scholarly journals Tight Bounds for Restricted Grid Scheduling

2019 ◽  
Vol 30 (03) ◽  
pp. 375-405
Author(s):  
Joan Boyar ◽  
Faith Ellen
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Competitive Ratio ◽  
Upper And Lower Bounds ◽  
Grid Scheduling ◽  
Total Size

The following problem is considered: Items with integer sizes are given and variable sized bins arrive online. A bin must be used if there is still an item remaining which fits in it when the bin arrives. The goal is to minimize the total size of all the bins used. Previously, a lower bound of [Formula: see text] on the competitive ratio of this problem was achieved using items of size [Formula: see text] and [Formula: see text]. For these item sizes and maximum bin size [Formula: see text], we obtain asymptotically matching upper and lower bounds, which vary depending on the ratio of the number of small items to the number of large items.

scholarly journals Online Graph Coloring Against a Randomized Adversary

2018 ◽  
Vol 29 (04) ◽  
pp. 551-569 ◽  
Author(s):  
Elisabet Burjons ◽  
Juraj Hromkovič ◽  
Rastislav Královič ◽  
Richard Královič ◽  
Xavier Muñoz ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Online Algorithms ◽  
Graph Coloring ◽  
Competitive Ratio ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Single Instance ◽  
Graph Coloring Problem

We consider an online model where an adversary constructs a set of [Formula: see text] instances [Formula: see text] instead of one single instance. The algorithm knows [Formula: see text] and the adversary will choose one instance from [Formula: see text] at random to present to the algorithm. We further focus on adversaries that construct sets of [Formula: see text]-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in [Formula: see text] and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.

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.

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.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

LOWER BOUNDS FOR STREETS AND GENERALIZED STREETS

2001 ◽  
Vol 11 (04) ◽  
pp. 401-421 ◽  
Author(s):  
ALEJANDRO LÓPEZ-ORTIZ ◽  
SVEN SCHUIERER
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Search Strategy ◽  
The Real ◽  
Simple Polygons ◽  
On Line ◽  
Special Classes

We present lower bounds for on-line searching problems in two special classes of simple polygons called streets and generalized streets. In streets we assume that the location of the target is known to the robot in advance and prove a lower bound of [Formula: see text] on the competitive ratio of any deterministic search strategy—which can be shown to be tight. For generalized streets we show that if the location of the target is not known, then there is a class of orthogonal generalized streets for which the competitive ratio of any search strategy is at least [Formula: see text] in the L2-metric—again matching the competitive ratio of the best known algorithm. We also show that if the location of the target is known, then the competitive ratio for searching in generalized streets in the L1-metric is at least 9 which is tight as well. The former result is based on a lower bound on the average competitive ratio of searching on the real line if an upper bound of D to the target is given. We show that in this case the average competitive ratio is at least 9-O(1/ log D).

scholarly journals ON THE POSSIBLE RANKS AMONG MATRICES WITH A GIVEN PATTERN

2010 ◽  
Vol 02 (03) ◽  
pp. 363-377 ◽  
Author(s):  
CHARLES R. JOHNSON ◽  
YULIN ZHANG
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Null Space ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Given are tight upper and lower bounds for the minimum rank among all matrices with a prescribed zero–nonzero pattern. The upper bound is based upon solving for a matrix with a given null space and, with optimal choices, produces the correct minimum rank. It leads to simple, but often accurate, bounds based upon overt statistics of the pattern. The lower bound is also conceptually simple. Often, the lower and an upper bound coincide, but examples are given in which they do not.

BOUNDS ON THE PRICE OF STABILITY OF UNDIRECTED NETWORK DESIGN GAMES WITH THREE PLAYERS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 1-17 ◽  
Author(s):  
VITTORIO BILÒ ◽  
ROBERTA BOVE
Keyword(s):  
Lower Bound ◽  
Network Design ◽  
Lower Bounds ◽  
Upper Bound ◽  
Cost Allocation ◽  
Upper And Lower Bounds ◽  
Price Of Stability ◽  
Undirected Network ◽  
Basic Case

After almost seven years from its definition,2 the price of stability of undirected network design games with fair cost allocation remains to be elusive. Its exact characterization has been achieved only for the basic case of two players2,7 and, as soon as the number of players increases, the gap between the known upper and lower bounds becomes super-constant, even in the special variants of multicast and broadcast games. Motivated by the intrinsic difficulties that seem to characterize this problem, we analyze the already challenging case of three players and provide either new or improved bounds. For broadcast games, we prove an upper bound of 1.485 which exactly matches a lower bound given in Ref. 4; for multicast games, we show new upper and lower bounds which confine the price of stability in the interval [1.524; 1.532]; while, for the general case, we give an improved upper bound of 1.634. The techniques exploited in this paper are a refinement of those used in Ref. 7 and can be easily adapted to deal with all the cases involving a small number of players.

A fundamental inequality for the assessment of forest land values

1987 ◽  
Vol 17 (10) ◽  
pp. 1309-1311 ◽  
Author(s):  
Karl Gustaf Löfgren
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Value Function ◽  
Upper And Lower Bounds ◽  
Land Values ◽  
Land Value ◽  
Forest Land ◽  
Present Value

In this paper one well-known property and one less known property of the present value function are used to provide upper and lower bounds on the change in the value of forest land when land value is evaluated at any two different discounted price vectors. The lower bound is expressed in possibly known entities of the original harvesting program.

Lower bounds to eigenvalues

1969 ◽  
Vol 47 (17) ◽  
pp. 1877-1879 ◽  
Author(s):  
Maurice Cohen ◽  
Tova Feldmann
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Hamiltonian Operator ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Classical Procedure ◽  
Ritz Procedure ◽  
Reasonable Choice ◽  
Lowest Eigenvalue

The classical procedure of Weinstein has been employed to obtain rigorous upper and lower bounds to the eigenvalues E of a quantum mechanical Hamiltonian operator H. The new bounds represent an improvement over Weinstein's bounds for any reasonable choice of variational trial function. In the case of the lowest eigenvalue E0, for which the Rayleigh–Ritz procedure gives the optimum upper bound, the new lower bound is an improvement over the lower bound formula of Stevenson and Crawford.

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