International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms最新文献

{"title":"Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics","authors":"Stefan Klootwijk, B. Manthey","doi":"10.4230/LIPIcs.AofA.2020.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.19","url":null,"abstract":"Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, “beyond worst-case analysis” of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős–Rényi random graphs) have been used so far. In this paper we extend these findings to sparse graphs, with a focus on sparse graphs with ‘fast growing cut sizes’, i.e. graphs for which $$|delta (U)|=Omega (|U|^varepsilon )$$\u0000 \u0000 |\u0000 δ\u0000 \u0000 (\u0000 U\u0000 )\u0000 \u0000 |\u0000 =\u0000 Ω\u0000 (\u0000 |\u0000 U\u0000 \u0000 |\u0000 ε\u0000 \u0000 )\u0000 \u0000 for some constant $$varepsilon in (0,1)$$\u0000 \u0000 ε\u0000 ∈\u0000 (\u0000 0\u0000 ,\u0000 1\u0000 )\u0000 \u0000 for all subsets U of the vertices, where $$delta (U)$$\u0000 \u0000 δ\u0000 (\u0000 U\u0000 )\u0000 \u0000 is the set of edges connecting U to the remaining vertices. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a sparse graph with fast growing cut sizes, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115211384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Disordered Chinese Restaurant and Other Competing Growth Processes","authors":"Cécile Mailler, Peter Mörters, Anna Senkevich","doi":"10.4230/LIPIcs.AofA.2020.21","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.21","url":null,"abstract":"The disordered Chinese restaurant process is a partition-valued stochastic process where the elements of the partitioned set are seen as customers sitting at different tables (the sets of the partition) in a restaurant. Each table is assigned a positive number called its attractiveness. At every step a customer enters the restaurant and either joins a table with a probability proportional to the product of its attractiveness and the number of customers sitting at the table, or occupies a previously unoccupied table, which is then assigned an attractiveness drawn from a bounded distribution independently of everything else. When all attractivenesses are equal to the upper bound this process is the classical Chinese restaurant process; we show that the introduction of disorder can drastically change the behaviour of the system. Our main results are distributional limit theorems for the scaled number of customers at the busiest table, and for the ratio of occupants at the busiest and second busiest table. The limiting distributions are universal, i.e. they do not depend on the distribution of the attractiveness. They follow from two general Poisson limit theorems for a broad class of processes consisting of families growing with different rates from different birth times, which have further applications, for example to preferential attachment networks.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131984493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hidden independence in unstructured probabilistic models","authors":"Antony Pearson, M. Lladser","doi":"10.4230/LIPIcs.AofA.2020.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.23","url":null,"abstract":"We describe a novel way to represent the probability distribution of a random binary string as a mixture having a maximally weighted component associated with independent (though not necessarily identically distributed) Bernoulli characters. We refer to this as the latent independent weight of the probabilistic source producing the string, and derive a combinatorial algorithm to compute it. The decomposition we propose may serve as an alternative to the Boolean paradigm of hypothesis testing, or to assess the fraction of uncorrupted samples originating from a source with independent marginals. In this sense, the latent independent weight quantifies the maximal amount of independence contained within a probabilistic source, which, properly speaking, may not have independent marginals.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121630981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Giant Component and 2-Core in Sparse Random Outerplanar Graphs","authors":"Mihyun Kang, Michael Missethan","doi":"10.4230/LIPIcs.AofA.2020.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.18","url":null,"abstract":"Let $A(n,m)$ be a graph chosen uniformly at random from the class of all vertex-labelled outerplanar graphs with $n$ vertices and $m$ edges. We consider $A(n,m)$ in the sparse regime when $m=n/2+s$ for $s=o(n)$. We show that with high probability the giant component in $A(n,m)$ emerges at $m=n/2+Oleft(n^{2/3}right)$ and determine the typical order of the 2-core. In addition, we prove that if $s=omegaleft(n^{2/3}right)$, with high probability every edge in $A(n,m)$ belongs to at most one cycle.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124713571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes","authors":"J. Borga, M. Maazoun","doi":"10.4230/LIPICS.AOFA.2020.7","DOIUrl":"https://doi.org/10.4230/LIPICS.AOFA.2020.7","url":null,"abstract":"Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations. We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130375938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polyharmonic Functions And Random Processes in Cones","authors":"F. Chapon, Éric Fusy, K. Raschel","doi":"10.4230/LIPIcs.AofA.2020.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.9","url":null,"abstract":"We investigate polyharmonic functions associated to Brownian motion and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126920861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Greedy Maximal Independent Sets via Local Limits","authors":"M. Krivelevich, T. Mészáros, Peleg Michaeli, C. Shikhelman","doi":"10.4230/LIPIcs.AofA.2020.20","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2020.20","url":null,"abstract":"The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science -- and even in chemistry. The algorithm builds a maximal independent set by inspecting the vertices of the graph one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge. \u0000In this paper we present a natural and general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a useful notion of local convergence. We use this framework both to give short and simple proofs for results on previously studied families of graphs, such as paths and binomial random graphs, and to study new ones, such as random trees. \u0000We conclude our work by analysing the greedy algorithm more closely when the base graph is a tree. We show that in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123006485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Refined Asymptotics for the Number of Leaves of Random Point Quadtrees","authors":"Michael Fuchs, Noëla Müller, Henning Sulzbach","doi":"10.4230/LIPIcs.AofA.2018.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2018.23","url":null,"abstract":"In the early 2000s, several phase change results from distributional convergence to distributional non-convergence have been obtained for shape parameters of random discrete structures. Recently, for those random structures which admit a natural martingale process, these results have been considerably improved by obtaining refined asymptotics for the limit behavior. In this work, we propose a new approach which is also applicable to random discrete structures which do not admit a natural martingale process. As an example, we obtain refined asymptotics for the number of leaves in random point quadtrees. More applications, for example to shape parameters in generalized m-ary search trees and random gridtrees, will be discussed in the journal version of this extended abstract.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128741888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stationary Distribution Analysis of a Queueing Model with Local Choice","authors":"P. S. Dester, C. Fricker, Hanene Mohamed","doi":"10.4230/LIPIcs.AofA.2018.22","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2018.22","url":null,"abstract":"The paper deals with load balancing between one-server queues on a circle by a local choice policy. Each one-server queue has a Poissonian arrival of customers. When a customer arrives at a queue, he joins the least loaded queue between this queue and the next one, ties solved at random. Service times have exponential distribution. The system is stable if the arrival-to-service rate ratio called load is less than one. When the load tends to zero, we derive the first terms of the expansion in this parameter for the stationary probabilities that a queue has 0 to 3 customers. We investigate the error, comparing these expansion results to numerical values obtained by simulations. Then we provide the asymptotics, as the load tends to zero, for the stationary probabilities of the queue length, for a fixed number of queues. It quantifies the difference between policies with this local choice, no choice and the choice between two queues chosen at random.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125437876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Periodic Pólya Urns and an Application to Young Tableaux","authors":"C. Banderier, P. Marchal, M. Wallner","doi":"10.4230/LIPIcs.AofA.2018.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.AofA.2018.11","url":null,"abstract":"P{o}lya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time (mod p). We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial partial differential equation indeed leads to ordinary linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed form for the corresponding generating functions (giving the exact state of the urns at time n). When the time goes to infinity, we show that these periodic P{o}lya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions. En passant, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.","PeriodicalId":175372,"journal":{"name":"International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130108531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}