \F_2^h
that has significantly more than eigenvalues near 1, which have no short
linear dependencies among them and which "explain" all of the large
eigenvalues. This extends and gives a converse to a recent construction of
Barak et al. (2012), which showed that locally testable codes imply Cayley
graphs that are small-set expanders but have many large eigenvalues.
\end{enumerate}Comment: 22 page
Testing Booleanity and the Uncertainty Principle
Let f:{-1,1}^n -> R be a real function on the hypercube, given by its
discrete Fourier expansion, or, equivalently, represented as a multilinear
polynomial. We say that it is Boolean if its image is in {-1,1}.
We show that every function on the hypercube with a sparse Fourier expansion
must either be Boolean or far from Boolean. In particular, we show that a
multilinear polynomial with at most k terms must either be Boolean, or output
values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its
domain.
It follows that given oracle access to f, together with the guarantee that
its representation as a multilinear polynomial has at most k terms, one can
test Booleanity using O(k^2) queries. We show an \Omega(k) queries lower bound
for this problem.
Our proof crucially uses Hirschman's entropic version of Heisenberg's
uncertainty principle.Comment: 15 page
Boolean function monotonicity testing requires (almost) non-adaptive queries
We prove a lower bound of , for all , on the query
complexity of (two-sided error) non-adaptive algorithms for testing whether an
-variable Boolean function is monotone versus constant-far from monotone.
This improves a lower bound for the same problem that
was recently given in [CST14] and is very close to , which we
conjecture is the optimal lower bound for this model
Public projects, Boolean functions and the borders of Border's theorem
Border's theorem gives an intuitive linear characterization of the feasible
interim allocation rules of a Bayesian single-item environment, and it has
several applications in economic and algorithmic mechanism design. All known
generalizations of Border's theorem either restrict attention to relatively
simple settings, or resort to approximation. This paper identifies a
complexity-theoretic barrier that indicates, assuming standard complexity class
separations, that Border's theorem cannot be extended significantly beyond the
state-of-the-art. We also identify a surprisingly tight connection between
Myerson's optimal auction theory, when applied to public project settings, and
some fundamental results in the analysis of Boolean functions.Comment: Accepted to ACM EC 201
Negative weights make adversaries stronger
The quantum adversary method is one of the most successful techniques for
proving lower bounds on quantum query complexity. It gives optimal lower bounds
for many problems, has application to classical complexity in formula size
lower bounds, and is versatile with equivalent formulations in terms of weight
schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on
the principle that if an algorithm successfully computes a function then, in
particular, it is able to distinguish between inputs which map to different
values.
We present a stronger version of the adversary method which goes beyond this
principle to make explicit use of the stronger condition that the algorithm
actually computes the function. This new method, which we call ADV+-, has all
the advantages of the old: it is a lower bound on bounded-error quantum query
complexity, its square is a lower bound on formula size, and it behaves well
with respect to function composition. Moreover ADV+- is always at least as
large as the adversary method ADV, and we show an example of a monotone
function for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing
that ADV+- does not face limitations of ADV like the certificate complexity
barrier and the property testing barrier.Comment: 29 pages, v2: added automorphism principle, extended to non-boolean
functions, simplified examples, added matching upper bound for AD
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
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