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Documents authored by Gravin, Nick

Document
DOI: 10.4230/LIPIcs.ESA.2025.103
On the Satisfiability of Random 3-SAT Formulas with k-Wise Independent Clauses

Authors: Ioannis Caragiannis, Nick Gravin, and Zhile Jiang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
The problem of identifying the satisfiability threshold of random 3-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The classical assumption in this line of research is that, for a given set of n Boolean variables, each clause is drawn uniformly at random among all sets of three literals from these variables, independently from other clauses. Here, we keep the uniform distribution of each clause, but deviate significantly from the independence assumption and consider richer families of probability distributions. For integer parameters n, m, and k, we denote by ℱ_k(n,m) the family of probability distributions that produce formulas with m clauses, each selected uniformly at random from all sets of three literals from the n variables, so that the clauses are k-wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in ℱ_k(n,m) for different values of the parameters n, m, and k. Our technical results are as follows: First, all probability distributions in ℱ₂(n,m) with m ∈ Ω(n³) return unsatisfiable formulas with high probability. This result is tight. We show that there exists a probability distribution 𝒟 ∈ ℱ₃(n,m) with m ∈ O(n³) so that a random formula drawn from 𝒟 is almost always satisfiable. In contrast, for m ∈ Ω(n²), any probability distribution 𝒟 ∈ ℱ₄(n,m) returns an unsatisfiable formula with high probability. This is our most surprising and technically involved result. Finally, for any integer k ≥ 2, any probability distribution 𝒟 ∈ ℱ_k(n,m) with m ∈ O(n^{1-1/k}) returns a satisfiable formula with high probability.
Cite as

Ioannis Caragiannis, Nick Gravin, and Zhile Jiang. On the Satisfiability of Random 3-SAT Formulas with k-Wise Independent Clauses. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 103:1-103:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{caragiannis_et_al:LIPIcs.ESA.2025.103,
  author =	{Caragiannis, Ioannis and Gravin, Nick and Jiang, Zhile},
  title =	{{On the Satisfiability of Random 3-SAT Formulas with k-Wise Independent Clauses}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{103:1--103:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.103},
  URN =		{urn:nbn:de:0030-drops-245721},
  doi =		{10.4230/LIPIcs.ESA.2025.103},
  annote =	{Keywords: Random 3-SAT, k-wise independence, Random bipartite graph}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.4
A Bicriterion Concentration Inequality and Prophet Inequalities for k-Fold Matroid Unions

Authors: Noga Alon, Nick Gravin, Tristan Pollner, Aviad Rubinstein, Hongao Wang, S. Matthew Weinberg, and Qianfan Zhang

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We investigate prophet inequalities with competitive ratios approaching 1, seeking to generalize k-uniform matroids. We first show that large girth does not suffice: for all k, there exists a matroid of girth ≥ k and a prophet inequality instance on that matroid whose optimal competitive ratio is 1/2. Next, we show k-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio 1-O(√{(log k)/k}) for any k-fold matroid union. Our prophet inequality follows from an online contention resolution scheme. The key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone 1-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a 1-Lipschitz function that is not (approximately) self-bounding.
Cite as

Noga Alon, Nick Gravin, Tristan Pollner, Aviad Rubinstein, Hongao Wang, S. Matthew Weinberg, and Qianfan Zhang. A Bicriterion Concentration Inequality and Prophet Inequalities for k-Fold Matroid Unions. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alon_et_al:LIPIcs.ITCS.2025.4,
  author =	{Alon, Noga and Gravin, Nick and Pollner, Tristan and Rubinstein, Aviad and Wang, Hongao and Weinberg, S. Matthew and Zhang, Qianfan},
  title =	{{A Bicriterion Concentration Inequality and Prophet Inequalities for k-Fold Matroid Unions}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{4:1--4:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.4},
  URN =		{urn:nbn:de:0030-drops-226329},
  doi =		{10.4230/LIPIcs.ITCS.2025.4},
  annote =	{Keywords: Prophet Inequalities, Online Contention Resolution Schemes, Concentration Inequalities}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.74
Online Stochastic Matching with Edge Arrivals

Authors: Nick Gravin, Zhihao Gavin Tang, and Kangning Wang

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by Gamlath et al., who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than 0.5. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greedy online policies motivated by practical considerations from a multitude of online matching platforms. Any prune & greedy algorithm consists of two stages: first, it decreases the probabilities of some edges in the stochastic instance and then runs greedy algorithm on the pruned graph. We propose prune & greedy algorithms that are 0.552-competitive on the instances that can be pruned to a 2-regular stochastic bipartite graph, and 0.503-competitive on arbitrary stochastic bipartite graphs. The algorithms and our analysis significantly deviate from the prior work. We first obtain analytically manageable lower bound on the size of the matching, which leads to a non-linear optimization problem. We further reduce this problem to a continuous optimization with a constant number of parameters that can be solved using standard software tools.
Cite as

Nick Gravin, Zhihao Gavin Tang, and Kangning Wang. Online Stochastic Matching with Edge Arrivals. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 74:1-74:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gravin_et_al:LIPIcs.ICALP.2021.74,
  author =	{Gravin, Nick and Tang, Zhihao Gavin and Wang, Kangning},
  title =	{{Online Stochastic Matching with Edge Arrivals}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{74:1--74:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.74},
  URN =		{urn:nbn:de:0030-drops-141438},
  doi =		{10.4230/LIPIcs.ICALP.2021.74},
  annote =	{Keywords: online matching, graph algorithms, prophet inequality}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.48
Tight Lower Bounds for Multiplicative Weights Algorithmic Families

Authors: Nick Gravin, Yuval Peres, and Balasubramanian Sivan

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
We study the fundamental problem of prediction with expert advice and develop regret lower bounds for a large family of algorithms for this problem. We develop simple adversarial primitives, that lend themselves to various combinations leading to sharp lower bounds for many algorithmic families. We use these primitives to show that the classic Multiplicative Weights Algorithm (MWA) has a regret of (T*ln(k)/2)^{0.5} (where T is the time horizon and k is the number of experts), there by completely closing the gap between upper and lower bounds. We further show a regret lower bound of (2/3)* (T*ln(k)/2)^{0.5} for a much more general family of algorithms than MWA, where the learning rate can be arbitrarily varied over time, or even picked from arbitrary distributions over time. We also use our primitives to construct adversaries in the geometric horizon setting for MWA to precisely characterize the regret at 0.391/(\delta)^{0.5} for the case of 2 experts and a lower bound of (1/2)*(ln(k)/(2*\delta))^{0.5}, for the case of arbitrary number of experts k (here \delta is the probability that the game ends in any given round).
Cite as

Nick Gravin, Yuval Peres, and Balasubramanian Sivan. Tight Lower Bounds for Multiplicative Weights Algorithmic Families. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gravin_et_al:LIPIcs.ICALP.2017.48,
  author =	{Gravin, Nick and Peres, Yuval and Sivan, Balasubramanian},
  title =	{{Tight Lower Bounds for Multiplicative Weights Algorithmic Families}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{48:1--48:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.48},
  URN =		{urn:nbn:de:0030-drops-74997},
  doi =		{10.4230/LIPIcs.ICALP.2017.48},
  annote =	{Keywords: Multiplicative Weights, Lower Bounds, Adversarial Primitives}
}
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