scholarly journals 3-hitting set on Bounded Degree Hypergraphs: Upper and Lower Bounds on the Kernel Size

2011 ◽  
pp. 163-174 ◽  
Author(s):  
Iyad A. Kanj ◽  
Fenghui Zhang
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Kernel Size ◽  
Bounded Degree ◽  
Hitting Set

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.

ALGEBRAIC NUMBERS WITH BOUNDED DEGREE AND WEIL HEIGHT

2018 ◽  
Vol 98 (2) ◽  
pp. 212-220
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Algebraic Numbers ◽  
Bounded Degree ◽  
Nonnegative Number ◽  
Weil Height

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$, let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$, these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$, which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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 An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

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