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Adventures in Monotone Complexity and TFNP

Authors Mika Göös, Pritish Kamath, Robert Robere, Dmitry Sokolov

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ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Circuit complexity
  • Theory of computation → Proof complexity
Keywords
  • TFNP
  • Monotone Complexity
  • Communication Complexity
  • Proof Complexity

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Abstract

Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity.
Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).

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Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in Monotone Complexity and TFNP. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.38

Author Details

Mika Göös
  • Institute for Advanced Study, Princeton, NJ, USA
Pritish Kamath
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Robert Robere
  • Simons Institute, Berkeley, CA, USA
Dmitry Sokolov
  • KTH Royal Institute of Technology, Stockholm, Sweden

Funding

  • Göös, Mika: Work done while at Harvard University; supported by the Michael O. Rabin Postdoctoral Fellowship.
  • Kamath, Pritish: Supported in parts by NSF grants CCF-1650733, CCF-1733808, and IIS-1741137.
  • Robere, Robert: Work done while at University of Toronto; supported by NSERC.
  • Sokolov, Dmitry: Supported by the Swedish Research Council grant 2016-00782, Knut and Alice Wallenberg Foundation grants KAW 2016.0066 and KAW 2016.0433.

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