scholarly journals Greedy Matching in Bipartite Random Graphs

2021 ◽  
Author(s):  
Nick Arnosti
Keyword(s):  
Random Graphs ◽  
Bipartite Graphs ◽  
Approximation Ratio ◽  
Prior Work ◽  
Asymptotic Results ◽  
Worst Case ◽  
Average Performance ◽  
Finite Graphs ◽  
Random Priority ◽  
Priority Order

This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].

scholarly journals Average-case Analysis of the Assignment Problem with Independent Preferences

2019 ◽  
Author(s):  
Yansong Gao ◽  
Jie Zhang
Keyword(s):  
Uniform Distribution ◽  
Assignment Problem ◽  
Affirmative Answer ◽  
Approximation Ratio ◽  
Average Case Analysis ◽  
Worst Case ◽  
Average Case ◽  
Efficiency Loss ◽  
Random Priority ◽  
Independent Preferences

The fundamental assignment problem is in search of welfare maximization mechanisms to allocate items to agents when the private preferences over indivisible items are provided by self-interested agents. The mainstream mechanism \textit{Random Priority} is asymptotically the best mechanism for this purpose, when comparing its welfare  to the optimal social welfare using the canonical \textit{worst-case approximation ratio}.  Surprisingly, the efficiency loss indicated by the worst-case ratio does not have a constant bound \cite{FFZ:14}.Recently, \cite{DBLP:conf/mfcs/DengG017} shows that when the agents' preferences are drawn from a uniform distribution, its \textit{average-case approximation ratio} is upper bounded by 3.718. They left it as an open question of whether a constant ratio holds for general scenarios. In this paper, we offer an affirmative answer to this question by showing that the ratio is bounded by $1/\mu$ when the preference values are independent and identically distributed random variables, where $\mu$ is the expectation of the value distribution. This upper bound improves the results in \cite{DBLP:conf/mfcs/DengG017} for the Uniform distribution as well. Moreover, under mild conditions, the ratio has a \textit{constant} bound for any independent  random values. En route to these results, we develop powerful tools to show the insights that for most valuation inputs, the efficiency loss is small.

scholarly journals Combinatorial approximation of maximum k-vertex cover in bipartite graphs within ratio 0.7

2018 ◽  
Vol 52 (1) ◽  
pp. 305-314 ◽  
Author(s):  
Vangelis Th. Paschos
Keyword(s):  
Greedy Algorithm ◽  
Vertex Cover ◽  
Bipartite Graphs ◽  
Approximation Ratio ◽  
Combinatorial Algorithm

We propose and analyze a simple purely combinatorial algorithm for max k-vertex cover in bipartite graphs, achieving approximation ratio 0.7. The only combinatorial algorithm currently known until now for this problem is the natural greedy algorithm, that achieves ratio (e − 1)/e = 0.632.

scholarly journals Average-Case Approximation Ratio of Scheduling without Payments

Algorithmica ◽  
2021 ◽  
Author(s):  
Jie Zhang
Keyword(s):  
Mechanism Design ◽  
Case Analysis ◽  
Theoretical Computer Science ◽  
Approximation Ratio ◽  
Approximation Schemes ◽  
Worst Case Analysis ◽  
Algorithmic Mechanism Design ◽  
Worst Case ◽  
Average Case ◽  
Optimal Mechanism

AbstractApart from the principles and methodologies inherited from Economics and Game Theory, the studies in Algorithmic Mechanism Design typically employ the worst-case analysis and design of approximation schemes of Theoretical Computer Science. For instance, the approximation ratio, which is the canonical measure of evaluating how well an incentive-compatible mechanism approximately optimizes the objective, is defined in the worst-case sense. It compares the performance of the optimal mechanism against the performance of a truthful mechanism, for all possible inputs. In this paper, we take the average-case analysis approach, and tackle one of the primary motivating problems in Algorithmic Mechanism Design—the scheduling problem (Nisan and Ronen, in: Proceedings of the 31st annual ACM symposium on theory of computing (STOC), 1999). One version of this problem, which includes a verification component, is studied by Koutsoupias (Theory Comput Syst 54(3):375–387, 2014). It was shown that the problem has a tight approximation ratio bound of $$(n+1)/2$$ ( n + 1 ) / 2 for the single-task setting, where n is the number of machines. We show, however, when the costs of the machines to executing the task follow any independent and identical distribution, the average-case approximation ratio of the mechanism given by Koutsoupias (Theory Comput Syst 54(3):375–387, 2014) is upper bounded by a constant. This positive result asymptotically separates the average-case ratio from the worst-case ratio. It indicates that the optimal mechanism devised for a worst-case guarantee works well on average.

Robust worst-case design for optimizing average performance in OFDM using quantized CSI

2010 ◽  
Author(s):  
Ana B. Rodriguez-Gonzalez ◽  
Luis M. Lopez-Ramos ◽  
Antonio G. Marques ◽  
Javier Ramos ◽  
Antonio J. Caamano
Keyword(s):  
Worst Case ◽  
Average Performance ◽  
Case Design

scholarly journals The price of anarchy in routing games as a function of the demand

2021 ◽  
Author(s):  
Roberto Cominetti ◽  
Valerio Dose ◽  
Marco Scarsini
Keyword(s):  
Price Of Anarchy ◽  
Heavy Traffic ◽  
Scaling Law ◽  
Congestion Games ◽  
Social Optimum ◽  
Asymptotic Results ◽  
Worst Case ◽  
Traffic Demand ◽  
Break Points ◽  
Routing Games

AbstractThe price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite.

Dynamic Partitioned Cache Memory for Real-Time MPSoCs with Mixed Criticality

2016 ◽  
Vol 25 (06) ◽  
pp. 1650062 ◽  
Author(s):  
Gang Chen ◽  
Kai Huang ◽  
Long Cheng ◽  
Biao Hu ◽  
Alois Knoll
Keyword(s):  
Real Time ◽  
Cache Memory ◽  
Time System ◽  
Worst Case ◽  
Real Time System ◽  
Average Performance ◽  
Shared Cache ◽  
Worst Case Execution Time ◽  
The One ◽  
Major Factors

Shared cache interference in multi-core architectures has been recognized as one of major factors that degrade predictability of a mixed-critical real-time system. Due to the unpredictable cache interference, the behavior of shared cache is hard to predict and analyze statically in multi-core architectures executing mixed-critical tasks, which will not only result in difficulty of estimating the worst-case execution time (WCET) but also introduce significant worst-case timing penalties for critical tasks. Therefore, cache management in mixed-critical multi-core systems has become a challenging task. In this paper, we present a dynamic partitioned cache memory for mixed-critical real-time multi-core systems. In this architecture, critical tasks can dynamically allocate and release the cache resourse during the execution interval according to the real-time workload. This dynamic partitioned cache can, on the one hand, provide the predicable cache performance for critical tasks. On the other hand, the released cache can be dynamically used by non-critical tasks to improve their average performance. We demonstrate and prototype our system design on the embedded FPGA platform. Measurements from the prototype clearly demonstrate the benefits of the dynamic partitioned cache for mixed-critical real-time multi-core systems.

scholarly journals Randomness in classes of matroids

2021 ◽  
Author(s):  
◽  
William Critchlow
Keyword(s):  
Lower Bound ◽  
Bipartite Graphs ◽  
Random Model ◽  
Asymptotic Results ◽  
Mathematical Objects ◽  
Set Systems ◽  
Random Models ◽  
Special Subset ◽  
One To One ◽  
Almost All

<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids.  Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors.  Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like.  Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

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