3,172 research outputs found
Cutoff phenomenon for the simple exclusion process on the complete graph
We study the time that the simple exclusion process on the complete graph
needs to reach equilibrium in terms of total variation distance. For the graph
with n vertices and 1<<k<n/2 particles we show that the mixing time is of order
(n/2)\log \min(k, \sqrt{n}), and that around this time, for any small positive
epsilon the total variation distance drops from 1-epsilon to epsilon in a time
window whose width is of order n (i.e. in a much shorter time). Our proof is
purely probabilistic and self-contained.Comment: 16 pages, to appear in ALE
Optimal Lower Bound for Itemset Frequency Indicator Sketches
Given a database, a common problem is to find the pairs or -tuples of
items that frequently co-occur. One specific problem is to create a small space
"sketch" of the data that records which -tuples appear in more than an
fraction of rows of the database.
We improve the lower bound of Liberty, Mitzenmacher, and Thaler [LMT14],
showing that bits are necessary
even in the case of . This matches the sampling upper bound for all
, and (in the case of ) another trivial upper
bound for .Comment: 3 page
Quantum Algorithms for Learning and Testing Juntas
In this article we develop quantum algorithms for learning and testing
juntas, i.e. Boolean functions which depend only on an unknown set of k out of
n input variables. Our aim is to develop efficient algorithms:
- whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
- with no access to any classical or quantum membership ("black-box")
queries. Instead, our algorithms use only classical examples generated
uniformly at random and fixed quantum superpositions of such classical
examples;
- which require only a few quantum examples but possibly many classical
random examples (which are considered quite "cheap" relative to quantum
examples).
Our quantum algorithms are based on a subroutine FS which enables sampling
according to the Fourier spectrum of f; the FS subroutine was used in earlier
work of Bshouty and Jackson on quantum learning. Our results are as follows:
- We give an algorithm for testing k-juntas to accuracy that uses
quantum examples. This improves on the number of examples used
by the best known classical algorithm.
- We establish the following lower bound: any FS-based k-junta testing
algorithm requires queries.
- We give an algorithm for learning -juntas to accuracy that
uses quantum examples and
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum
Information Processin
Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses
We present a series of almost settled inapproximability results for three
fundamental problems. The first in our series is the subexponential-time
inapproximability of the maximum independent set problem, a question studied in
the area of parameterized complexity. The second is the hardness of
approximating the maximum induced matching problem on bounded-degree bipartite
graphs. The last in our series is the tight hardness of approximating the
k-hypergraph pricing problem, a fundamental problem arising from the area of
algorithmic game theory. In particular, assuming the Exponential Time
Hypothesis, our two main results are:
- For any r larger than some constant, any r-approximation algorithm for the
maximum independent set problem must run in at least
2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of
2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the
domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et
al., 2013)
- For any k larger than some constant, there is no polynomial time min
(k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph
pricing problem, where n is the number of vertices in an input graph. This
almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and
Blum, 2007 and an algorithm in this paper).
We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness
for polynomial-time algorithms, the k-hypergraph pricing problem admits
n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts
this problem in a rare approximability class in which approximability
thresholds can be improved significantly by allowing algorithms to run in
quasi-polynomial time.Comment: The full version of FOCS 201
Twins in words and long common subsequences in permutations
A large family of words must contain two words that are similar. We
investigate several problems where the measure of similarity is the length of a
common subsequence.
We construct a family of n^{1/3} permutations on n letters, such that LCS of
any two of them is only cn^{1/3}, improving a construction of Beame, Blais, and
Huynh-Ngoc. We relate the problem of constructing many permutations with small
LCS to the twin word problem of Axenovich, Person and Puzynina. In particular,
we show that every word of length n over a k-letter alphabet contains two
disjoint equal subsequences of length cnk^{-2/3}.
Many problems are left open.Comment: 18+epsilon page
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