scholarly journals Approximation Algorithms for the Submodular Load Balancing with Submodular Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1785
Author(s):  
Xiaofei Liu ◽  
Peiyin Xing ◽  
Weidong Li
Keyword(s):  
Approximation Algorithms ◽  
Load Balancing ◽  
Maximum Load ◽  
Submodular Functions ◽  
Combinatorial Algorithm ◽  
Worst Case ◽  
Rejection Penalty ◽  
Rounding Algorithm ◽  
Element Set

In this paper, we study the submodular load balancing problem with submodular penalties. The objective of this problem is to balance the load among sets, while some elements can be rejected by paying some penalties. Officially, given an element set V, we want to find a subset R of rejected elements, and assign other elements to one of m sets A1,A2,⋯,Am. The objective is to minimize the sum of the maximum load among A1,A2,⋯,Am and the rejection penalty of R, where the load and rejection penalty are determined by different submodular functions. We study the submodular load balancing problem with submodular penalties under two settings: heterogenous setting (load functions are not identical) and homogenous setting (load functions are identical). Moreover, we design a Lovász rounding algorithm achieving a worst-case guarantee of m+1 under the heterogenous setting and a min{m,⌈nm⌉+1}=O(n)-approximation combinatorial algorithm under the homogenous setting.

scholarly journals The Multiple-Orientability Thresholds for Random Hypergraphs

2015 ◽  
Vol 25 (6) ◽  
pp. 870-908 ◽  
Author(s):  
NIKOLAOS FOUNTOULAKIS ◽  
MEGHA KHOSLA ◽  
KONSTANTINOS PANAGIOTOU
Keyword(s):  
Load Balancing ◽  
Hard Disk ◽  
Maximum Load ◽  
Disk Arrays ◽  
Uniform Hypergraph ◽  
Cuckoo Hashing ◽  
Uniform Hypergraphs ◽  
Random Hypergraphs ◽  
Do So ◽  
Parallel Access

Ak-uniform hypergraphH= (V, E) is called ℓ-orientable if there is an assignment of each edgee∈Eto one of its verticesv∈esuch that no vertex is assigned more than ℓ edges. LetHn,m,kbe a hypergraph, drawn uniformly at random from the set of allk-uniform hypergraphs withnvertices andmedges. In this paper we establish the threshold for the ℓ-orientability ofHn,m,kfor allk⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantityc*k,ℓsuch that with probability 1 −o(1) the graphHn,cn,khas an ℓ-orientation ifc<c*k,ℓ, but fails to do so ifc>c*k,ℓ.Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

A New Conception of Load Balancing in Cloud Computing Using Tasks Classification Levels

2018 ◽  
Vol 8 (4) ◽  
pp. 118-133 ◽  
Author(s):  
Fahim Youssef ◽  
Ben Lahmar El Habib ◽  
Rahhali Hamza ◽  
Labriji El Houssine ◽  
Eddaoui Ahmed ◽  
...  
Keyword(s):  
Load Balancing ◽  
Data Center ◽  
Execution Time ◽  
Data Centers ◽  
Virtual Machines ◽  
Service Providers ◽  
Cloud Service ◽  
Mathematical Functions ◽  
Worst Case ◽  
Cloud Users

Cloud users can have access to the service based on “pay as you go.” The daily increase of cloud users may decrease the performance, the availability and the profitability of the material and software resources used in cloud service. These challenges were solved by several load balancing algorithms between the virtual machines of the data centers. In order to determine a new load balancing improvement; this article's discussions will be divided into two research axes. The first, the pre-classification of tasks depending on whether their characteristics are accomplished or not (Notion of Levels). This new technique relies on the modeling of tasks classification based on an ascending order using techniques that calculate the worst-case execution time (WCET). The second, the authors choose distributed datacenters between quasi-similar virtual machines and the modeling of relationship between virtual machines using the pre-scheduling levels is included in the data center in terms of standard mathematical functions that controls this relationship. The key point of the improvement, is considering the current load of the virtual machine of a data center and the pre-estimation of the execution time of a task before any allocation. This contribution allows cloud service providers to improve the performance, availability and maximize the use of virtual machines workload in their data centers.

Faulted geological contacts: constraining uncertainty of discontinuities orientation using triangulation and combinatorial algorithm

2020 ◽  
Author(s):  
Michał Michalak ◽  
Ryszard Kuzak ◽  
Paweł Gładki ◽  
Agnieszka Kulawik
Keyword(s):  
Limited Resource ◽  
New Method ◽  
Combinatorial Algorithm ◽  
Fault Surface ◽  
Dominant Orientation ◽  
Delaunay Triangles ◽  
Synthetic Models ◽  
Element Set ◽  
Modelling Approach ◽  
Genetic Trends

&lt;p&gt;Subsurface information is usually a limited resource in geological modelling. This is not the case, however, for the Krak&amp;#243;w-Silesian Homocline in central Poland. It was subject to rapid exploitation of ore-bearing clays in the second half of the 20&lt;sup&gt;th &lt;/sup&gt;century. Exhaustive geological documentation remained after this activity had ceased and it contains thousands of borehole records. A small part of this resource has recently been incorporated to propose a new method for determining the dominant orientation of a selected geological contact. This new method regarded Delaunay triangles as source of local orientations that were then analyzed on stereonets. The geological contacts in this region are inclined gently towards NE, but they are also faulted and indicate some stratigraphic noise which makes the extraction of dominant orientation a challenging task.&lt;/p&gt;&lt;p&gt;It is still unknown, however, to which extent the proposed modelling approach is capable of detecting faults and calculating their orientation. This is particularly important for the introduction of a new method for the recognition of faults based on investigating spatial distribution of orientation patterns. This expert-guided methodology assumes to relate orientation trends with genetic trends and investigate them on 2D maps.&lt;/p&gt;&lt;p&gt;In this research, we built synthetic models of faulted contacts to observe the behaviour of triangles intersecting the fault surface. To observe the variability of the orientation at larger scale, and perhaps to constrain it at the same time, we applied a combinatorial algorithm for creating all three-element subsets from an n-element set. The employment of this combinatorial approach allowed to achieve a better clustering effect around the expected orientation. The limitation of the proposed approach can be attributed to some unexpected and unintuitive orientations. Compared to previous studies these singularities seem to be geometrical and not numerical in nature.&lt;/p&gt;

scholarly journals ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

2009 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
AKIYOSHI SHIOURA
Keyword(s):  
Approximation Algorithm ◽  
Convex Analysis ◽  
Submodular Function ◽  
Submodular Functions ◽  
Concave Functions ◽  
Discrete Convex Analysis ◽  
Discrete Convex ◽  
Weighted Rank ◽  
Rounding Algorithm ◽  
Rank Functions

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M ♮-concave functions. M ♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

WORST-CASE PERFORMANCE EVALUATION ON MULTIPROCESSOR TASK SCHEDULING WITH RESOURCE AUGMENTATION

2011 ◽  
Vol 22 (04) ◽  
pp. 971-982
Author(s):  
DESHI YE ◽  
QINMING HE
Keyword(s):  
Approximation Algorithms ◽  
Task Scheduling ◽  
Processing Time ◽  
Time Algorithm ◽  
Resource Augmentation ◽  
Worst Case ◽  
Multiprocessor Task Scheduling ◽  
Lpt Algorithm ◽  
On Line ◽  
The Greedy Algorithm

We study the worst-case performance of approximation algorithms for the problem of multiprocessor task scheduling on m identical processors with resource augmentation, whose objective is to minimize the makespan. In this case, the approximation algorithms are given k (k ≥ 0) extra processors than the optimal off-line algorithm. For on-line algorithms, the Greedy algorithm and shelf algorithms are studied. For off-line algorithm, we consider the LPT (longest processing time) algorithm. Particularly, we prove that the schedule produced by the LPT algorithm is no longer than the optimal off-line algorithm if and only if k ≥ m - 2.

Two-Machine Flowshop Problem with Release Dates, Rejection and Non-Availability Interval on the First Machine

2014 ◽  
Vol 620 ◽  
pp. 80-83 ◽  
Author(s):  
Xie Xie ◽  
Xiang Yu Kong ◽  
Yong Yue Zheng ◽  
Kun Wei
Keyword(s):  
Heuristic Method ◽  
Strong Sense ◽  
Release Dates ◽  
Np Hard ◽  
Worst Case ◽  
Maximal Profit ◽  
Rejection Penalty ◽  
Flowshop Problem

This paper studies a two-machine flowshop problem with release dates, rejection and non-availability interval on the first machine. The non-availability interval often origins from equipments maintain or man-power. Usually, in order to pursue maximal profit, some jobs which can be rejected, and in this situation the rejection penalty should be paid. Our objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. For this demonstrated NP-hard in strong sense, we propose a heuristic method and further demonstrate that its worst case performance is 3.

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