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} }

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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} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Consider the problem where n jobs, each with a release time, a deadline and a required processing time are to be feasibly scheduled in a single- or multi-processor setting so as to minimize the total energy consumption of the schedule. A processor has two available states: a sleep state where no energy is consumed but also no processing can take place, and an active state which consumes energy at a rate of one, and in which jobs can be processed. Transitioning from the active to the sleep does not incur any further energy cost, but transitioning from the sleep to the active state requires q energy units. Jobs may be preempted and (in the multi-processor case) migrated.
The single-processor case of the problem is known to be solvable in polynomial time via an involved dynamic program, whereas the only known approximation algorithm for the multi-processor case attains an approximation factor of 3 and is based on rounding the solution to a linear programming relaxation of the problem. In this work, we present efficient and combinatorial approximation algorithms for both the single- and the multi-processor setting. Before, only an algorithm based on linear programming was known for the multi-processor case. Our algorithms build upon the concept of a skeleton, a basic (and not necessarily feasible) schedule that captures the fact that some processor(s) must be active at some time point during an interval. Finally, we further demonstrate the power of skeletons by providing a 2-approximation algorithm for the multiprocessor case, thus improving upon the recent breakthrough 3-approximation result. Our algorithm is based on a novel rounding scheme of a linear-programming relaxation of the problem which incorporates skeletons.

Antonios Antoniadis, Gunjan Kumar, and Nikhil Kumar. Skeletons and Minimum Energy Scheduling. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{antoniadis_et_al:LIPIcs.ISAAC.2021.51, author = {Antoniadis, Antonios and Kumar, Gunjan and Kumar, Nikhil}, title = {{Skeletons and Minimum Energy Scheduling}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {51:1--51:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.51}, URN = {urn:nbn:de:0030-drops-154849}, doi = {10.4230/LIPIcs.ISAAC.2021.51}, annote = {Keywords: scheduling, energy-efficiency, approximation algorithms, dynamic programming, combinatorial algorithms} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Partial function extension is a basic problem that underpins multiple research topics in optimization, including learning, property testing, and game theory. Here, we are given a partial function consisting of n points from a domain and a function value at each point. Our objective is to determine if this partial function can be extended to a function defined on the domain, that additionally satisfies a given property, such as linearity. We formally study partial function extension to fundamental properties in combinatorial optimization - subadditivity, XOS, and matroid independence. A priori, it is not clear if partial function extension for these properties even lies in NP (or coNP).
Our contributions are twofold. Firstly, for the properties studied, we give bounds on the complexity of partial function extension. For subadditivity and XOS, we give tight bounds on approximation guarantees as well. Secondly, we develop new connections between partial function extension and learning and property testing, and use these to give new results for these problems. In particular, for subadditive functions, we give improved lower bounds on learning, as well as the first subexponential-query tester.

Umang Bhaskar and Gunjan Kumar. Partial Function Extension with Applications to Learning and Property Testing. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhaskar_et_al:LIPIcs.ISAAC.2020.46, author = {Bhaskar, Umang and Kumar, Gunjan}, title = {{Partial Function Extension with Applications to Learning and Property Testing}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {46:1--46:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.46}, URN = {urn:nbn:de:0030-drops-133906}, doi = {10.4230/LIPIcs.ISAAC.2020.46}, annote = {Keywords: Partial function extension, subadditivity, matroid rank, approximation algorithms, learning, property testing} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Coverage functions are an important subclass of submodular functions, finding applications in machine learning, game theory, social networks, and facility location. We study the complexity of partial function extension to coverage functions. That is, given a partial function consisting of a family of subsets of [m] and a value at each point, does there exist a coverage function defined on all subsets of [m] that extends this partial function? Partial function extension is previously studied for other function classes, including boolean functions and convex functions, and is useful in many fields, such as obtaining bounds on learning these function classes.
We show that determining extendibility of a partial function to a coverage function is NP-complete, establishing in the process that there is a polynomial-sized certificate of extendibility. The hardness also gives us a lower bound for learning coverage functions. We then study two natural notions of approximate extension, to account for errors in the data set. The two notions correspond roughly to multiplicative point-wise approximation and additive L_1 approximation. We show upper and lower bounds for both notions of approximation. In the second case we obtain nearly tight bounds.

Umang Bhaskar and Gunjan Kumar. The Complexity of Partial Function Extension for Coverage Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bhaskar_et_al:LIPIcs.APPROX-RANDOM.2019.30, author = {Bhaskar, Umang and Kumar, Gunjan}, title = {{The Complexity of Partial Function Extension for Coverage Functions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {30:1--30:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.30}, URN = {urn:nbn:de:0030-drops-112457}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.30}, annote = {Keywords: Coverage Functions, PAC Learning, Approximation Algorithm, Partial Function Extension} }