Pub Date : 2013-04-06DOI: 10.1137/1.9781611973761.4
O. Bodini, Jérémie O. Lumbroso, N. Rolin
Boltzmann samplers, introduced by Duchon et al. in 2001, make it possible to uniformly draw approximate size objects from any class which can be specified through the symbolic method. This, through by evaluating the associated generating functions to obtain the correct branching probabilities. But these samplers require generating functions, in particular in the neighborhood of their sunglarity, which is a complex problem; they also require picking an appropriate tuning value to best control the size of generated objects. Although Pivoteau~etal have brought a sweeping question to the first question, with the introduction of their Newton oracle, questions remain. By adapting the rejection method, a classical tool from the random, we show how to obtain a variant of the Boltzmann sampler framework, which is tolerant of approximation, even large ones. Our goal for this is twofold: this allows for exact sampling with approximate values; but this also allows much more flexibility in tuning samplers. For the class of simple trees, we will try to show how this could be used to more easily calibrate samplers.
{"title":"Analytic Samplers and the Combinatorial Rejection Method","authors":"O. Bodini, Jérémie O. Lumbroso, N. Rolin","doi":"10.1137/1.9781611973761.4","DOIUrl":"https://doi.org/10.1137/1.9781611973761.4","url":null,"abstract":"Boltzmann samplers, introduced by Duchon et al. in 2001, make it possible to uniformly draw approximate size objects from any class which can be specified through the symbolic method. This, through by evaluating the associated generating functions to obtain the correct branching probabilities. \u0000But these samplers require generating functions, in particular in the neighborhood of their sunglarity, which is a complex problem; they also require picking an appropriate tuning value to best control the size of generated objects. Although Pivoteau~etal have brought a sweeping question to the first question, with the introduction of their Newton oracle, questions remain. \u0000By adapting the rejection method, a classical tool from the random, we show how to obtain a variant of the Boltzmann sampler framework, which is tolerant of approximation, even large ones. Our goal for this is twofold: this allows for exact sampling with approximate values; but this also allows much more flexibility in tuning samplers. For the class of simple trees, we will try to show how this could be used to more easily calibrate samplers.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127934450","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.5
C. Heuberger, Daniel Krenn, S. Wagner
For fixed t ≥ 2, we consider the class of representations of 1 as sum of unit fractions whose denominators are powers of t or equivalently the class of canonical compact t-ary Huffman codes or equivalently rooted t-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).
{"title":"Analysis of parameters of trees corresponding to Huffman codes and sums of unit fractions","authors":"C. Heuberger, Daniel Krenn, S. Wagner","doi":"10.1137/1.9781611973037.5","DOIUrl":"https://doi.org/10.1137/1.9781611973037.5","url":null,"abstract":"For fixed t ≥ 2, we consider the class of representations of 1 as sum of unit fractions whose denominators are powers of t or equivalently the class of canonical compact t-ary Huffman codes or equivalently rooted t-ary plane \"canonical\" trees. \u0000 \u0000We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"132 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122766915","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.7
A. Bacher, O. Bodini, A. Jacquot
Boltzmann samplers are a kind of random samplers; in 2004, Duchon, Flajolet, Louchard and Schaeffer showed that given a combinatorial class and a combinatorial specification for that class, one can automatically build a Boltzmann sampler. In this paper, we introduce a Boltzmann sampler for Motzkin trees built from a holonomic specification, that is, a specification that uses the pointing operator. This sampler is inspired by Remy's algorithm on binary trees. We show that our algorithm gives an exact size sampler with a linear time and space complexity in average.
{"title":"Exact-size Sampling for Motzkin Trees in Linear Time via Boltzmann Samplers and Holonomic Specification","authors":"A. Bacher, O. Bodini, A. Jacquot","doi":"10.1137/1.9781611973037.7","DOIUrl":"https://doi.org/10.1137/1.9781611973037.7","url":null,"abstract":"Boltzmann samplers are a kind of random samplers; in 2004, Duchon, Flajolet, Louchard and Schaeffer showed that given a combinatorial class and a combinatorial specification for that class, one can automatically build a Boltzmann sampler. In this paper, we introduce a Boltzmann sampler for Motzkin trees built from a holonomic specification, that is, a specification that uses the pointing operator. This sampler is inspired by Remy's algorithm on binary trees. We show that our algorithm gives an exact size sampler with a linear time and space complexity in average.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131291973","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.6
Jeffrey Gaither, Mark Daniel Ward
We derive an asymptotic expression for the variance of the number of 2-protected nodes (neither leaves nor parents of leaves) in a binary trie. In an unbiased trie on n leaves we find, for example, that the variance is approximately: 934n plus small fluctuations (also of order n); but our result covers the general (biased) case as well. Our proof relies on the asymptotic similarities between a trie and its Poissonized counterpart, whose behavior we glean via the Mellin transform and singularity analysis.
{"title":"The Variance of the Number of 2-Protected Nodes in a Trie","authors":"Jeffrey Gaither, Mark Daniel Ward","doi":"10.1137/1.9781611973037.6","DOIUrl":"https://doi.org/10.1137/1.9781611973037.6","url":null,"abstract":"We derive an asymptotic expression for the variance of the number of 2-protected nodes (neither leaves nor parents of leaves) in a binary trie. In an unbiased trie on n leaves we find, for example, that the variance is approximately: 934n plus small fluctuations (also of order n); but our result covers the general (biased) case as well. Our proof relies on the asymptotic similarities between a trie and its Poissonized counterpart, whose behavior we glean via the Mellin transform and singularity analysis.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133962027","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.10
Jérémie Du Boisberranger, Danièle Gardy, X. Lorca, C. Truchet
This article presents new work on analyzing the behaviour of a constraint solver, with a view towards optimization. In Constraint Programming, the propagation mechanism is one of the key tools for solving hard combinatorial problems. It is based on specific algorithms: propagators, that are called a large number of times during the resolution process. But in practice, these algorithms may often do nothing: their output is equal to their input. It is thus highly desirable to be able to recognize such situations, so as to avoid useless calls. We propose to quantify this phenomenon in the particular case of the AllDifferent constraint (bound consistency propagator). Our first contribution is the definition of a probabilistic model for the constraint and the variables it is working on. This model then allows us to compute the probability that a call to the propagation algorithm for AllDifferent does modify its input. We give an asymptotic approximation of this probability, depending on some macroscopic quantities related to the variables and the domains, that can be computed in constant time. This reveals two very different behaviors depending of the sharpness of the constraint. First experiments show that the approximation allows us to improve constraint propagation behaviour.
{"title":"When is it worthwhile to propagate a constraint? A probabilistic analysis of AllDifferent","authors":"Jérémie Du Boisberranger, Danièle Gardy, X. Lorca, C. Truchet","doi":"10.1137/1.9781611973037.10","DOIUrl":"https://doi.org/10.1137/1.9781611973037.10","url":null,"abstract":"This article presents new work on analyzing the behaviour of a constraint solver, with a view towards optimization. In Constraint Programming, the propagation mechanism is one of the key tools for solving hard combinatorial problems. It is based on specific algorithms: propagators, that are called a large number of times during the resolution process. But in practice, these algorithms may often do nothing: their output is equal to their input. It is thus highly desirable to be able to recognize such situations, so as to avoid useless calls. We propose to quantify this phenomenon in the particular case of the AllDifferent constraint (bound consistency propagator). Our first contribution is the definition of a probabilistic model for the constraint and the variables it is working on. This model then allows us to compute the probability that a call to the propagation algorithm for AllDifferent does modify its input. We give an asymptotic approximation of this probability, depending on some macroscopic quantities related to the variables and the domains, that can be computed in constant time. This reveals two very different behaviors depending of the sharpness of the constraint. First experiments show that the approximation allows us to improve constraint propagation behaviour.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122748157","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.1
M. Drmota, M. Noy
We analyze several extremal parameters like the diameter or the maximum degree in sub-critial graph classes. Subcritical graph classes cover several well-known classes of graphs like trees, outerplanar graph or series-parallel graphs which have been intensively studied during the last few years. However, this paper is the first one, where these kinds of parameters are studied from a general point of view.
{"title":"Extremal Parameters in Sub-Critical Graph Classes","authors":"M. Drmota, M. Noy","doi":"10.1137/1.9781611973037.1","DOIUrl":"https://doi.org/10.1137/1.9781611973037.1","url":null,"abstract":"We analyze several extremal parameters like the diameter or the maximum degree in sub-critial graph classes. Subcritical graph classes cover several well-known classes of graphs like trees, outerplanar graph or series-parallel graphs which have been intensively studied during the last few years. However, this paper is the first one, where these kinds of parameters are studied from a general point of view.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"172 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116459419","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.11
J. Cichon, Z. Golebiewski, Marcin Kardas, M. Klonowski
We discuss a general framework for determining asymptotics of the expected value of random variables of the form f(X) in terms of a function f and central moments of the random variable X. This method may be used for approximation of entropy, inverse moments, and some statistics of discrete random variables useful in analysis of some randomized algorithms. Our approach is based on some variant of the Delta Method of Moments. We formulate a general result for an arbitrary distribution and next we show its specific extension to random variables which are sums of identically distributed independent random variables. Our method simpli files previous proofs of results of several authors and can be automated to a large extent. We apply our method to the binomial, negative binomial, Poisson and hypergeometric distribution. We extend the class of functions for which our method is applicable to some subclass of exponential functions and double exponential functions for some cases.
{"title":"On Delta-Method of Moments and Probabilistic Sums","authors":"J. Cichon, Z. Golebiewski, Marcin Kardas, M. Klonowski","doi":"10.1137/1.9781611973037.11","DOIUrl":"https://doi.org/10.1137/1.9781611973037.11","url":null,"abstract":"We discuss a general framework for determining asymptotics of the expected value of random variables of the form f(X) in terms of a function f and central moments of the random variable X. This method may be used for approximation of entropy, inverse moments, and some statistics of discrete random variables useful in analysis of some randomized algorithms. Our approach is based on some variant of the Delta Method of Moments. We formulate a general result for an arbitrary distribution and next we show its specific extension to random variables which are sums of identically distributed independent random variables. Our method simpli files previous proofs of results of several authors and can be automated to a large extent. We apply our method to the binomial, negative binomial, Poisson and hypergeometric distribution. We extend the class of functions for which our method is applicable to some subclass of exponential functions and double exponential functions for some cases.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"515 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123081098","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}
Pub Date : 2013-01-06DOI: 10.1137/1.9781611973037.4
Klaus-Tycho Förster
In this paper we study the NP-complete problem of finding small k-dominating sets in general graphs, which allow k---1 nodes to fail and still dominate the graph. The classic minimum dominating set problem is a special case with k = 1. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. For each α > 1 it can give solutions k/α times larger than a minimum k-dominating set. We also study lower bounds on possible approximation ratios. We show that it is NP-hard to approximate the minimum k-dominating set problem with a factor better than (0.2267/k) ln(n/k). Furthermore, a result for special finite sums allows us to use a greedy approach for k-domination with an approximation ratio of ln(Δ + k) + 1 < ln(Δ) + 1.7, with Δ being the maximum node-degree. We also achieve an approximation ratio of ln(n)+1.7 for h-step k-domination, where nodes do not need to be direct neighbors of dominating nodes, but can be h steps away.
{"title":"Approximating Fault-Tolerant Domination in General Graphs","authors":"Klaus-Tycho Förster","doi":"10.1137/1.9781611973037.4","DOIUrl":"https://doi.org/10.1137/1.9781611973037.4","url":null,"abstract":"In this paper we study the NP-complete problem of finding small k-dominating sets in general graphs, which allow k---1 nodes to fail and still dominate the graph. The classic minimum dominating set problem is a special case with k = 1. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. For each α > 1 it can give solutions k/α times larger than a minimum k-dominating set. We also study lower bounds on possible approximation ratios. We show that it is NP-hard to approximate the minimum k-dominating set problem with a factor better than (0.2267/k) ln(n/k). Furthermore, a result for special finite sums allows us to use a greedy approach for k-domination with an approximation ratio of ln(Δ + k) + 1 < ln(Δ) + 1.7, with Δ being the maximum node-degree. We also achieve an approximation ratio of ln(n)+1.7 for h-step k-domination, where nodes do not need to be direct neighbors of dominating nodes, but can be h steps away.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116193510","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}
Pub Date : 2012-04-24DOI: 10.1137/1.9781611973037.9
Edyta Szymanska, Marek Karpinski, A. Rucinski
In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of matchings in k-uniform hypergraphs whose intersection graphs contain few claws. Our method gives a generalization of the canonical path method of Jerrum and Sinclair to hypergraphs satisfying a local restriction. The proof depends on an application of the Euler tour technique for the canonical paths of the underlying Markov chains. On the other hand, we prove that it is NP-hard to approximate the number of matchings even for the class of 2-regular, linear, k-uniform hypergraphs, for all k ≥ 6, without the above restriction.
{"title":"Approximate Counting of Matchings in Sparse Uniform Hypergraphs","authors":"Edyta Szymanska, Marek Karpinski, A. Rucinski","doi":"10.1137/1.9781611973037.9","DOIUrl":"https://doi.org/10.1137/1.9781611973037.9","url":null,"abstract":"In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of matchings in k-uniform hypergraphs whose intersection graphs contain few claws. Our method gives a generalization of the canonical path method of Jerrum and Sinclair to hypergraphs satisfying a local restriction. The proof depends on an application of the Euler tour technique for the canonical paths of the underlying Markov chains. On the other hand, we prove that it is NP-hard to approximate the number of matchings even for the class of 2-regular, linear, k-uniform hypergraphs, for all k ≥ 6, without the above restriction.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124039827","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}
Pub Date : 2012-02-10DOI: 10.1137/1.9781611973037.12
S. Kingan
Two r X n matrices A and A' representing the same matroid M over GF(q), where q is a prime power, are projective equivalent representations of M if one can be obtained from the other by elementary row operations and column scaling. Bounds for projective inequivalence are difficult to obtain and are known for only a few special classes of matroids. In this paper we define two matrices A and A' to be geometric equivalent if, in addition to row operations and column scaling, column permutations are also allowed. We show that the number of geometric inequivalent representations is at most the number of projective inequivalent representions and we give a polynomial time algorithm for determining if two projective inequivalent representations are geometrically equivalent. Thus, from a computational perspective there is no additional cost to altering the definition of equivalence in this manner. The benefit is that it could lead to a new set of theorems for inequivalence with better bounds.
{"title":"Unlabeled equivalence for matroids representable over finite fields","authors":"S. Kingan","doi":"10.1137/1.9781611973037.12","DOIUrl":"https://doi.org/10.1137/1.9781611973037.12","url":null,"abstract":"Two r X n matrices A and A' representing the same matroid M over GF(q), where q is a prime power, are projective equivalent representations of M if one can be obtained from the other by elementary row operations and column scaling. Bounds for projective inequivalence are difficult to obtain and are known for only a few special classes of matroids. In this paper we define two matrices A and A' to be geometric equivalent if, in addition to row operations and column scaling, column permutations are also allowed. We show that the number of geometric inequivalent representations is at most the number of projective inequivalent representions and we give a polynomial time algorithm for determining if two projective inequivalent representations are geometrically equivalent. Thus, from a computational perspective there is no additional cost to altering the definition of equivalence in this manner. The benefit is that it could lead to a new set of theorems for inequivalence with better bounds.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128890498","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}
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