15,103 research outputs found
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
The VC-Dimension of Graphs with Respect to k-Connected Subgraphs
We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its -connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is -complete and that it remains
-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases and . On the positive side, we
observe it can be decided in linear time for graphs of bounded clique-width
Dynamical properties of electrical circuits with fully nonlinear memristors
The recent design of a nanoscale device with a memristive characteristic has
had a great impact in nonlinear circuit theory. Such a device, whose existence
was predicted by Leon Chua in 1971, is governed by a charge-dependent
voltage-current relation of the form . In this paper we show that
allowing for a fully nonlinear characteristic in memristive
devices provides a general framework for modeling and analyzing a very broad
family of electrical and electronic circuits; Chua's memristors are particular
instances in which is linear in . We examine several dynamical
features of circuits with fully nonlinear memristors, accommodating not only
charge-controlled but also flux-controlled ones, with a characteristic of the
form . Our results apply in particular to Chua's
memristive circuits; certain properties of these can be seen as a consequence
of the special form of the elastance and reluctance matrices displayed by
Chua's memristors.Comment: 19 page
Universal Programmable Quantum Circuit Schemes to Emulate an Operator
Unlike fixed designs, programmable circuit designs support an infinite number
of operators. The functionality of a programmable circuit can be altered by
simply changing the angle values of the rotation gates in the circuit. Here, we
present a new quantum circuit design technique resulting in two general
programmable circuit schemes. The circuit schemes can be used to simulate any
given operator by setting the angle values in the circuit. This provides a
fixed circuit design whose angles are determined from the elements of the given
matrix-which can be non-unitary-in an efficient way. We also give both the
classical and quantum complexity analysis for these circuits and show that the
circuits require a few classical computations. They have almost the same
quantum complexities as non-general circuits. Since the presented circuit
designs are independent from the matrix decomposition techniques and the global
optimization processes used to find quantum circuits for a given operator, high
accuracy simulations can be done for the unitary propagators of molecular
Hamiltonians on quantum computers. As an example, we show how to build the
circuit design for the hydrogen molecule.Comment: combined with former arXiv:1207.174
Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension
Algorithmic mechanism design (AMD) studies the delicate interplay between
computational efficiency, truthfulness, and optimality. We focus on AMD's
paradigmatic problem: combinatorial auctions. We present a new generalization
of the VC dimension to multivalued collections of functions, which encompasses
the classical VC dimension, Natarajan dimension, and Steele dimension. We
present a corresponding generalization of the Sauer-Shelah Lemma and harness
this VC machinery to establish inapproximability results for deterministic
truthful mechanisms. Our results essentially unify all inapproximability
results for deterministic truthful mechanisms for combinatorial auctions to
date and establish new separation gaps between truthful and non-truthful
algorithms
A Variant of the VC-Dimension with Applications to Depth-3 Circuits
We introduce the following variant of the VC-dimension. Given and a positive integer , we define to be the
size of the largest subset such that the projection of on
every subset of of size is the -dimensional cube. We show that
determining the largest cardinality of a set with a given
dimension is equivalent to a Tur\'an-type problem related to the total number
of cliques in a -uniform hypergraph. This allows us to beat the
Sauer--Shelah lemma for this notion of dimension. We use this to obtain several
results on -circuits, i.e., depth- circuits with top gate OR and
bottom fan-in at most :
* Tight relationship between the number of satisfying assignments of a
-CNF and the dimension of the largest projection accepted by it, thus
improving Paturi, Saks, and Zane (Comput. Complex. '00).
* Improved -circuit lower bounds for affine dispersers for
sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture
under which we get further improvement.
* We make progress towards settling the complexity of the inner
product function and all degree- polynomials over in general.
The question of determining the complexity of IP was recently
posed by Golovnev, Kulikov, and Williams (ITCS'21)
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