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About 2,571 papers for “Asymptotically optimal algorithm”

An Asymptotically Optimal Algorithm for the

Dutch National Flag, James R. Bitnet · 2007

We develop an algorithm for the Dutch National Flag problem that has an adjustable integer parameter smax 0 that allows a time-space trade-off.

Cited by 0

Asymptotically Optimal Algorithm for Stochastic Adwords

Nikhil R. Devanur, Balasubramanian Sivan, Yossi Azar · 2012

In this paper we consider the adwords problem in the unknown distribution model. We consider the case where the budget to bid ratio k is at least 2, and give improved competitive ratios. Earlier results had competitive ratios better than 1 − 1/e only for “large enough ” k, while our competitive ratio increases continuously with k. For k = 2 the competitive ratio we get is 0.729 and it is 0.9 for k = 16. We also improve the asymptotic competitive ratio for large k from 1 − O ( p log n/k) to 1 − O ( p 1/k), thus removing any dependence on n, the number of advertisers. This ratio is optimal, even with known distributions. That is, even if an algorithm is tailored to the distribution, it cannot get a competitive ratio of 1 − o ( p 1/k), whereas our algorithm does not depend on the distribution. The algorithm is rather simple, it computes a score for every advertiser based on his original budget, the remaining budget and the remaining number of steps in the algorithm and assigns a query to the advertiser with the highest bid plus his score. The analysis is based on a “hybrid argument ” that considers algorithms that are part actual, part hypothetical, to prove that our (actual) algorithm is better than a completely hypothetical algorithm whose performance is easy to analyze.

Cited by 0

An asymptotically optimal algorithm for online stacking

Martin Olsen, Lars Nørvang Andersen, Allan E. Gross · Mathematical Methods of Operations Research · 2023

Consider a storage area where arriving items are stored temporarily in bounded capacity stacks until their departure. We look into the problem of deciding where to put an arriving item with the objective of minimizing the maximum number of stacks used over time. The decision has to be made as soon as an item arrives, and we assume that we only have information on the departure times for the arriving item and the items currently at the storage area. We are only allowed to put an item on top of another item if the item below departs at a later time. We refer to this problem as online stacking. We assume that the storage time intervals are picked i.i.d. from $$[0, 1] \times [0, 1]$$ using an unknown distribution with a bounded probability density function. Under this mild condition, we present a simple polynomial time online algorithm and show that the competitive ratio converges to 1 in probability. The result holds if the stack capacity is $$o(\sqrt{n})$$ , where n is the number of items, including the realistic case where the capacity is a constant. Our experiments show that our results also have practical relevance.

Cited by 1Open access

An executable and asymptotically optimal adaptive RLS algorithm

Jiro Morimoto, Y. Yamamoto, Ikunori Kobayashi et al. · 2002

A setting method for the forgetting factor in recursive least squares (RLS) algorithms is presented. In a general adaptive setting method, to obtain the optimal value of the forgetting factor, the cost function constructed with the mean squared prediction errors is differentiated and the derivative that is set to zero is solved numerically. The problem of this method is that just one of local minimums is obtained unless the cost function is confirmed as unimodal. The main object of this contribution is to show the unimodality of the cost function of the RLS algorithm.

Cited by 0

An asymptotically optimal algorithm for job shop scheduling

Dimitris Bertsimas, David Gamarnik · 2003

We propose asymptotically optimal algorithms for job shop scheduling. We propose a fluid relaxation for the job shop scheduling problem, in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound C/sub max/ to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most C/sub max/+O(/spl radic/C/sub max/), where the constant in the O(/spl middot/) notation is independent of the number of jobs. However, it depends on the processing times of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C/sub max/+O(1). In computational experiments our algorithms produce schedules which are within 1% of optimality even for moderately sized problems.

Cited by 0

Asymptotically Optimal Contextual Bandit Algorithm Using Hierarchical Structures

Mohammadreza Mohaghegh Neyshabouri, Kaan Gökcesu, Hakan Gökcesu et al. · IEEE Transactions on Neural Networks and Learning Systems · 2018

We propose an online algorithm for sequential learning in the contextual multiarmed bandit setting. Our approach is to partition the context space and, then, optimally combine all of the possible mappings between the partition regions and the set of bandit arms in a data-driven manner. We show that in our approach, the best mapping is able to approximate the best arm selection policy to any desired degree under mild Lipschitz conditions. Therefore, we design our algorithm based on the optimal adaptive combination and asymptotically achieve the performance of the best mapping as well as the best arm selection policy. This optimality is also guaranteed to hold even in adversarial environments since we do not rely on any statistical assumptions regarding the contexts or the loss of the bandit arms. Moreover, we design an efficient implementation for our algorithm using various hierarchical partitioning structures, such as lexicographical or arbitrary position splitting and binary trees (BTs) (and several other partitioning examples). For instance, in the case of BT partitioning, the computational complexity is only log-linear in the number of regions in the finest partition. In conclusion, we provide significant performance improvements by introducing upper bounds (with respect to the best arm selection policy) that are mathematically proven to vanish in the average loss per round sense at a faster rate compared to the state of the art. Our experimental work extensively covers various scenarios ranging from bandit settings to multiclass classification with real and synthetic data. In these experiments, we show that our algorithm is highly superior to the state-of-the-art techniques while maintaining the introduced mathematical guarantees and a computationally decent scalability.

Cited by 15Open access

An asymptotically optimal algorithm for pickup and delivery problems

Kyle Treleaven, Marco Pavone, Emilio Frazzoli · 2011

Pickup and delivery problems (PDPs), in which objects or people have to be transported between specific locations, are among the most common combinatorial problems in real-world operations. One particular PDP is the Stacker Crane problem (SCP), where each commodity/customer is associated with a pickup location and a delivery location, and the objective is to find a minimum-length tour visiting all locations with the constraint that each pickup location and its associated delivery location are visited in consecutive order. The SCP is a route optimization problem behind several transportation systems, e.g., Transportation-On-Demand (TOD) systems. The SCP is NP-Hard and the best know approximation algorithm only provides a 9/5 approximation ratio. We present an algorithm for the stochastic SCP which: (i) is asymptotically optimal, i.e., it produces a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity O(n2+e), where n is the number of pickup/delivery pairs and e is an arbitrarily small positive constant. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations.

Cited by 0

An asymptotically optimal algorithm for pickup and delivery problems

Pavone, Marco, Frazzoli, Emilio, Treleaven, Kyle Ballantyne · DSpace@MIT (Massachusetts Institute of Technology) · 2011

Pickup and delivery problems (PDPs), in which objects or people have to be transported between specific locations, are among the most common combinatorial problems in real-world operations. One particular PDP is the Stacker Crane problem (SCP), where each commodity/customer is associated with a pickup location and a delivery location, and the objective is to find a minimum-length tour visiting all locations with the constraint that each pickup location and its associated delivery location are visited in consecutive order. The SCP is a route optimization problem behind several transportation systems, e.g., Transportation-On-Demand (TOD) systems. The SCP is NP-Hard and the best know approximation algorithm only provides a 9/5 approximation ratio. We present an algorithm for the stochastic SCP which: (i) is asymptotically optimal, i.e., it produces a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity O(n[superscript 2+ϵ]), where n is the number of pickup/delivery pairs and ϵ is an arbitrarily small positive constant. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations.

Cited by 0Open access

An Asymptotically Optimal Batched Algorithm for the Dueling Bandit Problem

Arpit Agarwal, Rohan Ghuge, Viswanath Nagarajan · arXiv (Cornell University) · 2022

We study the $K$-armed dueling bandit problem, a variation of the traditional multi-armed bandit problem in which feedback is obtained in the form of pairwise comparisons. Previous learning algorithms have focused on the $\textit{fully adaptive}$ setting, where the algorithm can make updates after every comparison. The "batched" dueling bandit problem is motivated by large-scale applications like web search ranking and recommendation systems, where performing sequential updates may be infeasible. In this work, we ask: $\textit{is there a solution using only a few adaptive rounds that matches the asymptotic regret bounds of the best sequential algorithms for $K$-armed dueling bandits?}$ We answer this in the affirmative $\textit{under the Condorcet condition}$, a standard setting of the $K$-armed dueling bandit problem. We obtain asymptotic regret of $O(K^2\log^2(K)) + O(K\log(T))$ in $O(\log(T))$ rounds, where $T$ is the time horizon. Our regret bounds nearly match the best regret bounds known in the fully sequential setting under the Condorcet condition. Finally, in computational experiments over a variety of real-world datasets, we observe that our algorithm using $O(\log(T))$ rounds achieves almost the same performance as fully sequential algorithms (that use $T$ rounds).

Cited by 0Open access

Asymptotically Optimal and Liner-time Algorithm for Polygonal Curve Simplification

Long Chen, James Z. Wang, Jinchao Xu et al. · 2005

In many application domains involving shapes and curves, polygonal curve simplification is an important part of the computer analysis processes. In this work, we have developed asymptotically optimal and linear-time algorithms to approximate a polygonal curve by another polygonal curve whose vertices are a subset of the vertices of the original one. The algorithm developed in this paper can be applied to a vector map data reduction in geographical information system especially for large-scale data. The error of the approximation is measured by the area of the domain bounded by the two polygonal curves. Based on the equidistribution principle and local refinement/coarsening strategy, an efficient

Cited by 0

AN ASYMPTOTICALLY OPTIMAL ALGORITHM FOR THE DYNAMIC TRAVELING REPAIR PROBLEM

Xiong Lu, Amelia Regan, Sandy Irani · 2002

The dynamic traveling repair problem involves providing service to customers whose locations are uniformly distributed over a bounded area in the Euclidean plane. The authors assume that customer requests arrive according to a Poisson point process. Earlier research provided a conjecture that the asymptotically optimal algorithm for this problem under very heavy traffic intensity involves partitioning the bounded area into subregions, waiting for sufficient demand to accumulate in the sub-regions and then serving demands in the sub-regions according to the optimal traveling salesman problem (TSP) tour and visiting the sub-regions in first-come first-served order as in a GI/G/M queue. Further, the researchers conjectured that the optimal algorithm for the single server case can be extended to the m-server problem by simply partitioning the service region into m sub-regions of the same size. In this paper they prove that both of these conjectures are true under some reasonable conditions and thus they demonstrate the strong connection between optimal solutions for static and dynamic vehicle routing and scheduling problems.

Cited by 2

Asymptotically optimal algorithm for job shop scheduling and packet routing

Dimitris Bertsimas, David Gamarnik · 1999

We propose asymptotically optimal algorithms for the job shop scheduling and packet routing problems. We propose a fluid relaxation for the job shop scheduling problem in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound C,,, to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most C,,,,, + O ( c) , where the constant in the O(.) notation is independent of the number of jobs, but it depends on the processing time of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C,,, + O(1). For the packet routing problem with fixed paths the previous algorithm applies directly. For the general packet routing problem we propose a linear programming relaxation that provides a lower bound C,,,, and an asymptotically optirnal algorithm that uses the optimal solution of the relaxation with objective value at most C,,,, + O ( c). Unlike asymptotically optirnal algorithms that rely on probabilistic assumptions, our proposed algorithms make no probabilistic assumptions and they are asymptotically optimal for all instances with a large number of jobs (packets). In computational experiments our algorithms produce schedules which are within 1 % of optimality even for moderately sized problems.

Cited by 0

THE DYNAMIC TRAVELING REPAIR PROBLEM: EXAMINATION OF AN ASYMPTOTICALLY OPTIMAL ALGORITHM

Xiong Lu, Amelia Regan, Sandy Irani · 2001

The dynamic traveling repair problem involves providing service to customers whose locations are uniformly distributed over a bounded area in the Euclidean plane. The authors assume that customer requests arrive according to a Poisson point process. Earlier research provided a conjecture that the asymptotically optimal algorithm for this problem under very heavy traffic intensity involves the following: partition the bounded area into sub-regions, wait for sufficient demand to accumulate in the sub-regions, serve the demands in the sub-regions according to the optimal traveling salesman problem (TSP) tour, and visit the sub-regions in first-come first-served order as in a GI/G/m queue. Further, the researchers conjectured that the optimal algorithm for the single server case can be extended to the m-server problem by simply partitioning the service region into m sub-regions of the same size and assigning one vehicle to a sub-region. In this paper they define a class of algorithms which includes the above algorithm. They then demonstrate the asymptotic optimality of an algorithm in this class and that the above algorithm is optimal among the class. Therefore, they prove the first conjecture made by the researchers. Finally, they argue that the conjecture made about the multiple server case is also true.

Cited by 0

HySST: An Asymptotically Near-Optimal Motion Planning Algorithm for Hybrid Systems

Nan Wang, Ricardo G. Sanfelice · 2023

This paper proposes a stable sparse rapidly-exploring random trees (SST) algorithm to solve the optimal motion planning problem for hybrid systems. At each iteration, the proposed algorithm, called HySST, selects a vertex with minimal cost among all the vertices within the neighborhood of a random sample, subsequently extending the search tree through flow or jump, which is also chosen randomly when both regimes are possible. In addition, HySST maintains a static set of witness points where all vertices within each witness's neighborhood are pruned, except for the ones with lowest cost. We show that HySST is asymptotically near-optimal, namely, the probability of failing to find a motion plan with cost close to the optimal approaches zero as the number of iterations of the algorithm increases to infinity. The proposed algorithm is applied to a collision-resilient tensegrity multicopter system so as to highlight its generality and computational features.

Cited by 2

Combinatorial Optimization Problems For Which Almost Every Algorithm Is Asymptotically Optimal!

Wojciech Szpankowski · Purdue e-Pubs (Purdue University System) · 1994

Consider a class of optimization problems for which the cardinality of the set of feasible solutions is m and the size of every feasible solution is N . We prove in a general probabilistic framework that the value of the optimal solution and the value of the worst solution are asymptotically almost surely (a.s.) the same provided log m = o(N) as N and m become large. This result implies that for such a class of combinatorial optimization problems almost every algorithm finds asymptotically optimal solution! The quadratic assignment problem, the location problem on graphs, and a pattern matching problem fall into this class. This research was primary done while the author was visiting INRIA, Rocquencourt, France, and he wishes to thank INRIA (projects ALGO, MEVAL and REFLECS) for a generous support. In addition, support was provided by NSF Grants CCR-9201078, NCR-9206315 and INT-8912631, by Grant AFOSR-90-0107, and in part by NATO Collaborative Grant 0057/89. 1. INTRODUCTION We co...

Cited by 0Open access