Results 1  10
of
22
A Bad Instance for kmeans++
, 2011
"... kmeans++ is a seeding technique for the kmeans method with an expected approximation ratio of O(log k), where k denotes the number of clusters. Examples are known on which the expected approximation ratio of kmeans++ is Ω(log k), showing that the upper bound is asymptotically tight. However, it ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
kmeans++ is a seeding technique for the kmeans method with an expected approximation ratio of O(log k), where k denotes the number of clusters. Examples are known on which the expected approximation ratio of kmeans++ is Ω(log k), showing that the upper bound is asymptotically tight. However, it remained open whether kmeans++ yields an O(1)approximation with probability 1/poly(k) or even with constant probability. We settle this question and present instances on which kmeans++ achieves an approximation ratio of (2/3 − ε) · log k only with exponentially small probability.
DOI: 10.7155/jgaa.00310 Smoothed Analysis of Belief Propagation for MinimumCost Flow and Matching
, 2013
"... Belief propagation (BP) is a messagepassing heuristic for statistical inference in graphical models such as Bayesian networks and Markov random fields. BP is used to compute marginal distributions or maximum likelihood assignments and has applications in many areas, including machine learning, im ..."
Abstract
 Add to MetaCart
Belief propagation (BP) is a messagepassing heuristic for statistical inference in graphical models such as Bayesian networks and Markov random fields. BP is used to compute marginal distributions or maximum likelihood assignments and has applications in many areas, including machine learning, image processing, and computer vision. However, the theoretical understanding of the performance of BP remains limited. Recently, BP has been applied to combinatorial optimization problems. It has been proved that BP can be used to compute maximumweight matchings and minimumcost flows for instances with a unique optimum. The number of iterations needed for this is pseudopolynomial and hence BP is not efficient in general. We study BP in the framework of smoothed analysis and prove that with high probability the number of iterations needed to compute maximumweight matchings and minimumcost flows is bounded by a polynomial if the weights/costs of the edges are randomly perturbed. To prove our upper bounds, we use an isolation lemma by Beier and Vöcking (SIAM Journal on Computing, 2006) for the matching problem and we generalize an isolation lemma by Gamarnik, Shah, and Wei (Operations Research, 2012) for the mincost flow problem. We also prove lower tail bounds for the number of iterations that BP needs to converge that almost match our upper bounds. Submitted:
Lower Bounds for the Smoothed Number of Pareto optimal Solutions
, 2011
"... In 2009, Röglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multicriteria optimization problems is polynomially bounded in the number n of variables and the maximum density φ of the semirandom input model for any fixed number of objective functions. Their bound ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In 2009, Röglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multicriteria optimization problems is polynomially bounded in the number n of variables and the maximum density φ of the semirandom input model for any fixed number of objective functions. Their bound is, however, not very practical because the exponents grow exponentially in the number d+1 of objective functions. In a recent breakthrough, Moitra and O’Donnell improved this bound significantly to O ( n²d φ d(d+1)/2). An “intriguing problem”, which Moitra and O’Donnell formulate in their paper, is how much further this bound can be improved. The previous lower bounds do not exclude the possibility of a polynomial upper bound whose degree does not depend on d. In this paper we resolve this question by constructing a class of instances with Ω((nφ) (d−log(d))·(1−Θ(1/φ))) Pareto optimal solutions in expectation. For the bicriteria case we present a higher lower bound of Ω(n² φ 1−Θ(1/φ)), which almost matches the known upper bound of O(n² φ).
Path Trading: Fast Algorithms, Smoothed Analysis, and Hardness Results
"... Abstract. The Border Gateway Protocol (BGP) serves as the main routing protocol of the Internet and ensures network reachability among autonomous systems (ASes). When traffic is forwarded between the many ASes on the Internet according to that protocol, each AS selfishly routes the traffic inside it ..."
Abstract
 Add to MetaCart
Abstract. The Border Gateway Protocol (BGP) serves as the main routing protocol of the Internet and ensures network reachability among autonomous systems (ASes). When traffic is forwarded between the many ASes on the Internet according to that protocol, each AS selfishly routes the traffic inside its own network according to some internal protocol that supports the local objectives of the AS. We consider possibilities of achieving higher global performance in such systems while maintaining the objectives and costs of the individual ASes. In particular, we consider how path trading, i.e. deviations from routing the traffic using individually optimal protocols, can lead to a better global performance. Shavitt and Singer (“Limitations and Possibilities of Path Trading between Autonomous Systems”, INFOCOM 2010) were the first to consider the computational complexity of finding such path trading solutions. They show that the problem is weakly NPhard and provide a dynamic program to find path trades between pairs of ASes. In this paper we improve upon their results, both theoretically and practically. First, we show that finding path trades between sets of ASes is also strongly NPhard. Moreover, we provide an algorithm that finds all Paretooptimal path trades for a pair of two ASes. While in principal the number of Paretooptimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give theoretical evidence that this is a general phenomenon, and not only limited to the instances on which we performed experiments. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than the previous dynamic program. 1
Smoothed Analysis of the Successive Shortest Path Algorithm
, 2013
"... The minimumcost flow problem is a classic problem in combinatorial optimization with various applications. Several pseudopolynomial, polynomial, and strongly polynomial algorithms have been developed in the past decades, and it seems that both the problem and the algorithms are well understood. Ho ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The minimumcost flow problem is a classic problem in combinatorial optimization with various applications. Several pseudopolynomial, polynomial, and strongly polynomial algorithms have been developed in the past decades, and it seems that both the problem and the algorithms are well understood. However, some of the algorithms’ running times observed in empirical studies contrast the running times obtained by worstcase analysis not only in the order of magnitude but also in the ranking when compared to each other. For example, the Successive Shortest Path (SSP) algorithm, which has an exponential worstcase running time, seems to outperform the strongly polynomial MinimumMean Cycle Canceling algorithm. To explain this discrepancy, we study the SSP algorithm in the framework of smoothed analysis and establish a bound of O(mnφ(m + n log n)) for its smoothed running time. This shows that worstcase instances for the SSP algorithm are not robust and unlikely to be encountered in practice.
Finding short paths on polytopes by the shadow vertex algorithm
 In Automata, Languages, and Programming
, 2013
"... We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x ∈ Rn: Ax ≤ b} along the edges of P, where A ∈ Rm×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x ∈ Rn: Ax ≤ b} along the edges of P, where A ∈ Rm×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that is a measure for the flatness of the vertices of P. For integer matrices A ∈ Zm×n we show a connection between δ and the largest absolute value ∆ of any subdeterminant of A, yielding a bound of O(∆4mn4) for the length of the computed path. This bound is expressed in the same parameter ∆ as the recent nonconstructive bound of O(∆2n4 log(n∆)) by Bonifas et al. [1]. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(mn4), which significantly improves the previously best known constructive bound of O(m16n3 log3(mn)) by Dyer and Frieze [7]. 1
Smoothed Performance Guarantees for Local Search ∗
"... We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worstcase lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out ho ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worstcase lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lexjump algorithm are (in contrast to the worst case) independent of the number of machines. They are Θ(φ) and Θ(log φ), respectively, where 1/φ is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is Θ(log φ) for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of Θ(log φ). 1
1 Executive Summary
"... This report documents the program and the outcomes of Dagstuhl Seminar 14372 “Analysis of Algorithms Beyond the Worst Case”. The theory of algorithms has traditionally focused on worstcase analysis. This focus has led to both a deep theory and many beautiful and useful algorithms. However, there ar ..."
Abstract
 Add to MetaCart
This report documents the program and the outcomes of Dagstuhl Seminar 14372 “Analysis of Algorithms Beyond the Worst Case”. The theory of algorithms has traditionally focused on worstcase analysis. This focus has led to both a deep theory and many beautiful and useful algorithms. However, there are a number of important problems and algorithms for which worstcase analysis does not provide useful or empirically accurate results. This is due to the fact that worstcase inputs are often rather contrived and occur hardly ever in practical applications. Only in recent years a paradigm shift towards a more realistic and robust algorithmic theory has been initiated. The development of a more realistic theory hinges on finding models that measure the performance of an algorithm not only by its worstcase behavior but rather by its behavior on “typical ” inputs. In this seminar, we discussed various recent theoretical models and results that go beyond worstcase analysis. The seminar helped to consolidate the research and to foster collaborations among the researchers working in the different branches of analysis of algorithms beyond the worst case.
Results 1  10
of
22