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

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.

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)

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@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} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

The sign-rank of a matrix A with ±1 entries is the smallest rank of a real matrix with the same sign pattern as A. To the best of our knowledge, there are only three known methods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is at least the VC-dimension; (ii) Forster’s method, which states that sign-rank is at least the inverse of the largest possible average margin among the representations of the matrix by points and half-spaces; (iii) Sign-rank is at least a logarithmic function of the density of the largest monochromatic rectangle.
We prove several results regarding the limitations of these methods.
- We prove that, qualitatively, the monochromatic rectangle density is the strongest of these three lower bounds. If it fails to provide a super-constant lower bound for the sign-rank of a matrix, then the other two methods will fail as well.
- We show that there exist n × n matrices with sign-rank n^Ω(1) for which none of these methods can provide a super-constant lower bound.
- We show that sign-rank is at most an exponential function of the deterministic communication complexity with access to an equality oracle. We combine this result with Green and Sanders' quantitative version of Cohen’s idempotent theorem to show that for a large class of sign matrices (e.g., xor-lifts), sign-rank is at most an exponential function of the γ₂ norm of the matrix. We conjecture that this holds for all sign matrices.
- Towards answering a question of Linial, Mendelson, Schechtman, and Shraibman regarding the relation between sign-rank and discrepancy, we conjecture that sign-ranks of the ±1 adjacency matrices of hypercube graphs can be arbitrarily large. We prove that none of the three lower bound techniques can resolve this conjecture in the affirmative.

Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, and Rosie Zhao. Lower Bound Methods for Sign-Rank and Their Limitations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 22:1-22:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hatami_et_al:LIPIcs.APPROX/RANDOM.2022.22, author = {Hatami, Hamed and Hatami, Pooya and Pires, William and Tao, Ran and Zhao, Rosie}, title = {{Lower Bound Methods for Sign-Rank and Their Limitations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {22:1--22:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.22}, URN = {urn:nbn:de:0030-drops-171445}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.22}, annote = {Keywords: Average Margin, Communication complexity, margin complexity, monochromatic rectangle, Sign-rank, Unbounded-error communication complexity, VC-dimension} }

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

We show EOPL = PLS ∩ PPAD. Here the class EOPL consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse CLS = PLS ∩ PPAD by Fearnley et al. (STOC 2021). We also prove a companion result SOPL = PLS ∩ PPADS, where SOPL is the class associated with the Sink-of-Potential-Line problem.

Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Further Collapses in TFNP. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 33:1-33:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goos_et_al:LIPIcs.CCC.2022.33, author = {G\"{o}\"{o}s, Mika and Hollender, Alexandros and Jain, Siddhartha and Maystre, Gilbert and Pires, William and Robere, Robert and Tao, Ran}, title = {{Further Collapses in TFNP}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {33:1--33:15}, 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.2022.33}, URN = {urn:nbn:de:0030-drops-165954}, doi = {10.4230/LIPIcs.CCC.2022.33}, annote = {Keywords: TFNP, PPAD, PLS, EOPL} }

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