15,470 research outputs found
Classical and quantum partition bound and detector inefficiency
We study randomized and quantum efficiency lower bounds in communication
complexity. These arise from the study of zero-communication protocols in which
players are allowed to abort. Our scenario is inspired by the physics setup of
Bell experiments, where two players share a predefined entangled state but are
not allowed to communicate. Each is given a measurement as input, which they
perform on their share of the system. The outcomes of the measurements should
follow a distribution predicted by quantum mechanics; however, in practice, the
detectors may fail to produce an output in some of the runs. The efficiency of
the experiment is the probability that the experiment succeeds (neither of the
detectors fails).
When the players share a quantum state, this gives rise to a new bound on
quantum communication complexity (eff*) that subsumes the factorization norm.
When players share randomness instead of a quantum state, the efficiency bound
(eff), coincides with the partition bound of Jain and Klauck. This is one of
the strongest lower bounds known for randomized communication complexity, which
subsumes all the known combinatorial and algebraic methods including the
rectangle (corruption) bound, the factorization norm, and discrepancy.
The lower bound is formulated as a convex optimization problem. In practice,
the dual form is more feasible to use, and we show that it amounts to
constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for
eff*). We give an example of a quantum distribution where the violation can be
exponentially bigger than the previously studied class of normalized Bell
inequalities.
For one-way communication, we show that the quantum one-way partition bound
is tight for classical communication with shared entanglement up to arbitrarily
small error.Comment: 21 pages, extended versio
Streaming Lower Bounds for Approximating MAX-CUT
We consider the problem of estimating the value of max cut in a graph in the
streaming model of computation. At one extreme, there is a trivial
-approximation for this problem that uses only space, namely,
count the number of edges and output half of this value as the estimate for max
cut value. On the other extreme, if one allows space, then a
near-optimal solution to the max cut value can be obtained by storing an
-size sparsifier that essentially preserves the max cut. An
intriguing question is if poly-logarithmic space suffices to obtain a
non-trivial approximation to the max-cut value (that is, beating the factor
). It was recently shown that the problem of estimating the size of a
maximum matching in a graph admits a non-trivial approximation in
poly-logarithmic space.
Our main result is that any streaming algorithm that breaks the
-approximation barrier requires space even if the
edges of the input graph are presented in random order. Our result is obtained
by exhibiting a distribution over graphs which are either bipartite or
-far from being bipartite, and establishing that
space is necessary to differentiate between these
two cases. Thus as a direct corollary we obtain that
space is also necessary to test if a graph is bipartite or -far
from being bipartite.
We also show that for any , any streaming algorithm that
obtains a -approximation to the max cut value when edges arrive
in adversarial order requires space, implying that
space is necessary to obtain an arbitrarily good approximation to
the max cut value
Streaming Hardness of Unique Games
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by p, the alphabet size of the Unique Game, gives a trivial p-approximation that can be computed in O(log n) space. Meanwhile, with high probability, a sample of O~(n) constraints suffices to estimate the optimal value to (1+epsilon) accuracy. We prove that any single-pass streaming algorithm that achieves a (p-epsilon)-approximation requires Omega_epsilon(sqrt n) space. Our proof is via a reduction from lower bounds for a communication problem that is a p-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downstream hardness results for streaming approximability for other CSP-like problems
Streaming Verification of Graph Properties
Streaming interactive proofs (SIPs) are a framework for outsourced
computation. A computationally limited streaming client (the verifier) hands
over a large data set to an untrusted server (the prover) in the cloud and the
two parties run a protocol to confirm the correctness of result with high
probability. SIPs are particularly interesting for problems that are hard to
solve (or even approximate) well in a streaming setting. The most notable of
these problems is finding maximum matchings, which has received intense
interest in recent years but has strong lower bounds even for constant factor
approximations.
In this paper, we present efficient streaming interactive proofs that can
verify maximum matchings exactly. Our results cover all flavors of matchings
(bipartite/non-bipartite and weighted). In addition, we also present streaming
verifiers for approximate metric TSP. In particular, these are the first
efficient results for weighted matchings and for metric TSP in any streaming
verification model.Comment: 26 pages, 2 figure, 1 tabl
Better Non-Local Games from Hidden Matching
We construct a non-locality game that can be won with certainty by a quantum
strategy using log n shared EPR-pairs, while any classical strategy has winning
probability at most 1/2+O(log n/sqrt{n}). This improves upon a recent result of
Junge et al. in a number of ways.Comment: 11 pages, late
Exponential Separation of Quantum and Classical One-Way Communication Complexity for a Boolean Function
We give an exponential separation between one-way quantum and classical
communication complexity for a Boolean function. Earlier such a separation was
known only for a relation. A very similar result was obtained earlier but
independently by Kerenidis and Raz [KR06]. Our version of the result gives an
example in the bounded storage model of cryptography, where the key is secure
if the adversary has a certain amount of classical storage, but is completely
insecure if he has a similar amount of quantum storage.Comment: 8 pages, no figure
The Sketching Complexity of Graph and Hypergraph Counting
Subgraph counting is a fundamental primitive in graph processing, with
applications in social network analysis (e.g., estimating the clustering
coefficient of a graph), database processing and other areas. The space
complexity of subgraph counting has been studied extensively in the literature,
but many natural settings are still not well understood. In this paper we
revisit the subgraph (and hypergraph) counting problem in the sketching model,
where the algorithm's state as it processes a stream of updates to the graph is
a linear function of the stream. This model has recently received a lot of
attention in the literature, and has become a standard model for solving
dynamic graph streaming problems.
In this paper we give a tight bound on the sketching complexity of counting
the number of occurrences of a small subgraph in a bounded degree graph
presented as a stream of edge updates. Specifically, we show that the space
complexity of the problem is governed by the fractional vertex cover number of
the graph . Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
. Our main technical contribution lies in a new set of Fourier analytic
tools that we develop to analyze multiplayer communication protocols in the
simultaneous communication model, allowing us to prove a tight lower bound. We
believe that our techniques are likely to find applications in other settings.
Besides giving tight bounds for all graphs , both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity
String Matching: Communication, Circuits, and Learning
String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings.
- Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol.
- Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k.
- Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem
- …