5 research outputs found
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
Super-polylogarithmic hypergraph coloring hardness via low-degree long codes
We prove improved inapproximability results for hypergraph coloring using the
low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012])
and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate
this code for inapproximability results. In particular, we prove
quasi-NP-hardness of the following problems on -vertex hyper-graphs:
* Coloring a 2-colorable 8-uniform hypergraph with
colors.
* Coloring a 4-colorable 4-uniform hypergraph with
colors.
* Coloring a 3-colorable 3-uniform hypergraph with colors.
In each of these cases, the hardness results obtained are (at least)
exponentially stronger than what was previously known for the respective cases.
In fact, prior to this result, polylog n colors was the strongest quantitative
bound on the number of colors ruled out by inapproximability results for
O(1)-colorable hypergraphs.
The fundamental bottleneck in obtaining coloring inapproximability results
using the low- degree long code was a multipartite structural restriction in
the PCP construction of Dinur-Guruswami. We are able to get around this
restriction by simulating the multipartite structure implicitly by querying
just one partition (albeit requiring 8 queries), which yields our result for
2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform
hypergraphs is obtained via a 'query doubling' method. For 3-colorable
3-uniform hypergraphs, we exploit the ternary domain to design a test with an
additive (as opposed to multiplicative) noise function, and analyze its
efficacy in killing high weight Fourier coefficients via the pseudorandom
properties of an associated quadratic form.Comment: 25 page
Some closure features of locally testable affine-invariant properties
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 31-32).We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and "lifts". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The "lift" is a less well-studied property, which creates some interesting affine-invariant properties over large domains, from properties over smaller domains. Previously such results were known for "single-orbit characterized" affine-invariant properties, which are known to be a subclass of locally testable ones, and are potentially a strict subclass. The fact that the intersection of locally-testable affine-invariant properties are locally testable could have been derived from previously known general results on closure of property testing under set-theoretic operations, but was not explicitly observed before. The closure under sum and lifts is implied by an affirmative answer to a central question attempting to characterize locally testable affine-invariant properties, but the status of that question remains wide open. Affine-invariant properties are clean abstractions of commonly studied, and extensively used, algebraic properties such linearity and low-degree. Thus far it is not known what makes affine-invariant properties locally testable - no characterizations are known, and till this work it was not clear if they satisfied any closure properties. This work shows that the class of locally testable affine-invariant properties are closed under some very natural operations. Our techniques use ones previously developed for the study of "single-orbit characterized" properties, but manage to apply them to the potentially more general class of all locally testable ones via a simple connection that may be of broad interest in the study of affine-invariant properties.by Alan Xinyu Guo.S.M
Optimal testing of multivariate polynomials over small prime fields
Abstract — We consider the problem of testing if a given function f: F n q → Fq is close to a n-variate degree d polynomial over the finite field Fq of q elements. The natural, low-query, test for this property would be to pick the smallest dimension t = tq,d ≈ d/q such that every function of degree greater than d reveals this aspect on some t-dimensional affine subspace of F n q and to test that f when restricted to a random t-dimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q t queries, independent of n. Previous works, by Alon et al. [1], and Kaufman and Ron [6] and Jutla et al. [5], showed that this natural test rejected functions that were Ω(1)-far from degree d-polynomials with probability at least Ω(q −t). (The initial work [1] considered only the case of q = 2, while the work [5] only considered the case of prime q. The results in [6] hold for all fields.) Thus to get a constant probability o