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FORMALIZING randomized MATCHING algorithms

LOGICAL METHODS IN COMPUTER SCIENCE 2012年第3期8卷 6-6页

作者： Le, Dai Tri Man Cook, Stephen A. Univ Toronto Dept Comp Sci Toronto ON Canada

Using Jerabek's framework for probabilistic reasoning, we formalize the correctness of two fundamental RNC2 algorithms for bipartite perfect matching within the theory VPV for polytime reasoning. The first algorithm is for testing if a bipartite graph has a perfect matching, and is based on the Schwartz-Zippel Lemma for polynomial identity testing applied to the Edmonds polynomial of the graph. The second algorithm, due to Mulmuley, Vazirani and Vazirani, is for finding a perfect matching, where the key ingredient of this algorithm is the Isolating Lemma.

关键词： bounded arithmetic computational complexity weak pigeonhole principle probabilistic reasoning randomized algorithms polynomial identity testing

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randomized Incremental algorithms for the PageRank Computation

Randomized Incremental Algorithms for the PageRank Computati...

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IEEE Annual Conference on Decision and Control

作者： Keyou You Roberto Tempo Li Qiu Department of Automation Tsinghua University 100084 China CNR-IEIIT Politecnico di Torino 10129 Torino Italy Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Hong Kong China

ISBN: (纸本)9781479978878

This paper proposes a new class of randomized algorithms to incrementally compute the Google PageRank of a large-scale number of webpages. First, we reformulate the PageRank computation as a least squares (LS) problem. Motivated by a random surfer model, the LS problem is solved in a randomized incremental way. Specifically, when visiting a page, the surfer incrementally updates an estimate of the PageRank by using the information from those pages connected by hyperlinks. Then, the surfer either randomly selects an outgoing link of the current page and moves to the page pointed by this link, or interrupts its search and jumps to an arbitrary page. Subsequently, the PageRank estimate is updated again. The transition between pages is naturally modeled as a Markov process. Under an ergodicity property, we prove the convergence of the proposed algorithm to the PageRank in both the almost sure and L~P sense. A comparison with a classical PageRank algorithm is discussed as well.

关键词： Markov chain Incremental randomized algorithms RANDOMISING Web pages New classes Hyperlinks algorithms current page Converge interruption

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randomized Gossip algorithms for Solving Laplacian Systems

Randomized Gossip Algorithms for Solving Laplacian Systems

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European Control Conference

作者： Anastasios Zouzias Nikolaos M. Freris department of Mathematical and Computational Sciences at IBM-Research Zurich Engineering Division of the New York University

ISBN: (纸本)9781467371605

We consider the problem of solving a Laplacian system of equations Lx = b in a distributed fashion, where L is the Laplacian of the communication graph. Solving Laplacian systems arises in a number of applications including consensus, distributed control, clock synchronization, localization and calculating effective resistances, to name a few. We leverage our analysis on a randomized variant of Kaczmarz's algorithm to propose a distributed asynchronous gossip algorithm with expected exponential convergence. We quantify the convergence rate depending solely on properties of the network topology, and further propose an accelerated version that scales favorably for larger networks. Our approach naturally extends to least-squares estimation of general linear systems where each row/column is assigned to nodes of a given network. Last but not least, we show that average consensus is a special case in our framework.

关键词： Gossip algorithms Laplacian systems Consensus Distributed algorithms Cyberphysical systems randomized algorithms Clock synchronization

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Adwords with Unknown Budgets and Beyond

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MANAGEMENT SCIENCE 2025年第2期71卷 iv-vi, 955页

作者： Udwani, Rajan Univ Calif Berkeley Ind Engn & Operat Res Berkeley CA 94720 USA

In the classic Adwords problem introduced by [Mehta A, Saberi A, Vazirani U, Vazirani V (2007) Adwords and generalized online matching. J. ACM 54(5):22-es.], we have a bipartite graph between advertisers and queries. Each advertiser has a maximum budget that is known a priori. Queries are unknown a priori and arrive sequentially. When a query arrives, advertisers make bids, and we (immediately and irrevocably) decide which (if any) Ad to display based on the bids and advertiser budgets. The winning advertiser for each query pays their bid up to their remaining budget. Our goal is to maximize total budget used without any foreknowledge of the arrival sequence (which could be adversarial). We consider the setting where the online algorithm does not know the advertisers' budgets a priori and the budget of an advertiser is revealed to the algorithm only when it is exceeded. A naive greedy algorithm is 0.5 competitive for this setting, and finding an algorithm with better performance remained an open problem. We show that no deterministic algorithm has competitive ratio better than 0.5 and give the first (randomized) algorithm with strictly better performance guarantee. We show that the competitive ratio of our algorithm is at least 0.522 but also strictly less than (1 - 1/e). We present novel applications of budget oblivious algorithms in search ads and beyond. In particular, we show that our algorithm achieves the best possible performance guarantee for deterministic online matching in the presence of multichannel traffic [Manshadi V, Rodilitz S, Saban D, Suresh A (2022) Online algorithms for matching platforms with multi-channel traffic. Proc. 23rd ACMConf. Econom. Comput. (ACM, NewYork), 986-987.].

关键词： Adwords randomized algorithms unknown budgets competitive ratio

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Identification of piecewise affine systems using a cluster refinement technique

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EUROPEAN JOURNAL OF CONTROL 2025年 83卷

作者： Yu, Miao Bianchi, Federico Piroddi, Luigi Politecn Milan Dipartimento Elettron Informaz & Bioingn I-20133 Milan Italy RSE SpA Ric Sistema Energet Via R Rubattino 54 I-20134 Milan Italy

The identification of piecewise affine (PWA) systems is a challenging mixed integer optimization problem that involves both the estimation of the dynamics associated to different modes of operation, and the partition of the state space in regions associated to said modes, the transition from one region to another corresponding to a mode switching. The challenges are mainly associated with the sample-mode assignment task, because the combinatorial complexity increases with the size of the dataset. Furthermore, some samples are consistent with more than one mode, making their classification 'ambiguous'. The identification problem is here addressed with a two-stage iterative method, alternating between an identification phase carried out over given clusters of data associated to regions in the state space (such that each cluster is assigned to a single mode), and a refinement phase, whereby the region borders are adjusted (by reassigning samples to other clusters) to improve the model quality. Operating on data clusters (as opposed to individual samples) significantly reduces the complexity of the combinatorial mode assignment problem, and naturally avoids region outliers (isolated samples surrounded by samples assigned to a different mode). However, this approach works properly only if accompanied by a cluster refinement procedure, responsible for reshaping the mode regions and reassigning stray samples to the correct modes. The combination of these two stages is ultimately successful in determining correctly both the local models and the associated state space regions, as shown here with reference to several benchmark examples.

关键词： System identification Piecewise affine systems randomized algorithms Hybrid systems

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Batched ranged random integer generation

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SOFTWARE-PRACTICE & EXPERIENCE 2025年第1期55卷 155-169页

作者： Brackett-Rozinsky, Nevin Lemire, Daniel Univ Quebec TELUQ Data Sci Res Ctr Montreal PQ H2S 3L5 Canada

Pseudorandom values are often generated as 64-bit binary words. These random words need to be converted into ranged values without statistical bias. We present an efficient algorithm to generate multiple independent uniformly-random bounded integers from a single uniformly-random binary word, without any bias. In the common case, our method uses one multiplication and no division operations per value produced. In practice, our algorithm can more than double the speed of unbiased random shuffling for small to moderately large arrays.

关键词： random number generation randomized algorithms rejection method

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Optimal matchings with one-sided preferences: fixed and cost-based quotas

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AUTONOMOUS AGENTS AND MULTI-AGENT SYSTEMS 2025年第1期39卷 1-36页

作者： Santhini, K. A. Sankar, Govind S. Nasre, Meghana Indian Inst Technol Madras Comp Sci & Engn Chennai India Duke Univ Comp Sci Durham NC 27708 USA

We consider the well-studied many-to-one bipartite matching problem of assigning applicants A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\mathcal {A}}}$$\end{document} to posts P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\mathcal {P}}}$$\end{document} where applicants rank posts in the order of preference. This setting models many important real-world allocation problems like assigning students to courses, applicants to jobs, amongst many others. In such scenarios, it is natural to ask for an allocation that satisfies guarantees of the form "match at least 80% of applicants to one of their top three choices" or "it is unacceptable to leave more than 10% of applicants unassigned". The well-studied notions of rank-maximality and fairness fail to capture such requirements due to their property of optimizing extreme ends of the signature of a matching. We, therefore, propose a novel optimality criterion, which we call the "weak dominance " of *** investigate the computational complexity of the new notion of optimality in the setting where posts have associated fixed quotas. We prove that under the fixed quota setting, the problem turns out to be NP-hard under natural restrictions. We provide randomized algorithms in the fixed quota setting when the number of ranks is constant. We also study the problem under a cost-based quota setting and show that a matching that weakly dominates the input signature and has minimum total cost can be computed efficiently. Apart from circumventing the hardness, the cost-based quota setting is motivated by real-world applications like course allocati

关键词： Matchings with preferences Network flows randomized algorithms NP-hardness

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The open-closed mod-minimizer algorithm

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algorithms FOR MOLECULAR BIOLOGY 2025年第1期20卷 1-17页

作者： Groot Koerkamp, Ragnar Liu, Daniel Pibiri, Giulio Ermanno Swiss Fed Inst Technol Zurich Switzerland Univ Calif Los Angeles Los Angeles CA USA Ca Foscari Univ Venice Venice Italy ISTI CNR Pisa Italy

Sampling algorithms that deterministically select a subset of k-mers are an important building block in bioinformat-ics applications. For example, they are used to index large textual collections, like DNA, and to compare sequences quickly. In such applications, a sampling algorithm is required to select one k-mer out of every window of w consecu-tive k-mers. The folklore and most used scheme is the random minimizer that selects the smallest k-mer in the win-dow according to some random order. This scheme is remarkably simple and versatile, and has a density (expected fraction of selected k-mers) of 2/(w+1) . In practice, lower density leads to faster methods and smaller indexes, and it turns out that the random minimizer is not the best one can do. Indeed, some schemes are known to approach optimal density 1/w when k ->infinity , like the recently introduced mod-minimizer (Groot Koerkamp and Pibiri, WABI 2024). In this work, we study methods that achieve low density when k <= w . In this small-k regime, a practical method with provably better density than the random minimizer is the miniception (Zheng et al., Bioinformatics 2021). This method can be elegantly described as sampling the smallest closed sycnmer (Edgar, PeerJ 2021) in the window according to some random order. We show that extending the miniception to prefer sampling open syncmers yields much better density. This new method-the open-closed minimizer-offers improved density for small k <= w while being as fast to compute as the random minimizer. Compared to methods based on decycling sets, that achieve very low density in the small-k regime, our method has comparable density while being computation-ally simpler and intuitive. Furthermore, we extend the mod-minimizer to improve density of any scheme that works well for small k to also work well when k>w is large. We hence obtain the open-closed mod-minimizer, a practical method that improves over the mod-minimizer for all k

关键词： Minimizers randomized algorithms Sketching Hashing

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Sampling methods for multi-stage robust optimization problems

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ANNALS OF OPERATIONS RESEARCH 2025年第3期347卷 1385-1423页

作者： Maggioni, Francesca Dabbene, Fabrizio Pflug, Georg Ch. Univ Bergamo Dept Management Informat & Prod Engn Viale G Marconi 5 I-24044 Dalmine Italy Politecn Torino CNR IEIIT Cso Duca Abruzzi N 24 I-10129 Turin Italy Univ Vienna Dept Stat & Operat Res Schloss pl 1 A-2361 Laxenburg Austria Int Inst Appl Syst Anal IIASA Schloss Pl 1 A-2361 Laxenburg Austria

In this paper, we consider multi-stage robust optimization problems of the minimax type. We assume that the total uncertainty set is the cartesian product of stagewise compact uncertainty sets and approximate the given problem by a sampled subproblem. Instead of looking for the worst case among the infinite and typically uncountable set of uncertain parameters, we consider only the worst case among a randomly selected subset of parameters. By adopting such a strategy, two main questions arise: (1) Can we quantify the error committed by the random approximation, especially as a function of the sample size? (2) If the sample size tends to infinity, does the optimal value converge to the "true" optimal value? Both questions will be answered in this paper. An explicit bound on the probability of violation is given and chain of lower bounds on the original multi-stage robust optimization problem provided. Numerical results dealing with a multi-stage inventory management problem show that the proposed approach works well for problems with two or three time periods while for larger ones the number of required samples is prohibitively large for computational tractability. Despite this, we believe that our results can be useful for problems with such small number of time periods, and it sheds some light on the challenge for problems with more time periods.

关键词： Multi-stage robust optimization Constraint sampling Scenario approach in optimization randomized algorithms

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randomized k-server algorithms for growth-rate bounded graphs

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JOURNAL OF algorithms-COGNITION INFORMATICS AND LOGIC 2005年第2期55卷 192-202页

作者： Bartal, Y Mendel, M Hebrew Univ Jerusalem Sch Comp Sci IL-91904 Jerusalem Israel

The k-server problem is a fundamental online problem where k mobile servers should be scheduled to answer a sequence of requests for points in a metric space as to minimize the total movement cost. While the deterministic competitive ratio is at least k, randomized k-server algorithms have the potential of reaching o (k)-competitive ratios. Prior to this work only few specific cases of this problem were solved. For arbitrary metric spaces, this goal may be approached by using probabilistic metric approximation techniques. This paper gives the first results in this direction, obtaining o(k)competitive ratio for a natural class of metric spaces, including d-dimensional grids, and wide range of k. (c) 2004 Elsevier Inc. All rights reserved.

关键词： k-server problem randomized algorithms metric embedding

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