scholarly journals Online Matching in Regular Bipartite Graphs

2018 ◽  
Vol 28 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Lali Barrière ◽  
Xavier Muñoz ◽  
Janosch Fuchs ◽  
Walter Unger

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.

Author(s):  
Toshiya Itoh ◽  
Shuichi Miyazaki ◽  
Makoto Satake

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.


2018 ◽  
Vol 29 (04) ◽  
pp. 505-527
Author(s):  
Maria Paola Bianchi ◽  
Hans-Joachim Böckenhauer ◽  
Tatjana Brülisauer ◽  
Dennis Komm ◽  
Beatrice Palano

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.


Author(s):  
Bernd Brüggenjürgen ◽  
Hans-Peter Stricker ◽  
Lilian Krist ◽  
Miriam Ortiz ◽  
Thomas Reinhold ◽  
...  

Abstract Aim To use a Delphi-panel-based assessment of the effectiveness of different non-pharmaceutical interventions (NPI) in order to retrospectively approximate and to prospectively predict the SARS-CoV-2 pandemic progression via a SEIR model (susceptible, exposed, infectious, removed). Methods We applied an evidence-educated Delphi-panel approach to elicit the impact of NPIs on the SARS-CoV-2 transmission rate R0 in Germany. Effectiveness was defined as the product of efficacy and compliance. A discrete, deterministic SEIR model with time step of 1 day, a latency period of 1.8 days, duration of infectiousness of 5 days, and a share of the total population of 15% assumed to be protected by immunity was developed in order to estimate the impact of selected NPI measures on the course of the pandemic. The model was populated with the Delphi-panel results and varied in sensitivity analyses. Results Efficacy and compliance estimates for the three most effective NPIs were as follows: test and isolate 49% (efficacy)/78% (compliance), keeping distance 42%/74%, personal protection masks (cloth masks or other face masks) 33%/79%. Applying all NPI effectiveness estimates to the SEIR model resulted in a valid replication of reported occurrence of the German SARS-CoV-2 pandemic. A combination of four NPIs at consented compliance rates might curb the CoViD-19 pandemic. Conclusion Employing an evidence-educated Delphi-panel approach can support SARS-CoV-2 modelling. Future curbing scenarios require a combination of NPIs. A Delphi-panel-based NPI assessment and modelling might support public health policy decision making by informing sequence and number of needed public health measures.


2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Daniel Vert ◽  
Renaud Sirdey ◽  
Stéphane Louise

AbstractThis paper experimentally investigates the behavior of analog quantum computers as commercialized by D-Wave when confronted to instances of the maximum cardinality matching problem which is specifically designed to be hard to solve by means of simulated annealing. We benchmark a D-Wave “Washington” (2X) with 1098 operational qubits on various sizes of such instances and observe that for all but the most trivially small of these it fails to obtain an optimal solution. Thus, our results suggest that quantum annealing, at least as implemented in a D-Wave device, falls in the same pitfalls as simulated annealing and hence provides additional evidences suggesting that there exist polynomial-time problems that such a machine cannot solve efficiently to optimality. Additionally, we investigate the extent to which the qubits interconnection topologies explains these latter experimental results. In particular, we provide evidences that the sparsity of these topologies which, as such, lead to QUBO problems of artificially inflated sizes can partly explain the aforementioned disappointing observations. Therefore, this paper hints that denser interconnection topologies are necessary to unleash the potential of the quantum annealing approach.


Algorithmica ◽  
2021 ◽  
Author(s):  
Matthias Englert ◽  
David Mezlaf ◽  
Matthias Westermann

AbstractIn the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.


2007 ◽  
Vol 8 (3) ◽  
pp. 439-446 ◽  
Author(s):  
Dagang Wang ◽  
Guiling Wang

Abstract Representation of the canopy hydrological processes has been challenging in land surface modeling due to the subgrid heterogeneity in both precipitation and surface characteristics. The Shuttleworth dynamic–statistical method is widely used to represent the impact of the precipitation subgrid variability on canopy hydrological processes but shows unwanted sensitivity to temporal resolution when implemented into land surface models. This paper presents a canopy hydrology scheme that is robust at different temporal resolutions. This scheme is devised by applying two physically based treatments to the Shuttleworth scheme: 1) the canopy hydrological processes within the rain-covered area are treated separately from those within the nonrain area, and the scheme tracks the relative rain location between adjacent time steps; and 2) within the rain-covered area, the canopy interception is so determined as to sustain the potential evaporation from the wetted canopy or is equal to precipitation, whichever is less, to maintain somewhat wet canopy during any rainy time step. When applied to the Amazon region, the new scheme establishes interception loss ratios of 0.3 at a 10-min time step and 0.23 at a 2-h time step. Compared to interception loss ratios of 0.45 and 0.09 at the corresponding time steps established by the original Shuttleworth scheme, the new scheme is much more stable under different temporal resolutions.


2019 ◽  
Vol 11 (3) ◽  
pp. 1463-1481 ◽  
Author(s):  
Ekaterina P. Rets ◽  
Viktor V. Popovnin ◽  
Pavel A. Toropov ◽  
Andrew M. Smirnov ◽  
Igor V. Tokarev ◽  
...  

Abstract. This study presents a dataset on long-term multidisciplinary glaciological, hydrological, and meteorological observations and isotope sampling in a sparsely monitored alpine zone of the North Caucasus in the Djankuat research basin. The Djankuat glacier, which is the largest in the basin, was chosen as representative of the central North Caucasus during the International Hydrological Decade and is one of 30 “reference” glaciers in the world that have annual mass balance series longer than 50 years (Zemp et al., 2009). The dataset features a comprehensive set of observations from 2007 to 2017 and contains yearly measurements of snow depth and density; measurements of dynamics of snow and ice melting; measurements of water runoff, conductivity, turbidity, temperature, δ18O, δD at the main gauging station (844 samples in total) with an hourly or sub-daily time step depending on the parameter; data on δ18O and δ2H sampling of liquid precipitation, snow, ice, firn, and groundwater in different parts of the watershed taken regularly during melting season (485 samples in total); measurements of precipitation amount, air temperature, relative humidity, shortwave incoming and reflected radiation, longwave downward and upward radiation, atmospheric pressure, and wind speed and direction – measured at several automatic weather stations within the basin with 15 min to 1 h time steps; gradient meteorological measurements to estimate turbulent fluxes of heat and moisture, measuring three components of wind speed at a frequency of 10 Hz to estimate the impulse of turbulent fluxes of sensible and latent heat over the glacier surface by the eddy covariance method. Data were collected during the ablation period (June–September). The observations were halted in winter. The dataset is available from PANGAEA (https://doi.org/10.1594/PANGAEA.894807, Rets et al., 2018a) and will be further updated. The dataset can be useful for developing and verifying hydrological, glaciological, and meteorological models for alpine areas, to study the impact of climate change on hydrology of mountain regions using isotopic and hydrochemical approaches in hydrology. As the dataset includes the measurements of hydrometeorological and glaciological variables during the catastrophic proglacial lake outburst in the neighboring Bashkara valley in September 2017, it is a valuable contribution to study lake outbursts.


10.37236/5756 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jakub Kozik ◽  
Grzegorz Matecki

We present a new model for the problem of on-line matching on bipartite graphs. Suppose that one part of a graph is given, but the vertices of the other part are presented in an on-line fashion. In the classical version, each incoming vertex is either irrevocably matched to a vertex from the other part or stays unmatched forever. In our version, an algorithm is allowed to match the new vertex to a group of elements (possibly empty). Later on, the algorithm can decide to remove some vertices from the group and assign them to another (just presented) vertex, with the restriction that each element belongs to at most one group. We present an optimal (deterministic) algorithm for this problem and prove that its competitive ratio equals $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$.


2014 ◽  
Vol 31 (04) ◽  
pp. 1450030 ◽  
Author(s):  
CHENGWEN JIAO ◽  
WENHUA LI ◽  
JINJIANG YUAN

We consider online scheduling of unit length jobs on m identical parallel-batch machines. Jobs arrive over time. The objective is to minimize maximum flow-time, with the flow-time of a job being the difference of its completion time and its release time. A parallel-batch machine can handle up to b jobs simultaneously as a batch. Here, the batch capacity is bounded, that is b < ∞. In this paper, we provide a best possible online algorithm for the problem with a competitive ratio of [Formula: see text].


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