100,123 research outputs found
A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs
A -birthday repetition of a
two-prover game is a game in which the two provers are sent
random sets of questions from of sizes and respectively.
These two sets are sampled independently uniformly among all sets of questions
of those particular sizes. We prove the following birthday repetition theorem:
when satisfies some mild conditions, decreases exponentially in where is the total number of
questions. Our result positively resolves an open question posted by Aaronson,
Impagliazzo and Moshkovitz (CCC 2014).
As an application of our birthday repetition theorem, we obtain new
fine-grained hardness of approximation results for dense CSPs. Specifically, we
establish a tight trade-off between running time and approximation ratio for
dense CSPs by showing conditional lower bounds, integrality gaps and
approximation algorithms. In particular, for any sufficiently large and for
every , we show the following results:
- We exhibit an -approximation algorithm for dense Max -CSPs
with alphabet size via -level of Sherali-Adams relaxation.
- Through our birthday repetition theorem, we obtain an integrality gap of
for -level Lasserre relaxation for fully-dense Max
-CSP.
- Assuming that there is a constant such that Max 3SAT cannot
be approximated to within of the optimal in sub-exponential
time, our birthday repetition theorem implies that any algorithm that
approximates fully-dense Max -CSP to within a factor takes
time, almost tightly matching the algorithmic
result based on Sherali-Adams relaxation.Comment: 45 page
Predictable arguments of knowledge
We initiate a formal investigation on the power of predictability for argument of knowledge systems for NP. Specifically, we consider private-coin argument systems where the answer of the prover can be predicted, given the private randomness of the verifier; we call such protocols Predictable Arguments of Knowledge (PAoK).
Our study encompasses a full characterization of PAoK, showing that such arguments can be made extremely laconic, with the prover sending a single bit, and assumed to have only one round (i.e., two messages) of communication without loss of generality.
We additionally explore PAoK satisfying additional properties (including zero-knowledge and the possibility of re-using the same challenge across multiple executions with the prover), present several constructions of PAoK relying on different cryptographic tools, and discuss applications to cryptography
Parallel repetition for entangled k-player games via fast quantum search
We present two parallel repetition theorems for the entangled value of
multi-player, one-round free games (games where the inputs come from a product
distribution). Our first theorem shows that for a -player free game with
entangled value , the -fold repetition of
has entangled value at most , where is the answer length of any
player. In contrast, the best known parallel repetition theorem for the
classical value of two-player free games is , due to Barak, et al. (RANDOM 2009). This
suggests the possibility of a separation between the behavior of entangled and
classical free games under parallel repetition.
Our second theorem handles the broader class of free games where the
players can output (possibly entangled) quantum states. For such games, the
repeated entangled value is upper bounded by . We also show that the dependence of the exponent
on is necessary: we exhibit a -player free game and such
that .
Our analysis exploits the novel connection between communication protocols
and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP
2014). We demonstrate that better communication protocols yield better parallel
repetition theorems: our first theorem crucially uses a quantum search protocol
by Aaronson and Ambainis, which gives a quadratic speed-up for distributed
search problems. Finally, our results apply to a broader class of games than
were previously considered before; in particular, we obtain the first parallel
repetition theorem for entangled games involving more than two players, and for
games involving quantum outputs.Comment: This paper is a significantly revised version of arXiv:1411.1397,
which erroneously claimed strong parallel repetition for free entangled
games. Fixed author order to alphabetica
Quantum hedging in two-round prover-verifier interactions
We consider the problem of a particular kind of quantum correlation that
arises in some two-party games. In these games, one player is presented with a
question they must answer, yielding an outcome of either 'win' or 'lose'.
Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a
perfect form of hedging, where the risk of losing a first game can completely
offset the corresponding risk for a second game. This is a non-classical
quantum phenomenon, and establishes the impossibility of performing strong
error-reduction for quantum interactive proof systems by parallel repetition,
unlike for classical interactive proof systems. We take a step in this article
towards a better understanding of the hedging phenomenon by giving a complete
characterization of when perfect hedging is possible for a natural
generalization of the game in arXiv:1104.1140. Exploring in a different
direction the subject of quantum hedging, and motivated by implementation
concerns regarding loss-tolerance, we also consider a variation of the protocol
where the player who receives the question can choose to restart the game
rather than return an answer. We show that in this setting there is no possible
hedging for any game played with state spaces corresponding to
finite-dimensional complex Euclidean spaces.Comment: 34 pages, 1 figure. Added work on connections with other result
Quantum Interactive Proofs with Competing Provers
This paper studies quantum refereed games, which are quantum interactive
proof systems with two competing provers: one that tries to convince the
verifier to accept and the other that tries to convince the verifier to reject.
We prove that every language having an ordinary quantum interactive proof
system also has a quantum refereed game in which the verifier exchanges just
one round of messages with each prover. A key part of our proof is the fact
that there exists a single quantum measurement that reliably distinguishes
between mixed states chosen arbitrarily from disjoint convex sets having large
minimal trace distance from one another. We also show how to reduce the
probability of error for some classes of quantum refereed games.Comment: 13 pages, to appear in STACS 200
A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian
In this work we show a barrier towards proving a randomness-efficient
parallel repetition, a promising avenue for achieving many tight
inapproximability results. Feige and Kilian (STOC'95) proved an impossibility
result for randomness-efficient parallel repetition for two prover games with
small degree, i.e., when each prover has only few possibilities for the
question of the other prover. In recent years, there have been indications that
randomness-efficient parallel repetition (also called derandomized parallel
repetition) might be possible for games with large degree, circumventing the
impossibility result of Feige and Kilian. In particular, Dinur and Meir
(CCC'11) construct games with large degree whose repetition can be derandomized
using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However,
obtaining derandomized parallel repetition theorems that would yield optimal
inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by
proving a limitation on derandomized parallel repetition. We formalize two
properties which we call "fortification-friendliness" and "yields robust
embeddings." We show that any proof of derandomized parallel repetition
achieving almost-linear blow-up cannot both (a) be fortification-friendly and
(b) yield robust embeddings. Unlike Feige and Kilian, we do not require the
small degree assumption.
Given that virtually all existing proofs of parallel repetition, including
the derandomized parallel repetition result of Dinur and Meir, share these two
properties, our no-go theorem highlights a major barrier to achieving
almost-linear derandomized parallel repetition
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