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APPROX

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Estimating the size of the union of a stream of sets S₁, S₂, …, S_M where each set is a subset of a known universe Ω is a fundamental problem in data streaming. This problem naturally generalizes the well-studied 𝖥₀ estimation problem in the streaming literature, where each set contains a single element from the universe. We consider the general case when the sets S_i can be succinctly represented and allow efficient membership, cardinality, and sampling queries (called a Delphic family of sets). A notable example in this framework is the Klee’s Measure Problem (KMP), where every set S_i is an axis-parallel rectangle in d-dimensional spaces (Ω = [Δ]^d where [Δ] := {1, … ,Δ} and Δ ∈ ℕ). Recently, Meel, Chakraborty, and Vinodchandran (PODS-21, PODS-22) designed a streaming algorithm for (ε,δ)-estimation of the size of the union of set streams over Delphic family with space and update time complexity O((log³|Ω|)/ε² ⋅ log 1/δ) and Õ((log⁴|Ω|)/ε² ⋅ log 1/(δ)), respectively.
This work presents a new, sampling-based algorithm for estimating the size of the union of Delphic sets that has space and update time complexity Õ((log²|Ω|)/ε² ⋅ log 1/(δ)). This improves the space complexity bound by a log|Ω| factor and update time complexity bound by a log² |Ω| factor.
A critical question is whether quadratic dependence of log|Ω| on space and update time complexities is necessary. Specifically, can we design a streaming algorithm for estimating the size of the union of sets over Delphic family with space and complexity linear in log|Ω| and update time poly(log|Ω|)? While this appears technically challenging, we show that establishing a lower bound of ω(log|Ω|) with poly(log|Ω|) update time is beyond the reach of current techniques. Specifically, we show that under certain hard-to-prove computational complexity hypothesis, there is a streaming algorithm for the problem with optimal space complexity O(log|Ω|) and update time poly(log(|Ω|)). Thus, establishing a space lower bound of ω(log|Ω|) will lead to break-through complexity class separation results.

Mridul Nandi, N. V. Vinodchandran, Arijit Ghosh, Kuldeep S. Meel, Soumit Pal, and Sourav Chakraborty. Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 26:1-26:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{nandi_et_al:LIPIcs.APPROX/RANDOM.2024.26, author = {Nandi, Mridul and Vinodchandran, N. V. and Ghosh, Arijit and Meel, Kuldeep S. and Pal, Soumit and Chakraborty, Sourav}, title = {{Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {26:1--26:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.26}, URN = {urn:nbn:de:0030-drops-210191}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.26}, annote = {Keywords: Sampling, Streaming, Klee’s Measure Problem} }

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**Published in:** LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Query evaluation over probabilistic databases is a notoriously intractable problem - not only in combined complexity, but for many natural queries in data complexity as well [Antoine Amarilli et al., 2017; Nilesh N. Dalvi and Dan Suciu, 2012]. This motivates the study of probabilistic query evaluation through the lens of approximation algorithms, and particularly of combined FPRASes, whose runtime is polynomial in both the query and instance size. In this paper, we focus on tuple-independent probabilistic databases over binary signatures, which can be equivalently viewed as probabilistic graphs. We study in which cases we can devise combined FPRASes for probabilistic query evaluation in this setting.
We settle the complexity of this problem for a variety of query and instance classes, by proving both approximability and (conditional) inapproximability results. This allows us to deduce many corollaries of possible independent interest. For example, we show how the results of [Marcelo Arenas et al., 2021] on counting fixed-length strings accepted by an NFA imply the existence of an FPRAS for the two-terminal network reliability problem on directed acyclic graphs: this was an open problem until now [Rico Zenklusen and Marco Laumanns, 2011]. We also show that one cannot extend a recent result [Timothy van Bremen and Kuldeep S. Meel, 2023] that gives a combined FPRAS for self-join-free conjunctive queries of bounded hypertree width on probabilistic databases: neither the bounded-hypertree-width condition nor the self-join-freeness hypothesis can be relaxed. Finally, we complement all our inapproximability results with unconditional lower bounds, showing that DNNF provenance circuits must have at least moderately exponential size in combined complexity.

Antoine Amarilli, Timothy van Bremen, and Kuldeep S. Meel. Conjunctive Queries on Probabilistic Graphs: The Limits of Approximability. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{amarilli_et_al:LIPIcs.ICDT.2024.15, author = {Amarilli, Antoine and van Bremen, Timothy and Meel, Kuldeep S.}, title = {{Conjunctive Queries on Probabilistic Graphs: The Limits of Approximability}}, booktitle = {27th International Conference on Database Theory (ICDT 2024)}, pages = {15:1--15:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-312-6}, ISSN = {1868-8969}, year = {2024}, volume = {290}, editor = {Cormode, Graham and Shekelyan, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.15}, URN = {urn:nbn:de:0030-drops-197978}, doi = {10.4230/LIPIcs.ICDT.2024.15}, annote = {Keywords: Probabilistic query evaluation, tuple-independent databases, approximation} }

Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We consider the problem of estimating the support size of a distribution D. Our investigations are pursued through the lens of distribution testing and seek to understand the power of conditional sampling (denoted as COND), wherein one is allowed to query the given distribution conditioned on an arbitrary subset S. The primary contribution of this work is to introduce a new approach to lower bounds for the COND model that relies on using powerful tools from information theory and communication complexity.
Our approach allows us to obtain surprisingly strong lower bounds for the COND model and its extensions.
- We bridge the longstanding gap between the upper bound O(log log n + 1/ε²) and the lower bound Ω(√{log log n}) for the COND model by providing a nearly matching lower bound. Surprisingly, we show that even if we get to know the actual probabilities along with COND samples, still Ω(log log n + 1/{ε² log (1/ε)}) queries are necessary.
- We obtain the first non-trivial lower bound for the COND equipped with an additional oracle that reveals the actual as well as the conditional probabilities of the samples (to the best of our knowledge, this subsumes all of the models previously studied): in particular, we demonstrate that Ω(log log log n + 1/{ε² log (1/ε)}) queries are necessary.

Diptarka Chakraborty, Gunjan Kumar, and Kuldeep S. Meel. Support Size Estimation: The Power of Conditioning. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 33:1-33:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2023.33, author = {Chakraborty, Diptarka and Kumar, Gunjan and Meel, Kuldeep S.}, title = {{Support Size Estimation: The Power of Conditioning}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {33:1--33:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.33}, URN = {urn:nbn:de:0030-drops-185675}, doi = {10.4230/LIPIcs.MFCS.2023.33}, annote = {Keywords: Support-size estimation, Distribution testing, Conditional sampling, Lower bound} }

Document

**Published in:** LIPIcs, Volume 271, 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)

The past three decades have witnessed notable success in designing efficient SAT solvers, with modern solvers capable of solving industrial benchmarks containing millions of variables in just a few seconds. The success of modern SAT solvers owes to the widely-used CDCL algorithm, which lacks comprehensive theoretical investigation. Furthermore, it has been observed that CDCL solvers still struggle to deal with specific classes of benchmarks comprising only hundreds of variables, which contrasts with their widespread use in real-world applications. Consequently, there is an urgent need to uncover the inner workings of these seemingly weak yet powerful black boxes.
In this paper, we present a first step towards this goal by introducing an approach called {CausalSAT}, which employs causal reasoning to gain insights into the functioning of modern SAT solvers. {CausalSAT} initially generates observational data from the execution of SAT solvers and learns a structured graph representing the causal relationships between the components of a SAT solver. Subsequently, given a query such as whether a clause with low literals blocks distance (LBD) has a higher clause utility, {CausalSAT} calculates the causal effect of LBD on clause utility and provides an answer to the question. We use {CausalSAT} to quantitatively verify hypotheses previously regarded as "rules of thumb" or empirical findings, such as the query above or the notion that clauses with high LBD experience a rapid drop in utility over time. Moreover, {CausalSAT} can address previously unexplored questions, like which branching heuristic leads to greater clause utility in order to study the relationship between branching and clause management. Experimental evaluations using practical benchmarks demonstrate that {CausalSAT} effectively fits the data, verifies four "rules of thumb", and provides answers to three questions closely related to implementing modern solvers.

Jiong Yang, Arijit Shaw, Teodora Baluta, Mate Soos, and Kuldeep S. Meel. Explaining SAT Solving Using Causal Reasoning. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{yang_et_al:LIPIcs.SAT.2023.28, author = {Yang, Jiong and Shaw, Arijit and Baluta, Teodora and Soos, Mate and Meel, Kuldeep S.}, title = {{Explaining SAT Solving Using Causal Reasoning}}, booktitle = {26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)}, pages = {28:1--28:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-286-0}, ISSN = {1868-8969}, year = {2023}, volume = {271}, editor = {Mahajan, Meena and Slivovsky, Friedrich}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2023.28}, URN = {urn:nbn:de:0030-drops-184909}, doi = {10.4230/LIPIcs.SAT.2023.28}, annote = {Keywords: Satisfiability, Causality, SAT solver, Clause management} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Given a Boolean formula ϕ over n variables, the problem of model counting is to compute the number of solutions of ϕ. Model counting is a fundamental problem in computer science with wide-ranging applications in domains such as quantified information leakage, probabilistic reasoning, network reliability, neural network verification, and more. Owing to the #P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that log n calls to an NP oracle are necessary and sufficient to achieve (ε,δ) guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting.
The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide n bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

Diptarka Chakraborty, Sourav Chakraborty, Gunjan Kumar, and Kuldeep S. Meel. Approximate Model Counting: Is SAT Oracle More Powerful Than NP Oracle?. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 123:1-123:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chakraborty_et_al:LIPIcs.ICALP.2023.123, author = {Chakraborty, Diptarka and Chakraborty, Sourav and Kumar, Gunjan and Meel, Kuldeep S.}, title = {{Approximate Model Counting: Is SAT Oracle More Powerful Than NP Oracle?}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {123:1--123:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.123}, URN = {urn:nbn:de:0030-drops-181750}, doi = {10.4230/LIPIcs.ICALP.2023.123}, annote = {Keywords: Model counting, Approximation, Satisfiability, NP oracle, SAT oracle} }

Document

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Given a data stream 𝒟 = ⟨ a₁, a₂, …, a_m ⟩ of m elements where each a_i ∈ [n], the Distinct Elements problem is to estimate the number of distinct elements in 𝒟. Distinct Elements has been a subject of theoretical and empirical investigations over the past four decades resulting in space optimal algorithms for it. All the current state-of-the-art algorithms are, however, beyond the reach of an undergraduate textbook owing to their reliance on the usage of notions such as pairwise independence and universal hash functions. We present a simple, intuitive, sampling-based space-efficient algorithm whose description and the proof are accessible to undergraduates with the knowledge of basic probability theory.

Sourav Chakraborty, N. V. Vinodchandran¹, and Kuldeep S. Meel. Distinct Elements in Streams: An Algorithm for the (Text) Book. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 34:1-34:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakraborty_et_al:LIPIcs.ESA.2022.34, author = {Chakraborty, Sourav and Vinodchandran¹, N. V. and Meel, Kuldeep S.}, title = {{Distinct Elements in Streams: An Algorithm for the (Text) Book}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {34:1--34:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.34}, URN = {urn:nbn:de:0030-drops-169725}, doi = {10.4230/LIPIcs.ESA.2022.34}, annote = {Keywords: F₀ Estimation, Streaming, Sampling} }

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Complete Volume

**Published in:** LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

LIPIcs, Volume 236, SAT 2022, Complete Volume

25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 1-618, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{meel_et_al:LIPIcs.SAT.2022, title = {{LIPIcs, Volume 236, SAT 2022, Complete Volume}}, booktitle = {25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)}, pages = {1--618}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-242-6}, ISSN = {1868-8969}, year = {2022}, volume = {236}, editor = {Meel, Kuldeep S. and Strichman, Ofer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022}, URN = {urn:nbn:de:0030-drops-166736}, doi = {10.4230/LIPIcs.SAT.2022}, annote = {Keywords: LIPIcs, Volume 236, SAT 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

Front Matter, Table of Contents, Preface, Conference Organization

25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{meel_et_al:LIPIcs.SAT.2022.0, author = {Meel, Kuldeep S. and Strichman, Ofer}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-242-6}, ISSN = {1868-8969}, year = {2022}, volume = {236}, editor = {Meel, Kuldeep S. and Strichman, Ofer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.0}, URN = {urn:nbn:de:0030-drops-166746}, doi = {10.4230/LIPIcs.SAT.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 235, 28th International Conference on Principles and Practice of Constraint Programming (CP 2022)

The problem of uniform sampling is, given a formula F, sample solutions of F uniformly at random from the solution space of F. Uniform sampling is a fundamental problem with widespread applications, including configuration testing, bug synthesis, function synthesis, and many more. State-of-the-art approaches for uniform sampling have a trade-off between scalability and theoretical guarantees. Many state of the art uniform samplers do not provide any theoretical guarantees on the distribution of samples generated, however, empirically they have shown promising results. In such cases, the main challenge is to test whether the distribution according to which samples are generated is indeed uniform or not.
Recently, Chakraborty and Meel (2019) designed the first scalable sampling tester, Barbarik, based on a grey-box sampling technique for testing if the distribution, according to which the given sampler is sampling, is close to the uniform or far from uniform. They were able to show that many off-the-self samplers are far from a uniform sampler. The availability of Barbarik increased the test-driven development of samplers. More recently, Golia, Soos, Chakraborty and Meel (2021), designed a uniform like sampler, CMSGen, which was shown to be accepted by Barbarik on all the instances. However, CMSGen does not provide any theoretical analysis of the sampling quality.
CMSGen leads us to observe the need for a tester to provide a quantitative answer to determine the quality of underlying samplers instead of merely a qualitative answer of Accept or Reject. Towards this goal, we design a computational hardness-based tester ScalBarbarik that provides a more nuanced analysis of the quality of a sampler. ScalBarbarik allows more expressive measurement of the quality of the underlying samplers. We empirically show that the state-of-the-art sampler, CMSGen is not accepted as a uniform-like sampler by ScalBarbarik. Furthermore, we show that ScalBarbarik can be used to design a sampler that can achieve balance between scalability and uniformity.

Mate Soos, Priyanka Golia, Sourav Chakraborty, and Kuldeep S. Meel. On Quantitative Testing of Samplers. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 235, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{soos_et_al:LIPIcs.CP.2022.36, author = {Soos, Mate and Golia, Priyanka and Chakraborty, Sourav and Meel, Kuldeep S.}, title = {{On Quantitative Testing of Samplers}}, booktitle = {28th International Conference on Principles and Practice of Constraint Programming (CP 2022)}, pages = {36:1--36:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-240-2}, ISSN = {1868-8969}, year = {2022}, volume = {235}, editor = {Solnon, Christine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2022.36}, URN = {urn:nbn:de:0030-drops-166655}, doi = {10.4230/LIPIcs.CP.2022.36}, annote = {Keywords: SAT Sampling, Testing of Samplers, SAT Solvers} }

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**Published in:** LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)

Despite the NP-completeness of Boolean satisfiability, modern SAT solvers are routinely able to handle large practical instances, and consequently have found wide ranging applications. The primary workhorse behind the success of SAT solvers is the widely acclaimed Conflict Driven Clause Learning (CDCL) paradigm, which was originally proposed in the context of Boolean formulas in CNF. The wide ranging applications of SAT solvers have highlighted that for several domains, CNF is not a natural representation and the reliance of modern SAT solvers on resolution proof system limit their ability to efficiently solve several families of constraints. Consequently, the past decade has witnessed the design of solvers with native support for constraints such as Pseudo-Boolean (PB) and CNF-XOR.
The primary contribution of our work is an efficient solver engineered for PB-XOR formulas, i.e., formulas consisting of a conjunction of PB and XOR constraints. We first observe that a simple adaption of CNF-XOR architecture does not provide an improvement over baseline; our analysis highlights the need for careful engineering of the order of propagations. To this end, we propose three different tactics, all of which achieve significant performance improvements over the baseline. Our work is motivated by applications arising from binarized neural network verification where the verification of properties such as robustness, fairness, trojan attacks can be reduced to model counting queries; the state of the art model counters reduce counting to polynomially many SAT queries over the original formula conjuncted with randomly generated XOR constraints. To this end, we augment ApproxMC with LinPB and we call the resulting counter as ApproxMCPB. In an extensive empirical comparison over 1076 benchmarks, we observe that ApproxMCPB can solve 912 instances while the baseline version of ApproxMC4 (augmented with CryptoMiniSat) can solve only 802 instances.

Jiong Yang and Kuldeep S. Meel. Engineering an Efficient PB-XOR Solver. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 58:1-58:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{yang_et_al:LIPIcs.CP.2021.58, author = {Yang, Jiong and Meel, Kuldeep S.}, title = {{Engineering an Efficient PB-XOR Solver}}, booktitle = {27th International Conference on Principles and Practice of Constraint Programming (CP 2021)}, pages = {58:1--58:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-211-2}, ISSN = {1868-8969}, year = {2021}, volume = {210}, editor = {Michel, Laurent D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.58}, URN = {urn:nbn:de:0030-drops-153499}, doi = {10.4230/LIPIcs.CP.2021.58}, annote = {Keywords: PB-XOR Solving, Pseudo-Boolean, XOR, Gauss Jordan Elimination, SAT-Solving, Model Counting} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

Propositional model counting is a fundamental problem in artificial intelligence with a wide variety of applications, such as probabilistic inference, decision making under uncertainty, and
probabilistic databases. Consequently, the problem is of theoretical as well as practical interest. When the constraints are expressed as DNF formulas, Monte Carlo-based techniques have been shown to provide a fully polynomial randomized approximation scheme (FPRAS). For CNF constraints, hashing-based approximation techniques have been demonstrated to be highly successful. Furthermore, it was shown that hashing-based techniques also yield an FPRAS for DNF counting without usage of Monte Carlo sampling. Our analysis, however, shows that the proposed hashing-based approach to DNF counting provides poor time complexity compared to the Monte Carlo-based DNF counting techniques. Given the success of hashing-based techniques for CNF constraints, it is natural to ask: Can hashing-based techniques provide an efficient FPRAS for DNF counting? In this paper, we provide a positive answer to this question. To this end, we introduce two novel algorithmic techniques: Symbolic Hashing and Stochastic Cell Counting, along
with a new hash family of Row-Echelon hash functions. These innovations allow us to design a hashing-based FPRAS for DNF counting of similar complexity (up to polylog factors) as that
of prior works. Furthermore, we expect these techniques to have potential applications beyond DNF counting.

Kuldeep S. Meel, Aditya A. Shrotri, and Moshe Y. Vardi. On Hashing-Based Approaches to Approximate DNF-Counting. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{meel_et_al:LIPIcs.FSTTCS.2017.41, author = {Meel, Kuldeep S. and Shrotri, Aditya A. and Vardi, Moshe Y.}, title = {{On Hashing-Based Approaches to Approximate DNF-Counting}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.41}, URN = {urn:nbn:de:0030-drops-84073}, doi = {10.4230/LIPIcs.FSTTCS.2017.41}, annote = {Keywords: Model Counting, Approximation, DNF, Hash Functions} }

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