scholarly journals Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

2021 ◽  
Vol 17 (2) ◽  
pp. 1-68
Author(s):  
Rajesh Chitnis ◽  
Andreas Emil Feldmann ◽  
Pasin Manurangsi
Keyword(s):  
Approximation Algorithm ◽  
Approximation Scheme ◽  
Upper And Lower Bounds ◽  
Important Special Case ◽  
Bidirected Graph ◽  
Negative Edge ◽  
Network Problems ◽  
Special Cases ◽  
Bidirected Graphs ◽  
Special Case

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for  k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by  k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter  k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

scholarly journals On the number of excursion sets of planar Gaussian fields

2020 ◽  
Vol 178 (3-4) ◽  
pp. 655-698
Author(s):  
Dmitry Beliaev ◽  
Michael McAuley ◽  
Stephen Muirhead
Keyword(s):  
Plane Wave ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Gaussian Fields ◽  
Important Special Case ◽  
Level Densities ◽  
Excursion Sets ◽  
Nodal Set ◽  
Different Types ◽  
Special Case

Abstract The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

APPROXIMATION ALGORITHMS FOR SCHEDULING MALLEABLE TASKS UNDER PRECEDENCE CONSTRAINTS

2002 ◽  
Vol 13 (04) ◽  
pp. 613-627 ◽  
Author(s):  
RENAUD LEPÈRE ◽  
DENIS TRYSTRAM ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Precedence Constraints ◽  
Performance Guarantee ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Special Cases ◽  
Special Case ◽  
Polynomial Time Approximation

This work presents approximation algorithms for scheduling the tasks of a parallel application that are subject to precedence constraints. The considered tasks are malleable which means that they may be executed on a varying number of processors in parallel. The considered objective criterion is the makespan, i.e., the largest task completion time. We demonstrate a close relationship between this scheduling problem and one of its subproblems, the allotment problem. By exploiting this relationship, we design a polynomial time approximation algorithm with performance guarantee arbitrarily close to [Formula: see text] for the special case of series parallel precedence constraints and for the special case of precedence constraints of bounded width. These special cases cover the important situation of tree structured precedence constraints. For arbitrary precedence constraints, we give a polynomial time approximation algorithm with performance guarantee [Formula: see text].

scholarly journals Further results on non-self-centrality measures of graphs

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5137-5148 ◽  
Author(s):  
Mahdieh Azari
Keyword(s):  
Structural Parameters ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Centrality Measures ◽  
Eccentric Connectivity Index ◽  
The Third ◽  
Eccentricity Index ◽  
Strong Product ◽  
Special Cases ◽  
Special Case

For indicating the non-self-centrality extent of graphs, two new eccentricity-based measures namely third Zagreb eccentricity index E3(G) and non-self-centrality number N(G) of a connected graph G have recently been introduced as E3(G) = ?uv?E(G)|?G(u)-?G(v)| and N(G) = ? {u,v}?V(G) |?G(u)-?G(v)|, where ?G(u) denotes the eccentricity of a vertex u in G. In this paper, we find relation between the third Zagreb eccentricity index of graphs with some eccentricity-based invariants such as second Zagreb eccentricity index and second eccentric connectivity index. We also give sharp upper and lower bounds on the nonself-centrality number of graphs in terms of some structural parameters and relate it to two well-known eccentricity-based invariants namely total eccentricity and first Zagreb eccentricity index. Furthermore, we present exact expressions or sharp upper bounds on the third Zagreb eccentricity index and non-selfcentrality number of several graph operations such as join, disjunction, symmetric difference, lexicographic product, strong product, and generalized hierarchical product. The formulae for Cartesian product and rooted product as two important special cases of generalized hierarchical product and the formulae for corona product as a special case of rooted product are also given.

Competitive Agents in Certain and Uncertain Markets

2020 ◽  
Author(s):  
Robert G. Chambers
Keyword(s):  
Convex Analysis ◽  
Duality Theorem ◽  
Optimization Theory ◽  
Economic Behavior ◽  
Important Special Case ◽  
Special Cases ◽  
Equilibrium Analyses ◽  
Fenchel's Duality Theorem ◽  
Special Case ◽  
Decision Settings

This book uses concepts from optimization theory to develop an integrated analytic framework for treating consumer, producer, and market equilibrium analyses as special cases of a generic optimization problem. The same framework applies to both stochastic and non-stochastic decision settings, so that the latter is recognized as an (important) special case of the former. The analytic techniques are borrowed from convex analysis and variational analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. The book shows how virtually identical conjugate analyses form the basis for modeling economic behavior in each of the areas studied. The basic analytic concepts are borrowed from convex analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. It is demonstrated how virtually identical conjugate analyses form the basis for modelling economic behaviour in each of the areas studied.

SOME NEW RELIABILITY BOUNDS FOR SUMS OF NBUE RANDOM VARIABLES

2010 ◽  
Vol 25 (1) ◽  
pp. 83-102
Author(s):  
Steven G. From
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Important Special Case ◽  
Lifetime Distributions ◽  
Survivor Function ◽  
Numerical Studies ◽  
Special Case ◽  
Better Than

In this article, we discuss some new upper and lower bounds for the survivor function of the sum of n independent random variables each of which has an NBUE (new better than used in expectation) distribution. In some cases, only the means of the random variables are assumed known. These bounds are compared to the sharp bounds given in Cheng and Lam [6], which requires both means and variances known. Although the new bounds are not sharp, they often produce better upper bounds for the survivor function in the extreme right tail of many NBUE lifetime distributions, an important special case in applications. Moreover, a lower bound exists in one case not handled by the lower bounds of Theorem 3 in Cheng and Lam [6]. Numerical studies are presented along with theoretical discussions.

scholarly journals Aperture referral in dioptric systems with stigmatic elements

2012 ◽  
Vol 71 (1) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
General Theory ◽  
Focal Point ◽  
Positive Zero ◽  
Mathematical Form ◽  
Linear Optics ◽  
Important Special Case ◽  
Vector Algebra ◽  
Special Cases ◽  
Special Case ◽  
Projective Field

A previous paper develops the general theory of aperture referral in linear optics and shows how several ostensibly distinct concepts, including the blur patch on the retina, the effective cornealpatch, the projective field and the field of view, are now unified as particular applications of the general theory.  The theory allows for astigmatism and heterocentricity.  Symplecticity and the generality of the approach, however, make it difficult to gain insight and mean that the material is not accessible to readers unfamiliar with matrices and linear algebra. The purpose of this paper is to examine whatis, perhaps, the most important special case, that in which astigmatism is ignored.  Symplecticity and, hence, the mathematics become greatly simplified. The mathematics reduces largely to elementary vector algebra and, in some places, simple scalar algebra and yet retains the mathematical form of the general approach.  As a result the paper allows insight into and provides a stepping stone to the general theory.  Under referral an aperture under-goes simple scalar magnification and transverse translation.  The paper pays particular attention to referral to transverse planes in the neighbourhood of a focal point where the magnification may be positive, zero or negative.  Circular apertures are treated as special cases of elliptical apertures and the meaning of referred apertures of negative radius is explained briefly. (S Afr Optom 2012 71(1) 3-11)

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

Optimal Order Picking in Carousels Storage System

2012 ◽  
Vol 601 ◽  
pp. 347-353
Author(s):  
Xiong Zhi Wang ◽  
Guo Qing Wang
Keyword(s):  
Approximation Algorithm ◽  
General Problem ◽  
Storage System ◽  
Order Picking ◽  
Total Order ◽  
Experimental Results ◽  
Optimal Order ◽  
Np Hard ◽  
Special Case

We study the order picking problem in carousels system with a single picker. The objective is to find a picking scheduling to minimizing the total order picking time. After showing the problem being strongly in NP-Hard and finding two characteristics, we construct an approximation algorithm for a special case (two carousels) and a heuristics for the general problem. Experimental results verify that the solutions are quickly and steadily achieved and show its better performance.

An Elementary Proof of Suslin Reciprocity

2005 ◽  
Vol 48 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Matt Kerr
Keyword(s):  
Elementary Proof ◽  
Function Fields ◽  
Algebraic Cycles ◽  
Important Special Case ◽  
Special Case

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

scholarly journals The Nonrelativistic Scattering States of the Deng-Fan Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Bentol Hoda Yazarloo ◽  
Liangliang Lu ◽  
Guanghui Liu ◽  
Saber Zarrinkamar ◽  
Hassan Hassanabadi
Keyword(s):  
Approximation Scheme ◽  
Wave Functions ◽  
Energy Eigenvalues ◽  
Scattering States ◽  
Special Cases ◽  
Scattering State ◽  
Term Energy ◽  
Scattering Phase ◽  
Scattering Phase Shifts ◽  
Nonrelativistic Scattering

The approximately analytical scattering state solution of the Schrodinger equation is obtained for the Deng-Fan potential by using an approximation scheme to the centrifugal term. Energy eigenvalues, normalized wave functions, and scattering phase shifts are calculated. We consider and verify two special cases: thel=0and thes-wave Hulthén potential.

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