The problem of estimating the number of cycles in a graph is one of the most widely studied graph problems in the data stream model. Three relevant variants of the data stream model include: the arbitrary order model in which the stream consists of the edges of the graph in arbitrary order, the random order model in which the edges are randomly permuted, and the adjacency list order model in which all edges incident to the same vertex appear consecutively. In this paper, we focus on the problem of triangle and four-cycle counting in these models. We improve over the state-of-the-art results as follows, where n is the number of vertices, m is the number of edges and T is the number of triangles/four-cycles in the graph (i.e., the quantity being estimated): Random Order Model: We present a single-pass algorithm that (1+ε)-approximates the number of triangles using ~O(ε-2 m/√T) space and prove that this is optimal in the range T ≤ √m. The best previous result, a (3+ε)-approximation using ~O(ε-4.5 m/√T) space, was presented by Cormode and Jowhari~(Theor. Comput. Sci. 2017). Adjacency List Model: We present an algorithm that returns a (1+ε)-approximation of the number of 4-cycles using two passes and ~O(ε-4 m/√T) space. The best previous result, a constant approximation using ~O(m/T3/8) space, was presented by Kallaugher et al. (PODS~2019). We also show that (1+ε)-approximation in a single pass is possible in a) polylog(n) space if T=Ω(n2) and b) ~O(n) space if T=Ω(n). Arbitrary Order Model: We present a three-pass algorithm that (1+ε)-approximates the number of 4-cycles using ~O(ε-2 m/T1/4) space and a one-pass algorithm that uses ~O(ε-2 n) space when T=Ω(n2). The best existing result, a (1+ε)-approximation using ~O(ε-2 m2/T) space, was presented by Bera and Chakrabarti (STACS~2017). We also show a multi-pass lower bound and another algorithm for distinguishing graphs with no four cycles and graphs with many 4-cycles.
{"title":"Triangle and Four Cycle Counting in the Data Stream Model","authors":"A. Mcgregor, Sofya Vorotnikova","doi":"10.1145/3375395.3387652","DOIUrl":"https://doi.org/10.1145/3375395.3387652","url":null,"abstract":"The problem of estimating the number of cycles in a graph is one of the most widely studied graph problems in the data stream model. Three relevant variants of the data stream model include: the arbitrary order model in which the stream consists of the edges of the graph in arbitrary order, the random order model in which the edges are randomly permuted, and the adjacency list order model in which all edges incident to the same vertex appear consecutively. In this paper, we focus on the problem of triangle and four-cycle counting in these models. We improve over the state-of-the-art results as follows, where n is the number of vertices, m is the number of edges and T is the number of triangles/four-cycles in the graph (i.e., the quantity being estimated): Random Order Model: We present a single-pass algorithm that (1+ε)-approximates the number of triangles using ~O(ε-2 m/√T) space and prove that this is optimal in the range T ≤ √m. The best previous result, a (3+ε)-approximation using ~O(ε-4.5 m/√T) space, was presented by Cormode and Jowhari~(Theor. Comput. Sci. 2017). Adjacency List Model: We present an algorithm that returns a (1+ε)-approximation of the number of 4-cycles using two passes and ~O(ε-4 m/√T) space. The best previous result, a constant approximation using ~O(m/T3/8) space, was presented by Kallaugher et al. (PODS~2019). We also show that (1+ε)-approximation in a single pass is possible in a) polylog(n) space if T=Ω(n2) and b) ~O(n) space if T=Ω(n). Arbitrary Order Model: We present a three-pass algorithm that (1+ε)-approximates the number of 4-cycles using ~O(ε-2 m/T1/4) space and a one-pass algorithm that uses ~O(ε-2 n) space when T=Ω(n2). The best existing result, a (1+ε)-approximation using ~O(ε-2 m2/T) space, was presented by Bera and Chakrabarti (STACS~2017). We also show a multi-pass lower bound and another algorithm for distinguishing graphs with no four cycles and graphs with many 4-cycles.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133389318","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}
This paper is devoted to a complexity study of various tasks related to query answering such as deciding if a Boolean query is true or not, counting the size of the answer set or enumerating the results. It is a survey of some of the many tools from complexity measures trough algorithmic methods to conditional lower bounds that have been designed in the domain over the last years.
{"title":"Fine-Grained Complexity Analysis of Queries: From Decision to Counting and Enumeration","authors":"Arnaud Durand","doi":"10.1145/3375395.3389130","DOIUrl":"https://doi.org/10.1145/3375395.3389130","url":null,"abstract":"This paper is devoted to a complexity study of various tasks related to query answering such as deciding if a Boolean query is true or not, counting the size of the answer set or enumerating the results. It is a survey of some of the many tools from complexity measures trough algorithmic methods to conditional lower bounds that have been designed in the domain over the last years.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131314235","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}
We consider three modern roles for logic in artificial intelligence, which are based on the theory of tractable Boolean circuits: (1) logic as a basis for computation, (2) logic for learning from a combination of data and knowledge, and (3) logic for reasoning about the behavior of machine learning systems.
{"title":"Three Modern Roles for Logic in AI","authors":"Adnan Darwiche","doi":"10.1145/3375395.3389131","DOIUrl":"https://doi.org/10.1145/3375395.3389131","url":null,"abstract":"We consider three modern roles for logic in artificial intelligence, which are based on the theory of tractable Boolean circuits: (1) logic as a basis for computation, (2) logic for learning from a combination of data and knowledge, and (3) logic for reasoning about the behavior of machine learning systems.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125013025","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}
Omri Ben-Eliezer, Rajesh Jayaram, David P. Woodruff, E. Yogev
We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems. In this work, we show that the answer is positive for various important streaming problems in the insertion-only model, including distinct elements and more generally $F_p$-estimation, Fp-heavy hitters, entropy estimation, and others. For all of these problems, we develop adversarially robust (1+ε)-approximation algorithms whose required space matches that of the best known non-robust algorithms up to a poly(log n, 1/ε) multiplicative factor (and in some cases even up to a constant factor). Towards this end, we develop several generic tools allowing one to efficiently transform a non-robust streaming algorithm into a robust one in various scenarios.
我们研究了流算法的对抗鲁棒性。在这种情况下，一个算法被认为是鲁棒的，如果它的性能保证保持，即使流是由对手自适应地选择的，对手沿着流观察算法的输出，并能以在线的方式作出反应。虽然确定性流算法具有固有的鲁棒性，但流文献中的许多核心问题不允许使用次线性空间确定性算法;另一方面，这些问题的经典空间高效随机算法通常不具有对抗性鲁棒性。这就提出了一个自然的问题，即是否存在有效的对抗鲁棒(随机)流算法来解决这些问题。在这项工作中，我们证明了对插入模型中各种重要的流问题的答案是肯定的，包括不同的元素和更普遍的$F_p$-估计、Fp-heavy hitters、熵估计等。对于所有这些问题，我们开发了对抗鲁棒(1+ε)逼近算法，其所需空间与已知的非鲁棒算法相匹配，可达一个多(log n, 1/ε)乘法因子(在某些情况下甚至可达一个常数因子)。为此，我们开发了几个通用工具，允许在各种场景中有效地将非鲁棒流算法转换为鲁棒流算法。
{"title":"A Framework for Adversarially Robust Streaming Algorithms","authors":"Omri Ben-Eliezer, Rajesh Jayaram, David P. Woodruff, E. Yogev","doi":"10.1145/3375395.3387658","DOIUrl":"https://doi.org/10.1145/3375395.3387658","url":null,"abstract":"We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems. In this work, we show that the answer is positive for various important streaming problems in the insertion-only model, including distinct elements and more generally $F_p$-estimation, Fp-heavy hitters, entropy estimation, and others. For all of these problems, we develop adversarially robust (1+ε)-approximation algorithms whose required space matches that of the best known non-robust algorithms up to a poly(log n, 1/ε) multiplicative factor (and in some cases even up to a constant factor). Towards this end, we develop several generic tools allowing one to efficiently transform a non-robust streaming algorithm into a robust one in various scenarios.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122940085","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}
We revisit the well-studied problem of triangle count estimation in graph streams. Given a graph represented as a stream of m edges, our aim is to compute a (1+-ε)-approximation to the triangle count T, using a small space algorithm. For arbitrary order and a constant number of passes, the space complexity is known to be essentially Θ(min(m3/2 /T, m/√T)) (McGregor et al., PODS 2016, Bera et al., STACS 2017). We give a (constant pass, arbitrary order) streaming algorithm that can circumvent this lower bound for low degeneracy graphs. The degeneracy, K, is a nuanced measure of density, and the class of constant degeneracy graphs is immensely rich (containing planar graphs, minor-closed families, and preferential attachment graphs). We design a streaming algorithm with space complexity ~O(mK/T). For constant degeneracy graphs, this bound is ~O(m/T), which is significantly smaller than both m3/2 /T and m/√T. We complement our algorithmic result with a nearly matching lower bound of Ω(mK/T).
{"title":"How the Degeneracy Helps for Triangle Counting in Graph Streams","authors":"Suman Kalyan Bera, Seshadhri Comandur","doi":"10.1145/3375395.3387665","DOIUrl":"https://doi.org/10.1145/3375395.3387665","url":null,"abstract":"We revisit the well-studied problem of triangle count estimation in graph streams. Given a graph represented as a stream of m edges, our aim is to compute a (1+-ε)-approximation to the triangle count T, using a small space algorithm. For arbitrary order and a constant number of passes, the space complexity is known to be essentially Θ(min(m3/2 /T, m/√T)) (McGregor et al., PODS 2016, Bera et al., STACS 2017). We give a (constant pass, arbitrary order) streaming algorithm that can circumvent this lower bound for low degeneracy graphs. The degeneracy, K, is a nuanced measure of density, and the class of constant degeneracy graphs is immensely rich (containing planar graphs, minor-closed families, and preferential attachment graphs). We design a streaming algorithm with space complexity ~O(mK/T). For constant degeneracy graphs, this bound is ~O(m/T), which is significantly smaller than both m3/2 /T and m/√T. We complement our algorithmic result with a nearly matching lower bound of Ω(mK/T).","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115449459","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}
Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of methods for generating such embeddings have been studied in the machine learning and knowledge representation literature. However, vector embeddings have received relatively little attention from a theoretical point of view. Starting with a survey of embedding techniques that have been used in practice, in this paper we propose two theoretical approaches that we see as central for understanding the foundations of vector embeddings. We draw connections between the various approaches and suggest directions for future research.
{"title":"word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data","authors":"Martin Grohe","doi":"10.1145/3375395.3387641","DOIUrl":"https://doi.org/10.1145/3375395.3387641","url":null,"abstract":"Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of methods for generating such embeddings have been studied in the machine learning and knowledge representation literature. However, vector embeddings have received relatively little attention from a theoretical point of view. Starting with a survey of embedding techniques that have been used in practice, in this paper we propose two theoretical approaches that we see as central for understanding the foundations of vector embeddings. We draw connections between the various approaches and suggest directions for future research.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"41 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":"121532969","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}
Michael Benedikt, S. Kikot, Piotr Ostropolski-Nalewaja, M. Romero
A query Q is monotonically determined over a set of views if Q can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for CQ views and queries. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.
{"title":"On Monotonic Determinacy and Rewritability for Recursive Queries and Views","authors":"Michael Benedikt, S. Kikot, Piotr Ostropolski-Nalewaja, M. Romero","doi":"10.1145/3375395.3387661","DOIUrl":"https://doi.org/10.1145/3375395.3387661","url":null,"abstract":"A query Q is monotonically determined over a set of views if Q can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for CQ views and queries. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131358206","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}
We consider static, external memory indexes for exact and approximate versions of the k-nearest neighbor (k-NN) problem, and show new lower bounds under a standard indivisibility assumption: Polynomial space indexing schemes for high-dimensional k-NN in Hamming space cannot take advantage of block transfers: í(k) block reads are needed to to answer a query. For the l∞ metric the lower bound holds even if we allow c-appoximate nearest neighbors to be returned, for c ∈ (1, 3). The restriction to c < 3 is necessary: For every metric there exists an indexing scheme in the indexability model of Hellerstein et al. using space O(kn), where n is the number of points, that can retrieve k 3-approximate nearest neighbors using optimal ⌈k/B⌉ I/Os, where B is the block size. For specific metrics, data structures with better approximation factors are possible. For k-NN in Hamming space and every approximation factor c>1 there exists a polynomial space data structure that returns k c-approximate nearest neighbors in ⌈k/B⌉ I/Os. To show these lower bounds we develop two new techniques: First, to handle that approximation algorithms have more freedom in deciding which result set to return we develop a relaxed version of the λ-set workload technique of Hellerstein et al. This technique allows us to show lower bounds that hold in d ≥ n dimensions. To extend the lower bounds down to d = O(k log(n/k)) dimensions, we develop a new deterministic dimension reduction technique that may be of independent interest.
{"title":"On the I/O Complexity of the k-Nearest Neighbors Problem","authors":"Mayank Goswami, R. Jacob, R. Pagh","doi":"10.1145/3375395.3387649","DOIUrl":"https://doi.org/10.1145/3375395.3387649","url":null,"abstract":"We consider static, external memory indexes for exact and approximate versions of the k-nearest neighbor (k-NN) problem, and show new lower bounds under a standard indivisibility assumption: Polynomial space indexing schemes for high-dimensional k-NN in Hamming space cannot take advantage of block transfers: í(k) block reads are needed to to answer a query. For the l∞ metric the lower bound holds even if we allow c-appoximate nearest neighbors to be returned, for c ∈ (1, 3). The restriction to c < 3 is necessary: For every metric there exists an indexing scheme in the indexability model of Hellerstein et al. using space O(kn), where n is the number of points, that can retrieve k 3-approximate nearest neighbors using optimal ⌈k/B⌉ I/Os, where B is the block size. For specific metrics, data structures with better approximation factors are possible. For k-NN in Hamming space and every approximation factor c>1 there exists a polynomial space data structure that returns k c-approximate nearest neighbors in ⌈k/B⌉ I/Os. To show these lower bounds we develop two new techniques: First, to handle that approximation algorithms have more freedom in deciding which result set to return we develop a relaxed version of the λ-set workload technique of Hellerstein et al. This technique allows us to show lower bounds that hold in d ≥ n dimensions. To extend the lower bounds down to d = O(k log(n/k)) dimensions, we develop a new deterministic dimension reduction technique that may be of independent interest.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131610041","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}
Martin Grohe, Benjamin Lucien Kaminski, J. Katoen, P. Lindner
Arguing for the need to combine declarative and probabilistic programming, Bárány et al. (TODS 2017) recently introduced a probabilistic extension of Datalog as a "purely declarative probabilistic programming language." We revisit this language and propose a more foundational approach towards defining its semantics. It is based on standard notions from probability theory known as stochastic kernels and Markov processes. This allows us to extend the semantics to continuous probability distributions, thereby settling an open problem posed by Bárány et al. We show that our semantics is fairly robust, allowing both parallel execution and arbitrary chase orders when evaluating a program. We cast our semantics in the framework of infinite probabilistic databases (Grohe and Lindner, ICDT 2020), and we show that the semantics remains meaningful even when the input of a probabilistic Datalog program is an arbitrary probabilistic database.
{"title":"Generative Datalog with Continuous Distributions","authors":"Martin Grohe, Benjamin Lucien Kaminski, J. Katoen, P. Lindner","doi":"10.1145/3375395.3387659","DOIUrl":"https://doi.org/10.1145/3375395.3387659","url":null,"abstract":"Arguing for the need to combine declarative and probabilistic programming, Bárány et al. (TODS 2017) recently introduced a probabilistic extension of Datalog as a \"purely declarative probabilistic programming language.\" We revisit this language and propose a more foundational approach towards defining its semantics. It is based on standard notions from probability theory known as stochastic kernels and Markov processes. This allows us to extend the semantics to continuous probability distributions, thereby settling an open problem posed by Bárány et al. We show that our semantics is fairly robust, allowing both parallel execution and arbitrary chase orders when evaluating a program. We cast our semantics in the framework of infinite probabilistic databases (Grohe and Lindner, ICDT 2020), and we show that the semantics remains meaningful even when the input of a probabilistic Datalog program is an arbitrary probabilistic database.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123512079","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}
The Shapley value is a conventional and well-studied function for determining the contribution of a player to the coalition in a cooperative game. Among its applications in a plethora of domains, it has recently been proposed to use the Shapley value for quantifying the contribution of a tuple to the result of a database query. In particular, we have a thorough understanding of the tractability frontier for the class of Conjunctive Queries (CQs) and aggregate functions over CQs. It has also been established that a tractable (randomized) multiplicative approximation exists for every union of CQs. Nevertheless, all of these results are based on the monotonicity of CQs. In this work, we investigate the implication of negation on the complexity of Shapley computation, in both the exact and approximate senses. We generalize a known dichotomy to account for negated atoms. We also show that negation fundamentally changes the complexity of approximation. We do so by drawing a connection to the problem of deciding whether a tuple is "relevant" to a query, and by analyzing its complexity.
{"title":"The Impact of Negation on the Complexity of the Shapley Value in Conjunctive Queries","authors":"A. Reshef, B. Kimelfeld, Ester Livshits","doi":"10.1145/3375395.3387664","DOIUrl":"https://doi.org/10.1145/3375395.3387664","url":null,"abstract":"The Shapley value is a conventional and well-studied function for determining the contribution of a player to the coalition in a cooperative game. Among its applications in a plethora of domains, it has recently been proposed to use the Shapley value for quantifying the contribution of a tuple to the result of a database query. In particular, we have a thorough understanding of the tractability frontier for the class of Conjunctive Queries (CQs) and aggregate functions over CQs. It has also been established that a tractable (randomized) multiplicative approximation exists for every union of CQs. Nevertheless, all of these results are based on the monotonicity of CQs. In this work, we investigate the implication of negation on the complexity of Shapley computation, in both the exact and approximate senses. We generalize a known dichotomy to account for negated atoms. We also show that negation fundamentally changes the complexity of approximation. We do so by drawing a connection to the problem of deciding whether a tuple is \"relevant\" to a query, and by analyzing its complexity.","PeriodicalId":412441,"journal":{"name":"Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115693193","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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