84 research outputs found
Short lists with short programs in short time - a short proof
Bauwens, Mahklin, Vereshchagin and Zimand [ECCC TR13-007] and Teutsch
[arxiv:1212.6104] have shown that given a string x it is possible to construct
in polynomial time a list containing a short description of it. We simplify
their technique and present a shorter proof of this result
Hardness of approximation for quantum problems
The polynomial hierarchy plays a central role in classical complexity theory.
Here, we define a quantum generalization of the polynomial hierarchy, and
initiate its study. We show that not only are there natural complete problems
for the second level of this quantum hierarchy, but that these problems are in
fact hard to approximate. Using these techniques, we also obtain hardness of
approximation for the class QCMA. Our approach is based on the use of
dispersers, and is inspired by the classical results of Umans regarding
hardness of approximation for the second level of the classical polynomial
hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of
approximation for include, among others, a quantum version of the Succinct Set
Cover problem, and a variant of the local Hamiltonian problem with hybrid
classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the
39th International Colloquium on Automata, Languages and Programming (ICALP),
pages 387-398, Springer, 201
Entanglement vs. gap for one-dimensional spin systems
We study the relationship between entanglement and spectral gap for local
Hamiltonians in one dimension. The area law for a one-dimensional system states
that for the ground state, the entanglement of any interval is upper-bounded by
a constant independent of the size of the interval. However, the possible
dependence of the upper bound on the spectral gap Delta is not known, as the
best known general upper bound is asymptotically much larger than the largest
possible entropy of any model system previously constructed for small Delta. To
help resolve this asymptotic behavior, we construct a family of one-dimensional
local systems for which some intervals have entanglement entropy which is
polynomial in 1/Delta, whereas previously studied systems, such as free fermion
systems or systems described by conformal field theory, had the entropy of all
intervals bounded by a constant times log(1/Delta).Comment: 16 pages. v2 is final published version with slight clarification
Noise-Resilient Group Testing: Limitations and Constructions
We study combinatorial group testing schemes for learning -sparse Boolean
vectors using highly unreliable disjunctive measurements. We consider an
adversarial noise model that only limits the number of false observations, and
show that any noise-resilient scheme in this model can only approximately
reconstruct the sparse vector. On the positive side, we take this barrier to
our advantage and show that approximate reconstruction (within a satisfactory
degree of approximation) allows us to break the information theoretic lower
bound of that is known for exact reconstruction of
-sparse vectors of length via non-adaptive measurements, by a
multiplicative factor .
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with measurements, that allow efficient
reconstruction of -sparse vectors up to false positives even in the
presence of false positives and false negatives within the
measurement outcomes, for any constant . We show that, information
theoretically, none of these parameters can be substantially improved without
dramatically affecting the others. Furthermore, we obtain several explicit
constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions
that allow fast reconstruction in time \poly(m), which would be sublinear in
for sufficiently sparse vectors. The main tool used in our construction is
the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the
same title) in proceedings of the 17th International Symposium on
Fundamentals of Computation Theory (FCT 2009
Evacuating Two Robots from a Disk: A Second Cut
We present an improved algorithm for the problem of evacuating two robots
from the unit disk via an unknown exit on the boundary. Robots start at the
center of the disk, move at unit speed, and can only communicate locally. Our
algorithm improves previous results by Brandt et al. [CIAC'17] by introducing a
second detour through the interior of the disk. This allows for an improved
evacuation time of . The best known lower bound of was shown by
Czyzowicz et al. [CIAC'15].Comment: 19 pages, 5 figures. This is the full version of the paper with the
same title accepted in the 26th International Colloquium on Structural
Information and Communication Complexity (SIROCCO'19
Byzantine Gathering in Networks
This paper investigates an open problem introduced in [14]. Two or more
mobile agents start from different nodes of a network and have to accomplish
the task of gathering which consists in getting all together at the same node
at the same time. An adversary chooses the initial nodes of the agents and
assigns a different positive integer (called label) to each of them. Initially,
each agent knows its label but does not know the labels of the other agents or
their positions relative to its own. Agents move in synchronous rounds and can
communicate with each other only when located at the same node. Up to f of the
agents are Byzantine. A Byzantine agent can choose an arbitrary port when it
moves, can convey arbitrary information to other agents and can change its
label in every round, in particular by forging the label of another agent or by
creating a completely new one.
What is the minimum number M of good agents that guarantees deterministic
gathering of all of them, with termination?
We provide exact answers to this open problem by considering the case when
the agents initially know the size of the network and the case when they do
not. In the former case, we prove M=f+1 while in the latter, we prove M=f+2.
More precisely, for networks of known size, we design a deterministic algorithm
gathering all good agents in any network provided that the number of good
agents is at least f+1. For networks of unknown size, we also design a
deterministic algorithm ensuring the gathering of all good agents in any
network but provided that the number of good agents is at least f+2. Both of
our algorithms are optimal in terms of required number of good agents, as each
of them perfectly matches the respective lower bound on M shown in [14], which
is of f+1 when the size of the network is known and of f+2 when it is unknown
A Lower Bound on List Size for List Decoding
A q-ary error-correcting code C â {1, 2,..., q} n is said to be list decodable to radius Ï with list size L if every Hamming ball of radius Ï contains at most L codewords of C. We prove that in order for a q-ary code to be list-decodable up to radius (1 â 1/q)(1 â Δ)n, we must have L = âŠ(1/Δ 2). Specifically, we prove that there exists a constant cq> 0 and a function fq such that for small enough Δ> 0, if C is list-decodable to radius (1 â 1/q)(1 â Δ)n with list size cq/Δ 2, then C has at most fq(Δ) codewords, independent of n. This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n) number of codewords are known for list size L = O(1/Δ 2). A result similar to ours is implicit in Blinovsky [Bli1] for the binary (q = 2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.
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