390 research outputs found
Quantum Circuits for the Unitary Permutation Problem
We consider the Unitary Permutation problem which consists, given unitary
gates and a permutation of , in
applying the unitary gates in the order specified by , i.e. in
performing . This problem has been
introduced and investigated by Colnaghi et al. where two models of computations
are considered. This first is the (standard) model of query complexity: the
complexity measure is the number of calls to any of the unitary gates in
a quantum circuit which solves the problem. The second model provides quantum
switches and treats unitary transformations as inputs of second order. In that
case the complexity measure is the number of quantum switches. In their paper,
Colnaghi et al. have shown that the problem can be solved within calls in
the query model and quantum switches in the new model. We
refine these results by proving that quantum switches
are necessary and sufficient to solve this problem, whereas calls
are sufficient to solve this problem in the standard quantum circuit model. We
prove, with an additional assumption on the family of gates used in the
circuits, that queries are required, for any
. The upper and lower bounds for the standard quantum circuit
model are established by pointing out connections with the permutation as
substring problem introduced by Karp.Comment: 8 pages, 5 figure
Systems of Linear Equations over and Problems Parameterized Above Average
In the problem Max Lin, we are given a system of linear equations
with variables over in which each equation is assigned a
positive weight and we wish to find an assignment of values to the variables
that maximizes the excess, which is the total weight of satisfied equations
minus the total weight of falsified equations. Using an algebraic approach, we
obtain a lower bound for the maximum excess.
Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin
introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75,
2009). In Max Lin AA all weights are integral and we are to decide whether the
maximum excess is at least , where is the parameter.
It is not hard to see that we may assume that no two equations in have
the same left-hand side and . Using our maximum excess results,
we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable
for a wide special case: for an arbitrary fixed function
.
Max -Lin AA is a special case of Max Lin AA, where each equation has at
most variables. In Max Exact -SAT AA we are given a multiset of
clauses on variables such that each clause has variables and asked
whether there is a truth assignment to the variables that satisfies at
least clauses. Using our maximum excess results, we
prove that for each fixed , Max -Lin AA and Max Exact -SAT AA can
be solved in time This improves
-time algorithms for the two problems obtained by Gutin et
al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Confirmatory factor analysis and examination of the psychometric properties of the eating beliefs questionnaire.
BACKGROUND: The Eating Beliefs Questionnaire (EBQ) is a 27-item self-report measure that assesses positive and negative beliefs about binge eating. It has been validated and its factor structure explored in a non-clinical sample. This study tested the psychometric properties of the EBQ in a clinical and a non-clinical sample. METHOD: A sample of 769 participants (573 participants recruited from the university and general community, 76 seeking treatment for an eating disorder and 120 participating in obesity research) completed a battery of questionnaires. A subset of clinical participants with a diagnosis of Bulimia Nervosa or Binge Eating Disorder completed the test-battery before and after receiving a psychological treatment (n = 27) or after allocation to a wait-list period (n = 28), and a subset of 35 community participants completed the test battery again after an interval of two-weeks. Confirmatory Factor Analysis (CFA) was performed. RESULTS: CFA found a two-factor structure that provided a good fit to the data, supporting the solution presented in the development paper. Items with poor psychometric properties were removed, resulting in a 16 item measure. EBQ scores were found to correlate with binge eating episode frequency, increases in body mass index (BMI), and measures of eating disorder behaviours and related psychopathology. The EBQ was found to have excellent internal consistency (α = .94), good test-retest reliability (r = .91) and sensitivity to treatment. CONCLUSION: These findings indicate that the EBQ is a psychometrically sound and clinically useful measure
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Partial Covering Arrays: Algorithms and Asymptotics
A covering array is an array with entries
in , for which every subarray contains each
-tuple of among its rows. Covering arrays find
application in interaction testing, including software and hardware testing,
advanced materials development, and biological systems. A central question is
to determine or bound , the minimum number of rows of
a . The well known bound
is not too far from being
asymptotically optimal. Sensible relaxations of the covering requirement arise
when (1) the set need only be contained among the rows
of at least of the subarrays and (2) the
rows of every subarray need only contain a (large) subset of . In this paper, using probabilistic methods, significant
improvements on the covering array upper bound are established for both
relaxations, and for the conjunction of the two. In each case, a randomized
algorithm constructs such arrays in expected polynomial time
Visibility Representations of Boxes in 2.5 Dimensions
We initiate the study of 2.5D box visibility representations (2.5D-BR) where
vertices are mapped to 3D boxes having the bottom face in the plane and
edges are unobstructed lines of sight parallel to the - or -axis. We
prove that: Every complete bipartite graph admits a 2.5D-BR; The
complete graph admits a 2.5D-BR if and only if ; Every
graph with pathwidth at most admits a 2.5D-BR, which can be computed in
linear time. We then turn our attention to 2.5D grid box representations
(2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit
square at integer coordinates. We show that an -vertex graph that admits a
2.5D-GBR has at most edges and this bound is tight. Finally,
we prove that deciding whether a given graph admits a 2.5D-GBR with a given
footprint is NP-complete. The footprint of a 2.5D-BR is the set of
bottom faces of the boxes in .Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
Optimal Fair Computation
A computation scheme among n parties is fair if no party obtains the computation result unless all other n-1 parties obtain the same result. A fair computation scheme is optimistic if n honest parties can obtain the computation result without resorting to a trusted third party. We prove, for the first time, a tight lower bound on the message complexity of optimistic fair computation for n parties among which n-1 can be malicious in an asynchronous network. We do so by relating the optimal message complexity of optimistic fair computation to the length of the shortest permutation sequence in combinatorics
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