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Documents authored by Brânzei, Simina

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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.52
Tarski Lower Bounds from Multi-Dimensional Herringbones

Authors: Simina Brânzei, Reed C. Phillips, and Nicholas J. Recker

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
Tarski’s theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the k-dimensional grid of side length n under the ≤ relation. In this setting, there is an unknown monotone function f: {0,1,…, n-1}^k → {0,1,…, n-1}^k and an algorithm must query a vertex v to learn f(v). The goal is to find a fixed point of f using as few oracle queries as possible. We show that the randomized query complexity of this problem is Ω((k⋅log²n)/log k) for all n,k ≥ 2. This unifies and improves upon two prior results: a lower bound of Ω(log²n) from [Etessami et al., 2020] and a lower bound of Ω((k⋅log(n)/log(k)) from [Brânzei et al., 2024], respectively.
Cite as

Simina Brânzei, Reed C. Phillips, and Nicholas J. Recker. Tarski Lower Bounds from Multi-Dimensional Herringbones. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 52:1-52:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{branzei_et_al:LIPIcs.APPROX/RANDOM.2025.52,
  author =	{Br\^{a}nzei, Simina and Phillips, Reed C. and Recker, Nicholas J.},
  title =	{{Tarski Lower Bounds from Multi-Dimensional Herringbones}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{52:1--52:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.52},
  URN =		{urn:nbn:de:0030-drops-244186},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.52},
  annote =	{Keywords: Tarski’s theorem, monotone functions, lattices, fixed points, computational complexity, oracle model, query complexity, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ITC.2021.16
P₄-free Partition and Cover Numbers & Applications

Authors: Alexander R. Block, Simina Brânzei, Hemanta K. Maji, Himanshi Mehta, Tamalika Mukherjee, and Hai H. Nguyen

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

Abstract
P₄-free graphs- also known as cographs, complement-reducible graphs, or hereditary Dacey graphs-have been well studied in graph theory. Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite P₄-free graphs. For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant. Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems. Interestingly, our covering problem is closely related to the well-studied problem of product (a.k.a., Prague) dimension of loopless undirected graphs, which allows us to employ algebraic lower-bounding techniques for the product/Prague dimension. We prove that computing these numbers is NP-complete, even for bipartite graphs. We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size-N partite sets such that these numbers are at least ε⋅N^{1-2ε}, for ε ∈ {1/3,1/4,1/5,...}. Finally, we accurately estimate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality. For applications in information theory and communication & cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the participants, Alice and Bob, their portion from this joint sample. Alice and Bob’s objective is to non-interactively establish a shared key and extract the left-over entropy from their portion of the samples as independent private randomness. A genie, who observes the joint sample, provides appropriate assistance to help Alice and Bob with their objective. Lower bounds to the minimum size of the genie’s assistance translate into communication and cryptographic lower bounds. We show that (the log₂ of) the P₄-free partition number of a graph encoding the joint distribution that the setup uses is equivalent to the size of the genie’s assistance. Consequently, the joint distributions corresponding to the bipartite graphs constructed above with high P₄-free partition numbers correspond to joint distributions requiring more assistance from the genie. As a representative application in non-deterministic communication complexity, we study the communication complexity of nondeterministic protocols augmented by access to the equality oracle at the output. We show that (the log₂ of) the P₄-free cover number of the bipartite graph encoding a Boolean function f is equivalent to the minimum size of the nondeterministic input required by the parties (referred to as the communication complexity of f in this model). Consequently, the functions corresponding to the bipartite graphs with high P₄-free cover numbers have high communication complexity. Furthermore, there are functions with communication complexity close to the naïve protocol where the nondeterministic input reveals a party’s input. Finally, the access to the equality oracle reduces the communication complexity of computing set disjointness by a constant factor in contrast to the model where parties do not have access to the equality oracle. To compute the inequality function, we show an exponential reduction in the communication complexity, and this bound is optimal. On the other hand, access to the equality oracle is (nearly) useless for computing set intersection.
Cite as

Alexander R. Block, Simina Brânzei, Hemanta K. Maji, Himanshi Mehta, Tamalika Mukherjee, and Hai H. Nguyen. P₄-free Partition and Cover Numbers & Applications. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 16:1-16:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{block_et_al:LIPIcs.ITC.2021.16,
  author =	{Block, Alexander R. and Br\^{a}nzei, Simina and Maji, Hemanta K. and Mehta, Himanshi and Mukherjee, Tamalika and Nguyen, Hai H.},
  title =	{{P₄-free Partition and Cover Numbers \& Applications}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{16:1--16:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.16},
  URN =		{urn:nbn:de:0030-drops-143357},
  doi =		{10.4230/LIPIcs.ITC.2021.16},
  annote =	{Keywords: Secure keys, Secure private randomness, Gray-Wyner system, Cryptographic complexity, Nondeterministic communication complexity, Leakage-resilience, Combinatorial optimization, Product dimension, Zarankiewicz problem, Algebraic lower-bounding techniques, P₄-free partition number, P₄-free cover number}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.80
Walrasian Pricing in Multi-Unit Auctions

Authors: Simina Brânzei, Aris Filos-Ratsikas, Peter Bro Miltersen, and Yulong Zeng

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources, and in particular, is incompatible with (Pareto) efficiency.
Cite as

Simina Brânzei, Aris Filos-Ratsikas, Peter Bro Miltersen, and Yulong Zeng. Walrasian Pricing in Multi-Unit Auctions. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{branzei_et_al:LIPIcs.MFCS.2017.80,
  author =	{Br\^{a}nzei, Simina and Filos-Ratsikas, Aris and Miltersen, Peter Bro and Zeng, Yulong},
  title =	{{Walrasian Pricing in Multi-Unit Auctions}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{80:1--80:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.80},
  URN =		{urn:nbn:de:0030-drops-81197},
  doi =		{10.4230/LIPIcs.MFCS.2017.80},
  annote =	{Keywords: mechanism design, multi-unit auctions, Walrasian pricing, market share}
}
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