8,637 research outputs found
Hardness and inapproximability results for minimum verification set and minimum path decision tree problems
Minimization of decision trees is a well studied problem. In this work, we introduce two new problems related to minimization of decision trees. The problems are called minimum verification set (MinVS) and minimum path decision tree (MinPathDT) problems. Decision tree problems ask the question "What is the unknown given object?". MinVS problem on the other hand asks the question "Is the unknown object z?", for a given object z. Hence it is not an identification, but rather a verification problem. MinPathDT problem aims to construct a decision tree where only the cost of the root-to-leaf path corresponding to a given object is minimized, whereas decision tree problems in general try to minimize the overall cost of decision trees considering all the
objects. Therefore, MinVS and MinPathDT are seemingly easier problems.
However, in this work we prove that MinVS and MinPathDT problems are both NP-complete and cannot be approximated within a factor in o(lg n) unless P = NP
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
AM with Multiple Merlins
We introduce and study a new model of interactive proofs: AM(k), or
Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known
MIP, here the assumption is that each Merlin receives an independent random
challenge from Arthur. One motivation for this model (which we explore in
detail) comes from the close analogies between it and the quantum complexity
class QMA(k), but the AM(k) model is also natural in its own right.
We illustrate the power of multiple Merlins by giving an AM(2) protocol for
3SAT, in which the Merlins' challenges and responses consist of only
n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the
Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP
with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms
nearly matching this lower bound are known, but their running times had never
been previously explained. Brandao and Harrow have also recently used our 3SAT
protocol to show quasipolynomial hardness for approximating the values of
certain entangled games.
In the other direction, we give a simple quasipolynomial-time approximation
algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT
protocol is essentially optimal. More generally, we show that multiple Merlins
never provide more than a polynomial advantage over one: that is, AM(k)=AM for
all k=poly(n). The key to this result is a subsampling theorem for free games,
which follows from powerful results by Alon et al. and Barak et al. on
subsampling dense CSPs, and which says that the value of any free game can be
closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page
Towards Fast Computation of Certified Robustness for ReLU Networks
Verifying the robustness property of a general Rectified Linear Unit (ReLU)
network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer
CAV17]. Although finding the exact minimum adversarial distortion is hard,
giving a certified lower bound of the minimum distortion is possible. Current
available methods of computing such a bound are either time-consuming or
delivering low quality bounds that are too loose to be useful. In this paper,
we exploit the special structure of ReLU networks and provide two
computationally efficient algorithms Fast-Lin and Fast-Lip that are able to
certify non-trivial lower bounds of minimum distortions, by bounding the ReLU
units with appropriate linear functions Fast-Lin, or by bounding the local
Lipschitz constant Fast-Lip. Experiments show that (1) our proposed methods
deliver bounds close to (the gap is 2-3X) exact minimum distortion found by
Reluplex in small MNIST networks while our algorithms are more than 10,000
times faster; (2) our methods deliver similar quality of bounds (the gap is
within 35% and usually around 10%; sometimes our bounds are even better) for
larger networks compared to the methods based on solving linear programming
problems but our algorithms are 33-14,000 times faster; (3) our method is
capable of solving large MNIST and CIFAR networks up to 7 layers with more than
10,000 neurons within tens of seconds on a single CPU core.
In addition, we show that, in fact, there is no polynomial time algorithm
that can approximately find the minimum adversarial distortion of a
ReLU network with a approximation ratio unless
=, where is the number of neurons in the network.Comment: Tsui-Wei Weng and Huan Zhang contributed equall
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
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