Worst-case performance of approximation algorithms for tool management problems

1999 ◽  
Vol 46 (5) ◽  
pp. 445-462 ◽  
Author(s):  
Yves Crama ◽  
Joris van de Klundert
Keyword(s):  
Approximation Algorithms ◽  
Worst Case ◽  
Tool Management ◽  
Management Problems

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.

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.

Linear-Time Approximation Algorithms for the Max Cut Problem

1993 ◽  
Vol 2 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Nguyen van Ngoc ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Approximation Algorithms ◽  
Lower Bounds ◽  
Connected Graph ◽  
Linear Time ◽  
Worst Case ◽  
Time Approximation ◽  
Bipartite Subgraph ◽  
Max Cut Problem ◽  
Multiple Edges

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

scholarly journals Worst-case optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks

2010 ◽  
Vol 8 (1) ◽  
pp. 65-75 ◽  
Author(s):  
Jaroslaw Byrka ◽  
Pawel Gawrychowski ◽  
Katharina T. Huber ◽  
Steven Kelk
Keyword(s):  
Approximation Algorithms ◽  
Optimal Approximation ◽  
Phylogenetic Networks ◽  
Worst Case

scholarly journals Randomized Strategies for Robust Combinatorial Optimization

2019 ◽  
Vol 33 ◽  
pp. 7876-7883 ◽  
Author(s):  
Yasushi Kawase ◽  
Hanna Sumita
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Independent Set ◽  
Objective Functions ◽  
Worst Case ◽  
Matroid Intersection ◽  
Independence System ◽  
Knapsack Constraint ◽  
Objective Value ◽  
Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory and optimization. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update (MWU) method. The direct application of the MWU method does not yield a strict multiplicative approximation algorithm but yield one with an additional additive error term. A key technique to overcome the issue is to introduce a new concept called (η,γ)-reductions for objective functions with parameters η and γ. We show that our scheme outputs a nearly α-approximate solution if there exists an α-approximation algorithm for a subproblem defined by (η,γ)-reductions. This improves approximation ratios in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.

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.

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