12 Search Results for "Risse, Kilian"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.151
Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity

Authors: Susanna F. de Rezende, David Engström, Yassine Ghannane, Duri Andrea Janett, and Artur Riazanov

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.
Cite as

Susanna F. de Rezende, David Engström, Yassine Ghannane, Duri Andrea Janett, and Artur Riazanov. Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 151:1-151:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{derezende_et_al:LIPIcs.ICALP.2026.151,
  author =	{de Rezende, Susanna F. and Engstr\"{o}m, David and Ghannane, Yassine and Janett, Duri Andrea and Riazanov, Artur},
  title =	{{Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{151:1--151:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.151},
  URN =		{urn:nbn:de:0030-drops-265407},
  doi =		{10.4230/LIPIcs.ICALP.2026.151},
  annote =	{Keywords: proof complexity, communication complexity, cutting planes, bounded-depth resolution over parities, clique problem, average-case hardness, binary encoding}
}
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.ITCS.2026.81
Supercritical Tradeoff Between Size and Depth for Resolution over Parities

Authors: Dmitry Itsykson and Alexander Knop

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Alekseev and Itsykson (STOC 2025) proved the existence of an unsatisfiable CNF formula such that any resolution over parities (Res(⊕)) refutation must either have exponential size (in the formula size) or superlinear depth (in the number of variables). In this paper, we extend this result by constructing a formula with the same hardness properties, but which additionally admits a resolution refutation of quasi-polynomial size. This establishes a supercritical tradeoff between size and depth for resolution over parities. The proof builds on the framework of Alekseev and Itsykson and relies on a lifting argument applied to the supercritical tradeoff between width and depth in resolution, proposed by Buss and Thapen (IPL 2026).
Cite as

Dmitry Itsykson and Alexander Knop. Supercritical Tradeoff Between Size and Depth for Resolution over Parities. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 81:1-81:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{itsykson_et_al:LIPIcs.ITCS.2026.81,
  author =	{Itsykson, Dmitry and Knop, Alexander},
  title =	{{Supercritical Tradeoff Between Size and Depth for Resolution over Parities}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{81:1--81:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.81},
  URN =		{urn:nbn:de:0030-drops-253680},
  doi =		{10.4230/LIPIcs.ITCS.2026.81},
  annote =	{Keywords: lifting theorems, resolution depth, resolution over parities, resolution width, supercritical tradeoff}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.60
Total Search Problems in ZPP

Authors: Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We initiate a systematic study of TFZPP, the class of total NP search problems solvable by polynomial time randomized algorithms. TFZPP contains a variety of important search problems such as Bertrand-Chebyshev (finding a prime between N and 2N), refuter problems for many circuit lower bounds, and Lossy-Code. The Lossy-Code problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas. While TFZPP collapses to FP under standard derandomization assumptions in the white-box setting, we are able to separate TFZPP from the major TFNP subclasses in the black-box setting. In fact, we are able to separate it from every uniform TFNP class assuming that NP is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box TFNP to randomized proof systems and randomized reductions. Next, we turn to developing a taxonomy of TFZPP problems. We highlight a problem called Nephew, originating from an infinity axiom in set theory. We show that Nephew is in PWPP∩ TFZPP and conjecture that it is not reducible to Lossy-Code. Intriguingly, except for some artificial examples, most other black-box TFZPP problems that we are aware of reduce to Lossy-Code: - We define a problem called Empty-Child capturing finding a leaf in a rooted (binary) tree, and show that this problem is equivalent to Lossy-Code. We also show that a variant of Empty-Child with "heights" is complete for the intersection of SOPL and Lossy-Code. - We strengthen Lossy-Code with several combinatorial inequalities such as the AM-GM inequality. Somewhat surprisingly, we show the resulting new problems are still reducible to Lossy-Code. A technical highlight of this result is that they are proved by formalizations in bounded arithmetic, specifically in Jeřábek’s theory APC₁ (JSL 2007). - Finally, we show that the Dense-Linear-Ordering problem reduces to Lossy-Code.
Cite as

Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan. Total Search Problems in ZPP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 60:1-60:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fleming_et_al:LIPIcs.ITCS.2026.60,
  author =	{Fleming, Noah and Grosser, Stefan and Jain, Siddhartha and Li, Jiawei and Ren, Hanlin and Shirley, Morgan and Yuan, Weiqiang},
  title =	{{Total Search Problems in ZPP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{60:1--60:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.60},
  URN =		{urn:nbn:de:0030-drops-253473},
  doi =		{10.4230/LIPIcs.ITCS.2026.60},
  annote =	{Keywords: TFNP, lossy code, randomized proof systems, query complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.111
Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits

Authors: Hanlin Ren, Yichuan Wang, and Yan Zhong

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Given a circuit G: {0, 1}ⁿ → {0, 1}^m with m > n, the range avoidance problem (Avoid) asks to output a string y ∈ {0, 1}^m that is not in the range of G. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related to the existence of proof complexity generators - circuits G: {0, 1}ⁿ → {0, 1}^m where m > n but for every y ∈ {0, 1}^m, it is infeasible to prove the statement "y ̸ ∈ Range(G)" in a given propositional proof system. This paper connects these two problems with the existence of demi-bits generators, a fundamental cryptographic primitive against nondeterministic adversaries introduced by Rudich (RANDOM '97). - We show that the existence of demi-bits generators implies Avoid is hard for nondeterministic algorithms. This resolves an open problem raised by Chen and Li (STOC '24). Furthermore, assuming the demi-hardness of certain LPN-style generators or Goldreich’s PRG, we prove the hardness of Avoid even when the instances are constant-degree polynomials over 𝔽₂. - We show that the dual weak pigeonhole principle is unprovable in Cook’s theory PV₁ under the existence of demi-bits generators secure against AM/_{O(1)}, thereby separating Jeřábek’s theory APC₁ from PV₁. Previously, Ilango, Li, and Williams (STOC '23) obtained the same separation under different (and arguably stronger) cryptographic assumptions. - We transform demi-bits generators to proof complexity generators that are pseudo-surjective in certain parameter regime. Pseudo-surjectivity is the strongest form of hardness considered in the literature for proof complexity generators. Our constructions are inspired by the recent breakthroughs on the hardness of Avoid by Ilango, Li, and Williams (STOC '23) and Chen and Li (STOC '24). We use randomness extractors to significantly simplify the construction and the proof.
Cite as

Hanlin Ren, Yichuan Wang, and Yan Zhong. Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 111:1-111:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ren_et_al:LIPIcs.ITCS.2026.111,
  author =	{Ren, Hanlin and Wang, Yichuan and Zhong, Yan},
  title =	{{Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{111:1--111:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.111},
  URN =		{urn:nbn:de:0030-drops-253982},
  doi =		{10.4230/LIPIcs.ITCS.2026.111},
  annote =	{Keywords: Range Avoidance, Proof Complexity Generators}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.64
Searching for Falsified Clause in Random (log{n})-CNFs Is Hard for Randomized Communication

Authors: Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov, and Weiqiang Yuan

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

Abstract
We show that for a randomly sampled unsatisfiable O(log n)-CNF over n variables the randomized two-party communication cost of finding a clause falsified by the given variable assignment is linear in n.
Cite as

Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov, and Weiqiang Yuan. Searching for Falsified Clause in Random (log{n})-CNFs Is Hard for Randomized Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{riazanov_et_al:LIPIcs.APPROX/RANDOM.2025.64,
  author =	{Riazanov, Artur and Sofronova, Anastasia and Sokolov, Dmitry and Yuan, Weiqiang},
  title =	{{Searching for Falsified Clause in Random (log\{n\})-CNFs Is Hard for Randomized Communication}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{64:1--64:17},
  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.2025.64},
  URN =		{urn:nbn:de:0030-drops-244306},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.64},
  annote =	{Keywords: communication complexity, proof complexity, random CNF}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.32
A Lower Bound for k-DNF Resolution on Random CNF Formulas via Expansion

Authors: Anastasia Sofronova and Dmitry Sokolov

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Random Δ-CNF formulas are one of the few candidates that are expected to be hard for proof systems and SAT algotirhms. Assume we sample m clauses over n variables. Here, the main complexity parameter is clause density, χ := m/n. For a fixed Δ, there exists a satisfiability threshold c_Δ such that for χ > c_Δ a formula is unsatisfiable with high probability. and for χ < c_Δ it is satisfiable with high probability. Near satisfiability threshold, there are various lower bounds for algorithms and proof systems [Eli Ben-Sasson, 2001; Eli Ben-Sasson and Russell Impagliazzo, 1999; Michael Alekhnovich and Alexander A. Razborov, 2003; Dima Grigoriev, 2001; Grant Schoenebeck, 2008; Pavel Hrubes and Pavel Pudlák, 2017; Noah Fleming et al., 2017; Dmitry Sokolov, 2024], and for high-density regimes, there exist upper bounds [Uriel Feige et al., 2006; Sebastian Müller and Iddo Tzameret, 2014; Jackson Abascal et al., 2021; Venkatesan Guruswami et al., 2022]. One of the frontiers in the direction of proving lower bounds on these formulas is the k-DNF Resolution proof system (aka Res(k)). There are several known results for k = 𝒪(√{log n}/{log log n}}) [Nathan Segerlind et al., 2004; Michael Alekhnovich, 2011], that are applicable only for density regime near the threshold. In this paper, we show the first Res(k) lower bound that is applicable in higher-density regimes. Our results work for slightly larger k = 𝒪(√{log n}).
Cite as

Anastasia Sofronova and Dmitry Sokolov. A Lower Bound for k-DNF Resolution on Random CNF Formulas via Expansion. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 32:1-32:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sofronova_et_al:LIPIcs.CCC.2025.32,
  author =	{Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{A Lower Bound for k-DNF Resolution on Random CNF Formulas via Expansion}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{32:1--32:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.32},
  URN =		{urn:nbn:de:0030-drops-237269},
  doi =		{10.4230/LIPIcs.CCC.2025.32},
  annote =	{Keywords: proof complexity, random CNFs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.24
Super-Critical Trade-Offs in Resolution over Parities via Lifting

Authors: Arkadev Chattopadhyay and Pavel Dvořák

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Razborov [Alexander A. Razborov, 2016] exhibited the following surprisingly strong trade-off phenomenon in propositional proof complexity: for a parameter k = k(n), there exists k-CNF formulas over n variables, having resolution refutations of O(k) width, but every tree-like refutation of width n^{1-ε}/k needs size exp(n^Ω(k)). We extend this result to tree-like Resolution over parities, commonly denoted by Res(⊕), with parameters essentially unchanged. To obtain our result, we extend the lifting theorem of Chattopadhyay, Mande, Sanyal and Sherif [Arkadev Chattopadhyay et al., 2023] to handle tree-like affine DAGs. We introduce additional ideas from linear algebra to handle forget nodes along long paths.
Cite as

Arkadev Chattopadhyay and Pavel Dvořák. Super-Critical Trade-Offs in Resolution over Parities via Lifting. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2025.24,
  author =	{Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
  title =	{{Super-Critical Trade-Offs in Resolution over Parities via Lifting}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{24:1--24:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.24},
  URN =		{urn:nbn:de:0030-drops-237186},
  doi =		{10.4230/LIPIcs.CCC.2025.24},
  annote =	{Keywords: Proof complexity, Lifting, Resolution over parities}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.8
Amortized Closure and Its Applications in Lifting for Resolution over Parities

Authors: Klim Efremenko and Dmitry Itsykson

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
The notion of closure of a set of linear forms, first introduced by Efremenko, Garlik, and Itsykson [Klim Efremenko et al., 2024], has proven instrumental in proving lower bounds on the sizes of regular and bounded-depth Res(⊕) refutations [Klim Efremenko et al., 2024; Yaroslav Alekseev and Dmitry Itsykson, 2025]. In this work, we present amortized closure, an enhancement that retains the properties of original closure [Klim Efremenko et al., 2024] but offers tighter control on its growth. Specifically, adding a new linear form increases the amortized closure by at most one. We explore two applications that highlight the power of this new concept. Utilizing our newly defined amortized closure, we extend and provide a succinct and elegant proof of the recent lifting theorem by Chattopadhyay and Dvorak [Arkadev Chattopadhyay and Pavel Dvorak, 2025]. Namely we show that for an unsatisfiable CNF formula φ and a 1-stifling gadget g: {0,1}^𝓁 → {0,1}, if the lifted formula φ∘g has a tree-like Res(⊕) refutation of size 2^d and width w, then φ has a resolution refutation of depth d and width w. The original theorem by Chattopadhyay and Dvorak [Arkadev Chattopadhyay and Pavel Dvorak, 2025] applies only to the more restrictive class of strongly stifling gadgets. As a more significant application of amortized closure, we show improved lower bounds for bounded-depth Res(⊕), extending the depth beyond that of Alekseev and Itsykson [Yaroslav Alekseev and Dmitry Itsykson, 2025]. Our result establishes an exponential lower bound for depth-Ω(n log n) Res(⊕) refutations of lifted Tseitin formulas, a notable improvement over the existing depth-Ω(n log log n) Res(⊕) lower bound.
Cite as

Klim Efremenko and Dmitry Itsykson. Amortized Closure and Its Applications in Lifting for Resolution over Parities. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{efremenko_et_al:LIPIcs.CCC.2025.8,
  author =	{Efremenko, Klim and Itsykson, Dmitry},
  title =	{{Amortized Closure and Its Applications in Lifting for Resolution over Parities}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{8:1--8:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.8},
  URN =		{urn:nbn:de:0030-drops-237023},
  doi =		{10.4230/LIPIcs.CCC.2025.8},
  annote =	{Keywords: lifting, resolution over parities, closure of linear forms, lower bounds, width, depth, size vs depth tradeoff}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2025.4
Some Recent Advancements in Monotone Circuit Complexity (Invited Talk)

Authors: Susanna F. de Rezende

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

Abstract
In 1985, Razborov [Razborov, 1985] proved the first superpolynomial size lower bound for monotone Boolean circuits for the perfect matching the clique functions, and, independently, Andreev [Andreev, 1985] obtained exponential size lower bounds. These breakthroughs were soon followed by further advancements in monotone complexity, including better lower bounds for clique [Alon and Boppana, 1987; Ingo Wegener, 1987], superlogarithmic depth lower bounds for connectivity by Karchmer and Wigderson [Karchmer and Wigderson, 1990], and the separations mon-NC ≠ mon-P and that mon-NC^i ≠ mon-NC^{i+1} by Raz and McKenzie [Ran Raz and Pierre McKenzie, 1999]. Karchmer and Wigderson [Karchmer and Wigderson, 1990] proved their result by establishing a relation between communication complexity and (monotone) circuit depth, and Raz and McKenzie [Ran Raz and Pierre McKenzie, 1999] introduced a new technique, now called lifting theorems, for obtaining communication lower bounds from query complexity lower bounds, In this talk, we will survey recent advancements in monotone complexity driven by query-to-communication lifting theorems. A decade ago, Göös, Pitassi, and Watson [Mika Göös et al., 2018] brought to light the generality of the result of Raz and McKenzie [Ran Raz and Pierre McKenzie, 1999] and reignited this line of work. A notable extension is the lifting theorem [Ankit Garg et al., 2020] for a model of DAG-like communication [Alexander A. Razborov, 1995; Dmitry Sokolov, 2017] that corresponds to circuit size. These powerful theorems, in their different flavours, have been instrumental in addressing many open questions in monotone circuit complexity, including: optimal 2^Ω(n) lower bounds on the size of monotone Boolean formulas computing an explicit function in NP [Toniann Pitassi and Robert Robere, 2017]; a complete picture of the relation between the mon-AC and mon-NC hierarchies [Susanna F. de Rezende et al., 2016]; a near optimal separation between monotone circuit and monotone formula size [Susanna F. de Rezende et al., 2020]; exponential separation between NC^2 and mon-P [Ankit Garg et al., 2020; Mika Göös et al., 2019]; and better lower bounds for clique [de Rezende and Vinyals, 2025; Lovett et al., 2022], improving on [Cavalar et al., 2021]. Very recently, lifting theorems were also used to prove supercritical trade-offs for monotone circuits showing that there are functions computable by small circuits for which any small circuit must have superlinear or even superpolynomial depth [de Rezende et al., 2024; Göös et al., 2024]. We will explore these results and their implications, and conclude by discussing some open problems.
Cite as

Susanna F. de Rezende. Some Recent Advancements in Monotone Circuit Complexity (Invited Talk). In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 4:1-4:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{derezende:LIPIcs.STACS.2025.4,
  author =	{de Rezende, Susanna F.},
  title =	{{Some Recent Advancements in Monotone Circuit Complexity}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{4:1--4:2},
  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.2025.4},
  URN =		{urn:nbn:de:0030-drops-228291},
  doi =		{10.4230/LIPIcs.STACS.2025.4},
  annote =	{Keywords: monotone circuit complexity, query complexity, lifting theorems}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.31
Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem

Authors: Per Austrin and Kilian Risse

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

Abstract
We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}ⁿ → {0,1}, SoS requires degree Ω(s^{1-ε}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ⊈ P/poly. We also show that for any 0 < α < 1 there are Boolean functions with circuit complexity larger than 2^{n^α} but SoS requires size 2^{2^Ω(n^α)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q.
Cite as

Per Austrin and Kilian Risse. Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{austrin_et_al:LIPIcs.CCC.2023.31,
  author =	{Austrin, Per and Risse, Kilian},
  title =	{{Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{31:1--31:21},
  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.2023.31},
  URN =		{urn:nbn:de:0030-drops-183011},
  doi =		{10.4230/LIPIcs.CCC.2023.31},
  annote =	{Keywords: Proof Complexity, Sum of Squares, Minimum Circuit Size Problem}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.28
Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

Authors: Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov

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

Abstract
We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov’s pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.
Cite as

Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2020.28,
  author =	{de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Risse, Kilian and Sokolov, Dmitry},
  title =	{{Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{28:1--28:24},
  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.2020.28},
  URN =		{urn:nbn:de:0030-drops-125804},
  doi =		{10.4230/LIPIcs.CCC.2020.28},
  annote =	{Keywords: proof complexity, resolution, weak pigeonhole principle, perfect matching, sparse graphs}
}
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