scholarly journals Competitive analysis for two variants of online metric matching problem

2021 ◽  
pp. 2150156
Author(s):  
Toshiya Itoh ◽  
Shuichi Miyazaki ◽  
Makoto Satake
Keyword(s):  
Greedy Algorithm ◽  
Lower Bounds ◽  
Metric Space ◽  
Competitive Analysis ◽  
Assignment Problem ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Upper And Lower Bounds ◽  
Matching Problem ◽  
Metric Matching

In the online metric matching problem, there are servers on a given metric space and requests are given one-by-one. The task of an online algorithm is to match each request immediately and irrevocably with one of the unused servers. In this paper, we pursue competitive analysis for two variants of the online metric matching problem. The first variant is a restriction where each server is placed at one of two positions, which is denoted by OMM([Formula: see text]). We show that a simple greedy algorithm achieves the competitive ratio of 3 for OMM([Formula: see text]). We also show that this greedy algorithm is optimal by showing that the competitive ratio of any deterministic online algorithm for OMM([Formula: see text]) is at least 3. The second variant is the online facility assignment problem on a line. In this problem, the metric space is a line, the servers have capacities, and the distances between any two consecutive servers are the same. We denote this problem by OFAL([Formula: see text]), where [Formula: see text] is the number of servers. We first observe that the upper and lower bounds for OMM([Formula: see text]) also hold for OFAL([Formula: see text]), so the competitive ratio for OFAL([Formula: see text]) is exactly 3. We then show lower bounds on the competitive ratio [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] for OFAL([Formula: see text]), OFAL([Formula: see text]) and OFAL([Formula: see text]), respectively.

scholarly journals Online Matching in Regular Bipartite Graphs

2018 ◽  
Vol 28 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Lali Barrière ◽  
Xavier Muñoz ◽  
Janosch Fuchs ◽  
Walter Unger
Keyword(s):  
Competitive Ratio ◽  
Online Algorithm ◽  
Upper And Lower Bounds ◽  
Partial Knowledge ◽  
Matching Problem ◽  
Time Step ◽  
Bipartite Matching ◽  
Advice Complexity ◽  
Binary Information ◽  
The Impact

In an online problem, the input is revealed one piece at a time. In every time step, the online algorithm has to produce a part of the output, based on the partial knowledge of the input. Such decisions are irrevocable, and thus online algorithms usually lead to nonoptimal solutions. The impact of the partial knowledge depends strongly on the problem. If the algorithm is allowed to read binary information about the future, the amount of bits read that allow the algorithm to solve the problem optimally is the so-called advice complexity. The quality of an online algorithm is measured by its competitive ratio, which compares its performance to that of an optimal offline algorithm. In this paper we study online bipartite matchings focusing on the particular case of bipartite matchings in regular graphs. We give tight upper and lower bounds on the competitive ratio of the online deterministic bipartite matching problem. The competitive ratio turns out to be asymptotically equal to the known randomized competitive ratio. Afterwards, we present an upper and lower bound for the advice complexity of the online deterministic bipartite matching problem.

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”.

Online Machine Scheduling with Family Setups

2016 ◽  
Vol 33 (04) ◽  
pp. 1650027
Author(s):  
Lele Zhang ◽  
Andrew Wirth
Keyword(s):  
Lower Bound ◽  
Single Machine ◽  
Competitive Analysis ◽  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Machine Scheduling ◽  
Processing Times ◽  
Family Setups ◽  
Special Case

We consider the problem of online scheduling a single machine with family setups under job availability. A setup must be scheduled when the next job comes from a different family from the last completed one, if any. The aim is to minimize the total completion time of all jobs. For the special case of identical processing times, we provide a lower bound for the competitive ratio and an online algorithm with its competitive analysis.

An Adaptive Heuristic Approach to Compute Upper and Lower Bounds for The Close-Enough Traveling Salesman Problem

2020 ◽  
Author(s):  
Francesco Carrabs ◽  
Carmine Cerrone ◽  
Raffaele Cerulli ◽  
Bruce Golden
Keyword(s):  
Greedy Algorithm ◽  
Traveling Salesman Problem ◽  
Lower Bounds ◽  
Radio Frequency Identification ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Upper And Lower Bounds ◽  
Target Point ◽  
Research Attention ◽  
Benchmark Instances

This paper addresses the close-enough traveling salesman problem, a variant of the Euclidean traveling salesman problem, in which the traveler visits a node if it passes through the neighborhood set of that node. We apply an effective strategy to discretize the neighborhoods of the nodes and the carousel greedy algorithm to appropriately select the neighborhoods that, step by step, are added to the partial solution until a feasible solution is generated. Our heuristic, based on these ingredients, is able to compute tight upper and lower bounds on the optimal solution relatively quickly. The computational results, carried out on benchmark instances, show that our heuristic often finds the optimal solution, on the instances where it is known, and in general, the upper bounds are more accurate than those from other algorithms available in the literature. Summary of Contribution: In this paper, we focus on the close-enough traveling salesman problem. This is a problem that has attracted research attention over the last 10 years; it has numerous real-world applications. For instance, consider the task of meter reading for utility companies. Homes and businesses have meters that measure the usage of gas, water, and electricity. Each meter transmits signals that can be read by a meter reader vehicle via radio-frequency identification (RFID) technology if the distance between the meter and the reader is less than r units. Each meter plays the role of a target point and the neighborhood is a disc of radius r centered at each target point. Now, suppose the meter reader vehicle is a drone and the goal is to visit each disc while minimizing the amount of energy expended by the drone. To solve this problem, we develop a metaheuristic approach, called (lb/ub)Alg, which computes both upper and lower bounds on the optimal solution value. This metaheuristic uses an innovative discretization scheme and the Carousel Greedy algorithm to obtain high-quality solutions. On benchmark instances where the optimal solution is known, (lb/ub)Alg obtains this solution 83% of the time. Over the remaining 17% of these instances, the deviation from the optimality is 0.05%, on average. On the instances with the highest overlap ratio, (lb/ub)Alg does especially well.

scholarly journals Efficient Mechanism Design for Online Scheduling

2016 ◽  
Vol 56 ◽  
pp. 429-461 ◽  
Author(s):  
Xujin Chen ◽  
Xiaodong Hu ◽  
Tie-Yan Liu ◽  
Weidong Ma ◽  
Tao Qin ◽  
...  
Keyword(s):  
Social Welfare ◽  
Mechanism Design ◽  
Lower Bounds ◽  
Competitive Analysis ◽  
Competitive Ratio ◽  
Online Scheduling ◽  
Constant Factor ◽  
Release Time ◽  
Incentive Compatible ◽  
Unequal Length

This paper concerns the mechanism design for online scheduling in a strategic setting. In this setting, each job is owned by a self-interested agent who may misreport the release time, deadline, length, and value of her job, while we need to determine not only the schedule of the jobs, but also the payment of each agent. We focus on the design of incentive compatible (IC) mechanisms, and study the maximization of social welfare (i.e., the aggregated value of completed jobs) by competitive analysis. We first derive two lower bounds on the competitive ratio of any deterministic IC mechanism to characterize the landscape of our research. We then propose a deterministic IC mechanism and show that such a simple mechanism works very well for both the preemption-restart model and the preemption-resume model. We show the mechanism can achieve the optimal competitive ratio of 5 for equal-length jobs and a near optimal competitive ratio (within a constant factor) for unequal-length jobs.

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.

WIN Algorithm for Discrete Online TSP

2011 ◽  
Vol 15 (9) ◽  
pp. 1199-1202
Author(s):  
Yonghua Wu ◽  
◽  
Guohun Zhu ◽  
Huaying Chen ◽  
Jucun Qin ◽  
...  
Keyword(s):  
Traveling Salesman Problem ◽  
Metric Space ◽  
Competitive Analysis ◽  
Competitive Ratio ◽  
Traveling Salesman ◽  
Complete Problem ◽  
Competitive Algorithm ◽  
Np Complete

Traveling Salesman Problem (TSP) which is proved as an NP-Complete problem is solved by many algorithms. In this paper, we propose online TSP which is based on general discrete metric space. A Waiting-If-Necessary (WIN) algorithm is proposed that involves with increasing cost caused by zealous algorithms and unnecessary waiting caused by cautious algorithms. We measure the performance of the WIN algorithm using competitive analysis and found that it is a 2-competitive algorithm. The competitive ratio of theWIN algorithm can be improved by setting parameterT0.

scholarly journals Favorite-Candidate Voting for Eliminating the Least Popular Candidate in a Metric Space

2020 ◽  
Vol 34 (02) ◽  
pp. 1894-1901
Author(s):  
Xujin Chen ◽  
Minming Li ◽  
Chenhao Wang
Keyword(s):  
Lower Bounds ◽  
Metric Space ◽  
Social Value ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
The Social ◽  
Special Cases ◽  
Worst Case Ratio ◽  
The One

We study single-candidate voting embedded in a metric space, where both voters and candidates are points in the space, and the distances between voters and candidates specify the voters' preferences over candidates. In the voting, each voter is asked to submit her favorite candidate. Given the collection of favorite candidates, a mechanism for eliminating the least popular candidate finds a committee containing all candidates but the one to be eliminated. Each committee is associated with a social value that is the sum of the costs (utilities) it imposes (provides) to the voters. We design mechanisms for finding a committee to optimize the social value. We measure the quality of a mechanism by its distortion, defined as the worst-case ratio between the social value of the committee found by the mechanism and the optimal one. We establish new upper and lower bounds on the distortion of mechanisms in this single-candidate voting, for both general metrics and well-motivated special cases.

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.

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