28 Search Results for "Robere, Robert"

Document
DOI: 10.4230/LIPIcs.CCC.2024.9
Lifting Dichotomies

Authors: Yaroslav Alekseev, Yuval Filmus, and Alexander Smal

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

Abstract
Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f. In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].
Cite as

Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{alekseev_et_al:LIPIcs.CCC.2024.9,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Smal, Alexander},
  title =	{{Lifting Dichotomies}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{9:1--9:18},
  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.2024.9},
  URN =		{urn:nbn:de:0030-drops-204051},
  doi =		{10.4230/LIPIcs.CCC.2024.9},
  annote =	{Keywords: decision trees, log-rank conjecture, lifting, parity decision trees}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.15
Quantum Automating TC⁰-Frege Is LWE-Hard

Authors: Noel Arteche, Gaia Carenini, and Matthew Gray

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

Abstract
We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate TC⁰-Frege. This extends the line of results of Krajíček and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and Raz (FOCS, 1997), and Bonet, Domingo, Gavaldà, Maciel, and Pitassi (Computational Complexity, 2004), who showed that Extended Frege, TC⁰-Frege and AC⁰-Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.
Cite as

Noel Arteche, Gaia Carenini, and Matthew Gray. Quantum Automating TC⁰-Frege Is LWE-Hard. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{arteche_et_al:LIPIcs.CCC.2024.15,
  author =	{Arteche, Noel and Carenini, Gaia and Gray, Matthew},
  title =	{{Quantum Automating TC⁰-Frege Is LWE-Hard}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{15:1--15:25},
  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.2024.15},
  URN =		{urn:nbn:de:0030-drops-204117},
  doi =		{10.4230/LIPIcs.CCC.2024.15},
  annote =	{Keywords: automatability, post-quantum cryptography, feasible interpolation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.22
Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP

Authors: Theodoros Papamakarios

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

Abstract
We show that for any integer d > 0, depth-d Frege systems are NP-hard to automate. Namely, given a set S of depth-d formulas, it is NP-hard to find a depth-d Frege refutation of S in time polynomial in the size of the shortest such refutation. This extends the result of Atserias and Müller [JACM, 2020] for the non-automatability of resolution - a depth-1 Frege system - to Frege systems of any depth d > 0.
Cite as

Theodoros Papamakarios. Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{papamakarios:LIPIcs.CCC.2024.22,
  author =	{Papamakarios, Theodoros},
  title =	{{Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{22:1--22:17},
  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.2024.22},
  URN =		{urn:nbn:de:0030-drops-204187},
  doi =		{10.4230/LIPIcs.CCC.2024.22},
  annote =	{Keywords: Proof complexity, Automatability, Bounded-depth Frege}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.33
Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It

Authors: Michal Garlík

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

Abstract
We show that for every integer k ≥ 2, the Res(k) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a result of Atserias and Müller [Atserias and Müller, 2019] to Res(k). We show that if NP is not included in P (resp. QP, SUBEXP) then for every integer k ≥ 1, Res(k) is not automatable in polynomial (resp. quasi-polynomial, subexponential) time.
Cite as

Michal Garlík. Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 33:1-33:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{garlik:LIPIcs.CCC.2024.33,
  author =	{Garl{\'\i}k, Michal},
  title =	{{Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{33:1--33:23},
  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.2024.33},
  URN =		{urn:nbn:de:0030-drops-204295},
  doi =		{10.4230/LIPIcs.CCC.2024.33},
  annote =	{Keywords: reflection principle, feasible disjunction property, k-DNF Resolution}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.12
From Proof Complexity to Circuit Complexity via Interactive Protocols

Authors: Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam

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

Abstract
Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.
Cite as

Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam. From Proof Complexity to Circuit Complexity via Interactive Protocols. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{arteche_et_al:LIPIcs.ICALP.2024.12,
  author =	{Arteche, Noel and Khaniki, Erfan and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{From Proof Complexity to Circuit Complexity via Interactive Protocols}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{12:1--12:20},
  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.2024.12},
  URN =		{urn:nbn:de:0030-drops-201557},
  doi =		{10.4230/LIPIcs.ICALP.2024.12},
  annote =	{Keywords: proof complexity, circuit complexity, interactive protocols}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.116
One-Way Communication Complexity of Partial XOR Functions

Authors: Vladimir V. Podolskii and Dmitrii Sluch

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

Abstract
Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the Fourier analytic technique. For total XOR functions, it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on the Boolean Fourier analysis and thus, the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.
Cite as

Vladimir V. Podolskii and Dmitrii Sluch. One-Way Communication Complexity of Partial XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 116:1-116:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{podolskii_et_al:LIPIcs.ICALP.2024.116,
  author =	{Podolskii, Vladimir V. and Sluch, Dmitrii},
  title =	{{One-Way Communication Complexity of Partial XOR Functions}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{116:1--116:16},
  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.2024.116},
  URN =		{urn:nbn:de:0030-drops-202591},
  doi =		{10.4230/LIPIcs.ICALP.2024.116},
  annote =	{Keywords: Partial functions, XOR functions, communication complexity, decision trees, covering codes}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.117
Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle

Authors: Aaron Potechin and Aaron Zhang

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

Abstract
We show that the minimum total coefficient size of a Nullstellensatz proof of the pigeonhole principle on n+1 pigeons and n holes is 2^{Θ(n)}. We also investigate the ordering principle and construct an explicit Nullstellensatz proof for the ordering principle on n elements with total coefficient size 2ⁿ - n.
Cite as

Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{potechin_et_al:LIPIcs.ICALP.2024.117,
  author =	{Potechin, Aaron and Zhang, Aaron},
  title =	{{Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{117:1--117:20},
  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.2024.117},
  URN =		{urn:nbn:de:0030-drops-202604},
  doi =		{10.4230/LIPIcs.ICALP.2024.117},
  annote =	{Keywords: Proof complexity, Nullstellensatz, pigeonhole principle, coefficient size}
}
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)

Copy BibTex To Clipboard

@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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.32
Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents

Authors: Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber

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

Abstract
We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the α-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our α-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.
Cite as

Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber. Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{borzechowski_et_al:LIPIcs.ICALP.2024.32,
  author =	{Borzechowski, Michaela and Fearnley, John and Gordon, Spencer and Savani, Rahul and Schnider, Patrick and Weber, Simon},
  title =	{{Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{32:1--32:18},
  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.2024.32},
  URN =		{urn:nbn:de:0030-drops-201751},
  doi =		{10.4230/LIPIcs.ICALP.2024.32},
  annote =	{Keywords: P-LCP, Unique Sink Orientation, \alpha-Ham Sandwich, search complexity, TFNP, UEOPL}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.63
TFNP Intersections Through the Lens of Feasible Disjunction

Authors: Pavel Hubáček, Erfan Khaniki, and Neil Thapen

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
The complexity class CLS was introduced by Daskalakis and Papadimitriou (SODA 2010) to capture the computational complexity of important TFNP problems solvable by local search over continuous domains and, thus, lying in both PLS and PPAD. It was later shown that, e.g., the problem of computing fixed points guaranteed by Banach’s fixed point theorem is CLS-complete by Daskalakis et al. (STOC 2018). Recently, Fearnley et al. (J. ACM 2023) disproved the plausible conjecture of Daskalakis and Papadimitriou that CLS is a proper subclass of PLS∩PPAD by proving that CLS = PLS∩PPAD. To study the possibility of other collapses in TFNP, we connect classes formed as the intersection of existing subclasses of TFNP with the phenomenon of feasible disjunction in propositional proof complexity; where a proof system has the feasible disjunction property if, whenever a disjunction F ∨ G has a small proof, and F and G have no variables in common, then either F or G has a small proof. Based on some known and some new results about feasible disjunction, we separate the classes formed by intersecting the classical subclasses PLS, PPA, PPAD, PPADS, PPP and CLS. We also give the first examples of proof systems which have the feasible interpolation property, but not the feasible disjunction property.
Cite as

Pavel Hubáček, Erfan Khaniki, and Neil Thapen. TFNP Intersections Through the Lens of Feasible Disjunction. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 63:1-63:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{hubacek_et_al:LIPIcs.ITCS.2024.63,
  author =	{Hub\'{a}\v{c}ek, Pavel and Khaniki, Erfan and Thapen, Neil},
  title =	{{TFNP Intersections Through the Lens of Feasible Disjunction}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{63:1--63:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.63},
  URN =		{urn:nbn:de:0030-drops-195917},
  doi =		{10.4230/LIPIcs.ITCS.2024.63},
  annote =	{Keywords: TFNP, feasible disjunction, proof complexity, TFNP intersection classes}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.74
Intersection Classes in TFNP and Proof Complexity

Authors: Yuhao Li, William Pires, and Robert Robere

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
A recent breakthrough in the theory of total NP search problems (TFNP) by Fearnley, Goldberg, Hollender, and Savani has shown that CLS = PLS ∩ PPAD, or, in other words, the class of problems reducible to gradient descent are exactly those problems in the intersection of the complexity classes PLS and PPAD. Since this result, two more intersection theorems have been discovered in this theory: EOPL = PLS ∩ PPAD and SOPL = PLS ∩ PPADS. It is natural to wonder if this exhausts the list of intersection classes in TFNP, or, if other intersections exist. In this work, we completely classify all intersection classes involved among the classical TFNP classes PLS, PPAD, and PPA, giving new complete problems for the newly-introduced intersections. Following the close links between the theory of TFNP and propositional proof complexity, we develop new proof systems - each of which is a generalization of the classical Resolution proof system - that characterize all of the classes, in the sense that a query total search problem is in the intersection class if and only if a tautology associated with the search problem has a short proof in the proof system. We complement these new characterizations with black-box separations between all of the newly introduced classes and prior classes, thus giving strong evidence that no further collapse occurs. Finally, we characterize arbitrary intersections and joins of the PPA_q classes for q ≥ 2 in terms of the Nullstellensatz proof systems.
Cite as

Yuhao Li, William Pires, and Robert Robere. Intersection Classes in TFNP and Proof Complexity. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 74:1-74:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{li_et_al:LIPIcs.ITCS.2024.74,
  author =	{Li, Yuhao and Pires, William and Robere, Robert},
  title =	{{Intersection Classes in TFNP and Proof Complexity}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{74:1--74:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.74},
  URN =		{urn:nbn:de:0030-drops-196023},
  doi =		{10.4230/LIPIcs.ITCS.2024.74},
  annote =	{Keywords: TFNP, Proof Complexity, Intersection Classes}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.75
Total NP Search Problems with Abundant Solutions

Authors: Jiawei Li

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We define a new complexity class TFAP to capture TFNP problems that possess abundant solutions for each input. We identify several problems across diverse fields that belong to TFAP, including WeakPigeon (finding a collision in a mapping from [2n] pigeons to [n] holes), Yamakawa-Zhandry’s problem [Takashi Yamakawa and Mark Zhandry, 2022], and all problems in TFZPP. Conversely, we introduce the notion of "semi-gluability" to characterize TFNP problems that could have a unique or a very limited number of solutions for certain inputs. We prove that there is no black-box reduction from any "semi-gluable" problems to any TFAP problems. Furthermore, it can be extended to rule out randomized black-box reduction in most cases. We identify that the majority of common TFNP subclasses, including PPA, PPAD, PPADS, PPP, PLS, CLS, SOPL, and UEOPL, are "semi-gluable". This leads to a broad array of oracle separation results within TFNP regime. As a corollary, UEOPL^O ⊈ PWPP^O relative to an oracle O.
Cite as

Jiawei Li. Total NP Search Problems with Abundant Solutions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 75:1-75:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{li:LIPIcs.ITCS.2024.75,
  author =	{Li, Jiawei},
  title =	{{Total NP Search Problems with Abundant Solutions}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{75:1--75:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.75},
  URN =		{urn:nbn:de:0030-drops-196031},
  doi =		{10.4230/LIPIcs.ITCS.2024.75},
  annote =	{Keywords: TFNP, Pigeonhole Principle}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.36
Colourful TFNP and Propositional Proofs

Authors: Ben Davis and Robert Robere

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
Recent work has shown that many of the standard TFNP classes - such as PLS, PPADS, PPAD, SOPL, and EOPL - have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C-PLS, C-PPADS, C-PPAD, C-SOPL, and C-EOPL. While C-PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that: - Each of the classes C-PLS, C-PPADS, and C-SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C-PLS is characterized by depth-1.5 Frege (Res(polylog(n)). - The classes C-PPAD and C-EOPL coincide exactly with the uncoloured classes PPADS and SOPL, respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively. - Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C-PLS ∩ C-PPADS = C-SOPL, generalizing the intersection theorem PLS ∩ PPADS = SOPL. However, while it is known in the uncoloured world that PLS ∩ PPAD = EOPL = CLS, we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C-PLS^O ∩ C-PPAD^O ⊋ C-EOPL^O. To prove our results, we introduce an abstract multivalued proof system - the Blockwise Calculus - which may be of independent interest.
Cite as

Ben Davis and Robert Robere. Colourful TFNP and Propositional Proofs. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{davis_et_al:LIPIcs.CCC.2023.36,
  author =	{Davis, Ben and Robere, Robert},
  title =	{{Colourful TFNP and Propositional Proofs}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{36:1--36:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.36},
  URN =		{urn:nbn:de:0030-drops-183066},
  doi =		{10.4230/LIPIcs.CCC.2023.36},
  annote =	{Keywords: oracle separations, TFNP, proof complexity, Res(k), lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.23
On Low-End Obfuscation and Learning

Authors: Elette Boyle, Yuval Ishai, Pierre Meyer, Robert Robere, and Gal Yehuda

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Most recent works on cryptographic obfuscation focus on the high-end regime of obfuscating general circuits while guaranteeing computational indistinguishability between functionally equivalent circuits. Motivated by the goals of simplicity and efficiency, we initiate a systematic study of "low-end" obfuscation, focusing on simpler representation models and information-theoretic notions of security. We obtain the following results. - Positive results via "white-box" learning. We present a general technique for obtaining perfect indistinguishability obfuscation from exact learning algorithms that are given restricted access to the representation of the input function. We demonstrate the usefulness of this approach by obtaining simple obfuscation for decision trees and multilinear read-k arithmetic formulas. - Negative results via PAC learning. A proper obfuscation scheme obfuscates programs from a class C by programs from the same class. Assuming the existence of one-way functions, we show that there is no proper indistinguishability obfuscation scheme for k-CNF formulas for any constant k ≥ 3; in fact, even obfuscating 3-CNF by k-CNF is impossible. This result applies even to computationally secure obfuscation, and makes an unexpected use of PAC learning in the context of negative results for obfuscation. - Separations. We study the relations between different information-theoretic notions of indistinguishability obfuscation, giving cryptographic evidence for separations between them.
Cite as

Elette Boyle, Yuval Ishai, Pierre Meyer, Robert Robere, and Gal Yehuda. On Low-End Obfuscation and Learning. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 23:1-23:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{boyle_et_al:LIPIcs.ITCS.2023.23,
  author =	{Boyle, Elette and Ishai, Yuval and Meyer, Pierre and Robere, Robert and Yehuda, Gal},
  title =	{{On Low-End Obfuscation and Learning}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{23:1--23:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.23},
  URN =		{urn:nbn:de:0030-drops-175265},
  doi =		{10.4230/LIPIcs.ITCS.2023.23},
  annote =	{Keywords: Indistinguishability obfuscation, cryptography, learning}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.8
Trading Time and Space in Catalytic Branching Programs

Authors: James Cook and Ian Mertz

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
An m-catalytic branching program (Girard, Koucký, McKenzie 2015) is a set of m distinct branching programs for f which are permitted to share internal (i.e. non-source non-sink) nodes. While originally introduced as a non-uniform analogue to catalytic space, this also gives a natural notion of amortized non-uniform space complexity for f, namely the smallest value |G|/m for an m-catalytic branching program G for f (Potechin 2017). Potechin (2017) showed that every function f has amortized size O(n), witnessed by an m-catalytic branching program where m = 2^(2ⁿ-1). We recreate this result by defining a catalytic algorithm for evaluating polynomials using a large amount of space but O(n) time. This allows us to balance this with previously known algorithms which are efficient with respect to space at the cost of time (Cook, Mertz 2020, 2021). We show that for any ε ≥ 2n^(-1), every function f has an m-catalytic branching program of size O_ε(mn), where m = 2^(2^(ε n)). We similarly recreate an improved result due to Robere and Zuiddam (2021), and show that for d ≤ n and ε ≥ 2d^(-1), the same result holds for m = 2^binom(n, ≤ ε d) as long as f is a degree-d polynomial over 𝔽₂. We also show that for certain classes of functions, m can be reduced to 2^(poly n) while still maintaining linear or quasi-linear amortized size. In the other direction, we bound the necessary length, and by extension the amortized size, of any permutation branching program for an arbitrary function between 3n and 4n-4.
Cite as

James Cook and Ian Mertz. Trading Time and Space in Catalytic Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{cook_et_al:LIPIcs.CCC.2022.8,
  author =	{Cook, James and Mertz, Ian},
  title =	{{Trading Time and Space in Catalytic Branching Programs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{8:1--8:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.8},
  URN =		{urn:nbn:de:0030-drops-165708},
  doi =		{10.4230/LIPIcs.CCC.2022.8},
  annote =	{Keywords: complexity theory, branching programs, amortized, space complexity, catalytic computation}
}
  • Refine by Author
  • 12 Robere, Robert
  • 5 Göös, Mika
  • 4 Fleming, Noah
  • 4 Pitassi, Toniann
  • 2 Arteche, Noel
  • Show More...

  • Refine by Classification
  • 15 Theory of computation → Proof complexity
  • 5 Theory of computation → Complexity classes
  • 4 Theory of computation → Communication complexity
  • 4 Theory of computation → Problems, reductions and completeness
  • 3 Theory of computation → Circuit complexity
  • Show More...

  • Refine by Keyword
  • 7 TFNP
  • 6 Proof Complexity
  • 5 proof complexity
  • 4 Cutting Planes
  • 2 Communication Complexity
  • Show More...

  • Refine by Type
  • 28 document

  • Refine by Publication Year
  • 12 2024
  • 5 2022
  • 3 2019
  • 3 2020
  • 2 2021
  • Show More...

© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact