Some Recent Advancements in Monotone Circuit Complexity (Invited Talk)

Author Susanna F. de Rezende



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Susanna F. de Rezende
  • Lund University, Sweden

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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) https://doi.org/10.4230/LIPIcs.STACS.2025.4

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Communication complexity
  • Theory of computation → Proof complexity
Keywords
  • monotone circuit complexity
  • query complexity
  • lifting theorems

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References

  1. Noga Alon and Ravi B. Boppana. The monotone circuit complexity of Boolean functions. Combinatorica, 7(1):1-22, March 1987. URL: https://doi.org/10.1007/bf02579196.
  2. Alexander E. Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Soviet Mathematics Doklady, 31(3):530-534, 1985. English translation of a paper in Doklady Akademii Nauk SSSR. Google Scholar
  3. Bruno Pasqualotto Cavalar, Mrinal Kumar, and Benjamin Rossman. Monotone circuit lower bounds from robust sunflowers. In Proceedings of the 14th Latin American Symposium on Theoretical Informatics (LATIN '20), volume 12118 of Lecture Notes in Computer Science, pages 311-322. Springer, January 2021. URL: https://doi.org/10.1007/978-3-030-61792-9_25.
  4. Susanna F. de Rezende, Noah Fleming, Duri Andrea Janett, Jakob Nordström, and Shuo Pang. Truly supercritical trade-offs for resolution, cutting planes, monotone circuits, and weisfeiler-leman. Technical Report 2411.14267, arXiv.org, November 2024. URL: https://doi.org/10.48550/arXiv.2411.14267.
  5. Susanna F. de Rezende, Or Meir, Jakob Nordstrom, Toniann Pitassi, Robert Robere, and Marc Vinyals. Lifting with simple gadgets and applications to circuit and proof complexity. In Proceedings of the 61st IEEE Annual Symposium on Foundations of Computer Science (FOCS '20), November 2020. URL: https://doi.org/10.1109/focs46700.2020.00011.
  6. Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals. How limited interaction hinders real communication (and what it means for proof and circuit complexity). In Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science (FOCS '16), October 2016. URL: https://doi.org/10.1109/focs.2016.40.
  7. Susanna F. de Rezende and Marc Vinyals. Lifting with colourful sunflowers. Manuscript, 2025. Google Scholar
  8. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. Theory of Computing, 16(13):1-30, 2020. Preliminary version in STOC '18. URL: https://doi.org/10.4086/toc.2020.v016a013.
  9. Mika Göös, Gilbert Maystre, Kilian Risse, and Dmitry Sokolov. Supercritical tradeoffs for monotone circuits. Technical Report 2411.14268, arXiv.org, November 2024. URL: https://doi.org/10.48550/arXiv.2411.14268.
  10. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. SIAM Journal on Computing, 47(6):2435-2450, January 2018. Preliminary version in FOCS '15. URL: https://doi.org/10.1137/16m1059369.
  11. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS '19), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:19, January 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.38.
  12. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics, 3(2):255-265, 1990. Preliminary version in STOC '88. URL: https://doi.org/10.1137/0403021.
  13. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with Sunflowers. In Proceedings of the 13th Innovations in Theoretical Computer Science Conference (ITCS '22), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), pages 104:1-104:24, January 2022. URL: https://doi.org/10.4230/LIPICS.ITCS.2022.104.
  14. Toniann Pitassi and Robert Robere. Strongly exponential lower bounds for monotone computation. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC '17), pages 1246-1255, June 2017. URL: https://doi.org/10.1145/3055399.3055478.
  15. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. Combinatorica, 19(3):403-435, March 1999. Preliminary version in FOCS '97. URL: https://doi.org/10.1007/s004930050062.
  16. Alexander A. Razborov. Lower bounds for the monotone complexity of some Boolean functions. Soviet Mathematics Doklady, 31(2):354-357, 1985. English translation of a paper in Doklady Akademii Nauk SSSR. Google Scholar
  17. Alexander A. Razborov. Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izvestiya: Mathematics, pages 205-227, February 1995. URL: https://doi.org/10.1070/im1995v059n01abeh000009.
  18. Dmitry Sokolov. Dag-like communication and its applications. In Proceedings of the 12th International Computer Science Symposium in Russia (CSR '17), volume 10304 of Lecture Notes in Computer Science, pages 294-307. Springer, June 2017. URL: https://doi.org/10.1007/978-3-319-58747-9_26.
  19. Ingo Wegener. The complexity of Boolean functions. Wiley-Teubner, 1987. URL: http://ls2-www.cs.uni-dortmund.de/monographs/bluebook/.
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