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Documents authored by Loff, Bruno

Document
DOI: 10.4230/LIPIcs.STACS.2026.35
A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

Authors: Pavel Dvořák, Bruno Loff, and Suhail Sherif

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We are interested in what happens when we take a Π₁ combinatorial statement, write its negation as a homogeneous quadratic feasibility problem (HQFP), and relax the problem into a positive semidefinite feasibility problem. This question is particularly interesting owing to the fact that any statement written as a PSD feasibility problem can be proven or disproven using a short proof. We investigate this for one very simple and one very complicated statement. The simple statement we look at is the pigeonhole principle. We prove that the relaxed negation of the PHP remains unsatisfiable and we thus obtain a new "quantum" pigeonhole principle (QPHP) which is a stronger statement than the vanilla PHP. It states that if we take n copies of the same state, and measure each copy using a measurement with only n-1 outcomes (the measurement can be different for different copies), then there will be an outcome j and two copies i₁, i₂ where the resulting states, obtained when the outcome is j for both copies, are not orthogonal. We then look at the statement "the deterministic communication complexity of f is ≤ k", where f could be either a function or a relation. We write this statement in two equivalent ways, using two different HQFPs. By relaxing to PSD feasibility, we increase the set of available protocols, and thus we always get a communication model which is stronger than deterministic communication complexity. An argument from proof complexity shows that any model obtained in this way will solve all Karchmer-Wigderson games efficiently. However, the argument is very indirect and does not give us an explicit protocol that solves the Karchmer-Wigderson games. We then work to find such protocols in the two communication models obtained by relaxing our two formulations. When relaxing the first of the two formulations we obtain a structured variant of the γ₂ norm. This communication model is to subunit γ₂ norm matrices like deterministic protocols are to rectangles, and so we call the protocols in this model γ₂ protocols. We show that log-inverse-discrepancy is a lower-bound for this model. We then show how to compute equality (deterministically) using O(1) bits of γ₂-communication, which implies that KW games are easy in the model. When relaxing the second of the two formulations we obtain what we call quantum lab protocols. This model happens to have a functional description, wherein Alice and Bob communicate solely via the outcomes of binary measurements of a shared quantum state (whose initial state is independent of the inputs). They are required to give the correct output with zero error probability. We use our QPHP to prove a lower-bound of n against two-round quantum lab protocols for equality. However we also show that any Boolean function f can be computed in three rounds and four measurements.
Cite as

Pavel Dvořák, Bruno Loff, and Suhail Sherif. A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2026.35,
  author =	{Dvo\v{r}\'{a}k, Pavel and Loff, Bruno and Sherif, Suhail},
  title =	{{A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.35},
  URN =		{urn:nbn:de:0030-drops-255243},
  doi =		{10.4230/LIPIcs.STACS.2026.35},
  annote =	{Keywords: Proofs, Semidefinite Programs, Quantum Pigeonhole Principle, Communication Complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.66
The Hardness of Decision Tree Complexity

Authors: Bruno Loff and Alexey Milovanov

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Let f be a Boolean function given as either a truth table or a circuit. How difficult is it to find the decision tree complexity, also known as deterministic query complexity, of f in both cases? We prove that this problem is NC¹-hard and PSPACE-hard, respectively. The second bound is tight, and the first bound is close to being tight.
Cite as

Bruno Loff and Alexey Milovanov. The Hardness of Decision Tree Complexity. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 66:1-66:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{loff_et_al:LIPIcs.STACS.2025.66,
  author =	{Loff, Bruno and Milovanov, Alexey},
  title =	{{The Hardness of Decision Tree Complexity}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{66:1--66:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.66},
  URN =		{urn:nbn:de:0030-drops-228913},
  doi =		{10.4230/LIPIcs.STACS.2025.66},
  annote =	{Keywords: Decision tree, Log-depth circuits}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.7
Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy

Authors: Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
The following question arises naturally in the study of graph streaming algorithms: Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes? Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems. Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes. The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds. Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions: - We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem. - We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.
Cite as

Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay. Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{assadi_et_al:LIPIcs.CCC.2024.7,
  author =	{Assadi, Sepehr and Ghosh, Prantar and Loff, Bruno and Mittal, Parth and Mukhopadhyay, Sagnik},
  title =	{{Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.7},
  URN =		{urn:nbn:de:0030-drops-204035},
  doi =		{10.4230/LIPIcs.CCC.2024.7},
  annote =	{Keywords: Graph streaming, Lower bounds, Communication complexity, k-Cores and degeneracy}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.147
Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games

Authors: Bruno Loff and Mateusz Skomra

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We devise a policy-iteration algorithm for deterministic two-player discounted and mean-payoff games, that runs in polynomial time with high probability, on any input where each payoff is chosen independently from a sufficiently random distribution and the underlying graph of the game is ergodic. This includes the case where an arbitrary set of payoffs has been perturbed by a Gaussian, showing for the first time that deterministic two-player games can be solved efficiently, in the sense of smoothed analysis. More generally, we devise a condition number for deterministic discounted and mean-payoff games played on ergodic graphs, and show that our algorithm runs in time polynomial in this condition number. Our result confirms a previous conjecture of Boros et al., which was claimed as a theorem [Boros et al., 2011] and later retracted [Boros et al., 2018]. It stands in contrast with a recent counter-example by Christ and Yannakakis [Christ and Yannakakis, 2023], showing that Howard’s policy-iteration algorithm does not run in smoothed polynomial time on stochastic single-player mean-payoff games. Our approach is inspired by the analysis of random optimal assignment instances by Frieze and Sorkin [Frieze and Sorkin, 2007], and the analysis of bias-induced policies for mean-payoff games by Akian, Gaubert and Hochart [Akian et al., 2018].
Cite as

Bruno Loff and Mateusz Skomra. Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 147:1-147:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{loff_et_al:LIPIcs.ICALP.2024.147,
  author =	{Loff, Bruno and Skomra, Mateusz},
  title =	{{Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{147:1--147:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.147},
  URN =		{urn:nbn:de:0030-drops-202908},
  doi =		{10.4230/LIPIcs.ICALP.2024.147},
  annote =	{Keywords: Mean-payoff games, discounted games, policy iteration, smoothed analysis}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.10
Memory Compression with Quantum Random-Access Gates

Authors: Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman

Published in: LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Abstract
In the classical RAM, we have the following useful property. If we have an algorithm that uses M memory cells throughout its execution, and in addition is sparse, in the sense that, at any point in time, only m out of M cells will be non-zero, then we may "compress" it into another algorithm which uses only m log M memory and runs in almost the same time. We may do so by simulating the memory using either a hash table, or a self-balancing tree. We show an analogous result for quantum algorithms equipped with quantum random-access gates. If we have a quantum algorithm that runs in time T and uses M qubits, such that the state of the memory, at any time step, is supported on computational-basis vectors of Hamming weight at most m, then it can be simulated by another algorithm which uses only O(m log M) memory, and runs in time Õ(T). We show how this theorem can be used, in a black-box way, to simplify the presentation in several papers. Broadly speaking, when there exists a need for a space-efficient history-independent quantum data-structure, it is often possible to construct a space-inefficient, yet sparse, quantum data structure, and then appeal to our main theorem. This results in simpler and shorter arguments.
Cite as

Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman. Memory Compression with Quantum Random-Access Gates. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buhrman_et_al:LIPIcs.TQC.2022.10,
  author =	{Buhrman, Harry and Loff, Bruno and Patro, Subhasree and Speelman, Florian},
  title =	{{Memory Compression with Quantum Random-Access Gates}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{10:1--10:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-237-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{232},
  editor =	{Le Gall, Fran\c{c}ois and Morimae, Tomoyuki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.10},
  URN =		{urn:nbn:de:0030-drops-165177},
  doi =		{10.4230/LIPIcs.TQC.2022.10},
  annote =	{Keywords: complexity theory, data structures, algorithms, quantum walk}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.31
Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks

Authors: Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n³)-time or O(n²)-time classic algorithm can be solved by a known O(n^{1.5})-time or O(n)-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro and Speelman [Buhrman et al., 2021], and by Aaronson, Chia, Lin, Wang, and Zhang [Aaronson et al., 2020]. In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time O(n^{2-ε}), for any ε > 0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-ε})-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems. These results are proven by adapting to the quantum setting known classical fine-grained reductions from the 3SUM problem. This adaptation is not trivial, however, since the original classical reductions require pre-processing the input in various ways, e.g. by sorting it according to some order, and this pre-processing (provably) cannot be done in sublinear quantum time. We overcome this bottleneck by combining a quantum walk with a classical dynamic data-structure having a certain "history-independence" property. This type of construction has been used in the past to prove upper bounds, and here we use it for the first time as part of a reduction. This general proof strategy allows us to prove tight lower bounds on several computational-geometry problems, on Convolution-3SUM and on the 0-Edge-Weight-Triangle problem, conditional on the Quantum-3SUM-Conjecture. We believe this proof strategy will be useful in proving tight (conditional) lower-bounds, and limits on quantum speed-ups, for many other problems.
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Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman. Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buhrman_et_al:LIPIcs.ITCS.2022.31,
  author =	{Buhrman, Harry and Loff, Bruno and Patro, Subhasree and Speelman, Florian},
  title =	{{Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.31},
  URN =		{urn:nbn:de:0030-drops-156273},
  doi =		{10.4230/LIPIcs.ITCS.2022.31},
  annote =	{Keywords: complexity theory, fine-grained complexity, 3SUM, computational geometry problems, data structures, quantum walk}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.31
Hardness of Constant-Round Communication Complexity

Authors: Shuichi Hirahara, Rahul Ilango, and Bruno Loff

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
How difficult is it to compute the communication complexity of a two-argument total Boolean function f:[N]×[N] → {0,1}, when it is given as an N×N binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard. In this work, we show that it is NP-hard to approximate the size (number of leaves) of the smallest constant-round protocol for a two-argument total Boolean function f:[N]×[N] → {0,1}, when it is given as an N×N binary matrix. Along the way to proving this, we show a new deterministic variant of the round elimination lemma, which may be of independent interest.
Cite as

Shuichi Hirahara, Rahul Ilango, and Bruno Loff. Hardness of Constant-Round Communication Complexity. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 31:1-31:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hirahara_et_al:LIPIcs.CCC.2021.31,
  author =	{Hirahara, Shuichi and Ilango, Rahul and Loff, Bruno},
  title =	{{Hardness of Constant-Round Communication Complexity}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{31:1--31:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.31},
  URN =		{urn:nbn:de:0030-drops-143055},
  doi =		{10.4230/LIPIcs.CCC.2021.31},
  annote =	{Keywords: NP-completeness, Communication Complexity, Round Elimination Lemma, Meta-Complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.20
Lower Bounds for Semi-adaptive Data Structures via Corruption

Authors: Pavel Dvořák and Bruno Loff

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove an unconditional polynomial lower bound of a trade-off between the update time and the query time of an adaptive dynamic data structure computing some explicit function. Ko and Weinstein provided such lower bound for a restricted class of semi-adaptive data structures, which compute the Disjointness function. There, the data are subsets x₁,… ,x_k and y of {1,… ,n}, the updates can modify y (by inserting and removing elements), and the queries are an index i ∈ {1,… ,k} (query i should answer whether x_i and y are disjoint, i.e., it should compute the Disjointness function applied to (x_i, y)). The semi-adaptiveness places a restriction in how the data structure can be accessed in order to answer a query. We generalize the lower bound of Ko and Weinstein to work not just for the Disjointness, but for any function having high complexity under the smooth corruption bound.
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Pavel Dvořák and Bruno Loff. Lower Bounds for Semi-adaptive Data Structures via Corruption. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dvorak_et_al:LIPIcs.FSTTCS.2020.20,
  author =	{Dvo\v{r}\'{a}k, Pavel and Loff, Bruno},
  title =	{{Lower Bounds for Semi-adaptive Data Structures via Corruption}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.20},
  URN =		{urn:nbn:de:0030-drops-132617},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.20},
  annote =	{Keywords: semi-adaptive dynamic data structure, polynomial lower bound, corruption bound, information theory}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.22
NP-Hardness of Circuit Minimization for Multi-Output Functions

Authors: Rahul Ilango, Bruno Loff, and Igor C. Oliveira

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n → {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless 𝖯 = 𝖭𝖯, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.
Cite as

Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 22:1-22:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ilango_et_al:LIPIcs.CCC.2020.22,
  author =	{Ilango, Rahul and Loff, Bruno and Oliveira, Igor C.},
  title =	{{NP-Hardness of Circuit Minimization for Multi-Output Functions}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{22:1--22:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.22},
  URN =		{urn:nbn:de:0030-drops-125744},
  doi =		{10.4230/LIPIcs.CCC.2020.22},
  annote =	{Keywords: MCSP, circuit minimization, communication complexity, Boolean circuit}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.50
Lifting Theorems for Equality

Authors: Bruno Loff and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We show a deterministic simulation (or lifting) theorem for composed problems f o Eq_n where the inner function (the gadget) is Equality on n bits. When f is a total function on p bits, it is easy to show via a rank argument that the communication complexity of f o Eq_n is Omega(deg(f) * n). However, there is a surprising counter-example of a partial function f on p bits, such that any completion f' of f has deg(f') = Omega(p), and yet f o Eq_n has communication complexity O(n). Nonetheless, we are able to show that the communication complexity of f o Eq_n is at least D(f) * n for a complexity measure D(f) which is closely related to the AND-query complexity of f and is lower-bounded by the logarithm of the leaf complexity of f. As a corollary, we also obtain lifting theorems for the set-disjointness gadget, and a lifting theorem in the context of parity decision-trees, for the NOR gadget. As an application, we prove a tight lower-bound for the deterministic communication complexity of the communication problem, where Alice and Bob are each given p-many n-bit strings, with the promise that either all of the strings are distinct, or all-but-one of the strings are distinct, and they wish to know which is the case. We show that the complexity of this problem is Theta(p * n).
Cite as

Bruno Loff and Sagnik Mukhopadhyay. Lifting Theorems for Equality. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{loff_et_al:LIPIcs.STACS.2019.50,
  author =	{Loff, Bruno and Mukhopadhyay, Sagnik},
  title =	{{Lifting Theorems for Equality}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{50:1--50:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.50},
  URN =		{urn:nbn:de:0030-drops-102892},
  doi =		{10.4230/LIPIcs.STACS.2019.50},
  annote =	{Keywords: Communication complexity, Query complexity, Simulation theorem, Equality function}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.21
Lower Bounds for Elimination via Weak Regularity

Authors: Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.
Cite as

Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21,
  author =	{Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik},
  title =	{{Lower Bounds for Elimination via Weak Regularity}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.21},
  URN =		{urn:nbn:de:0030-drops-70128},
  doi =		{10.4230/LIPIcs.STACS.2017.21},
  annote =	{Keywords: communication complexity, elimination, discrepancy, regularity, greater-than}
}
Document
DOI: 10.4230/LIPIcs.STACS.2016.24
Catalytic Space: Non-determinism and Hierarchy

Authors: Harry Buhrman, Michal Koucký, Bruno Loff, and Florian Speelman

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Abstract
Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.
Cite as

Harry Buhrman, Michal Koucký, Bruno Loff, and Florian Speelman. Catalytic Space: Non-determinism and Hierarchy. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{buhrman_et_al:LIPIcs.STACS.2016.24,
  author =	{Buhrman, Harry and Kouck\'{y}, Michal and Loff, Bruno and Speelman, Florian},
  title =	{{Catalytic Space: Non-determinism and Hierarchy}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.24},
  URN =		{urn:nbn:de:0030-drops-57258},
  doi =		{10.4230/LIPIcs.STACS.2016.24},
  annote =	{Keywords: catalytic computation, Immerman–Szelepcs\'{e}nyi theorem, space hierarchy}
}
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