scholarly journals 3-Hitting set on bounded degree hypergraphs: Upper and lower bounds on the kernel size

2015 ◽  
Vol 07 (02) ◽  
pp. 1550011
Author(s):  
Iyad Kanj ◽  
Fenghui Zhang
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Kernel Size ◽  
Bounded Degree ◽  
Hitting Set ◽  
Uniform Hypergraphs ◽  
Hitting Set Problem

We study upper and lower bounds on the vertex-kernel size for the 3-HITTING SET problem on hypergraphs of degree at most 3, denoted 3-3-HS. We first show that, unless P = NP, 3-3-HS on 3-uniform hypergraphs does not have a vertex-kernel of size at most 35k/19 > 1.8421k. We then give a 4k - k0.2692 vertex-kernel for 3-3-hs that is computable in time O(k2). We do not assume that the hypergraph is 3-uniform for the vertex-kernel upper bound results. This result improves the upper bound of 4k on the vertex-kernel size for 3-3-HS, implied by the results of Wahlström.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

Localization of Discrete Spectrum of Multiparticle Schrödinger Operators

1985 ◽  
Vol 40 (10) ◽  
pp. 1052-1058 ◽  
Author(s):  
Heinz K. H. Siedentop
Keyword(s):  
Discrete Spectrum ◽  
Lower Bounds ◽  
Upper Bound ◽  
Schrödinger Operators ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators

An upper bound on the dimension of eigenspaces of multiparticle Schrödinger operators is given. Its relation to upper and lower bounds on the eigenvalues is discussed.

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals When is non-trivial estimation possible for graphons and stochastic block models?‡

2017 ◽  
Vol 7 (2) ◽  
pp. 169-181
Author(s):  
Audra McMillan ◽  
Adam Smith
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Random Networks ◽  
Block Models

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

Full-Strength Reinforcement of a Cutout in a Cylindrical Shell

1964 ◽  
Vol 31 (4) ◽  
pp. 667-675 ◽  
Author(s):  
Philip G. Hodge
Keyword(s):  
Cylindrical Shell ◽  
Lower Bounds ◽  
Upper Bound ◽  
Circular Cylindrical Shell ◽  
Minimum Weight ◽  
Upper And Lower Bounds ◽  
Minimum Weight Design ◽  
Initial Strength ◽  
Weight Design ◽  
Circular Cutout

A long circular cylindrical shell is to be pierced with a circular cutout, and it is desired to design a plane annular reinforcing ring which will restore the shell to its initial strength. Upper and lower bounds on the design of the reinforcement are obtained. Although these bounds are far a part, it is conjectured that the upper bound, in addition to being safe, is reasonably close to the minimum weight design. Some suggestions for further work on the problem are advanced.

scholarly journals On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

2020 ◽  
Vol 36 (36) ◽  
pp. 124-133
Author(s):  
Shinpei Imori ◽  
Dietrich Von Rosen
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Degrees Of Freedom ◽  
Generalized Inverse ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Identity Matrix ◽  
Wishart Matrix ◽  
The Mean ◽  
Scale Matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution.

scholarly journals Sensitivity bounds on a GI/M/n/n queueing system

1989 ◽  
Vol 31 (2) ◽  
pp. 135-149 ◽  
Author(s):  
Andrew Coyle
Keyword(s):  
Lower Bounds ◽  
Performance Measures ◽  
Markov Processes ◽  
Simple Form ◽  
Upper Bound ◽  
Queueing System ◽  
Upper And Lower Bounds ◽  
Semi Markov Processes

AbstractA method for determining the upper and lower bounds for performance measures for certain types of Generalised Semi-Markov Processes has been described in Taylor and Coyle [8]. A brief description of this method and its use in finding an upper bound for the time congestion of a GI/M/n/n queueing system will be given. This bound turns out to have a simple form which is quickly calculated and easy to use in practice.

scholarly journals Best possible rates of distribution of dense lattice orbits in homogeneous spaces

2018 ◽  
Vol 2018 (745) ◽  
pp. 155-188 ◽  
Author(s):  
Anish Ghosh ◽  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Upper Bound ◽  
Homogeneous Spaces ◽  
Automorphic Representation ◽  
Upper And Lower Bounds ◽  
Duality Principle ◽  
Wide Range ◽  
Dense Orbits

Abstract This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group G, acting on suitable homogeneous spaces G/H. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G/Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on L^{2} (Γ \setminus G). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

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