Document Open Access Logo

Adventures in Monotone Complexity and TFNP

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



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2019.38.pdf
  • Filesize: 0.77 MB
  • 19 pages

Document Identifiers

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

Cite AsGet BibTex

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

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).

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, and Avi Wigderson. Pseudorandom Generators in Propositional Proof Complexity. SIAM Journal on Computing, 34(1):67-88, 2004. URL: http://dx.doi.org/10.1137/S0097539701389944.
  2. László Babai, Peter Frankl, and Janos Simon. Complexity Classes in Communication Complexity Theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337-347. IEEE, 1986. URL: http://dx.doi.org/10.1109/SFCS.1986.15.
  3. László Babai, Anna Gál, and Avi Wigderson. Superpolynomial Lower Bounds for Monotone Span Programs. Combinatorica, 19(3):301-319, 1999. URL: http://dx.doi.org/10.1007/s004930050058.
  4. Yakov Babichenko, Shahar Dobzinski, and Noam Nisan. The Communication Complexity of Local Search. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1804.02676.
  5. Yakov Babichenko and Aviad Rubinstein. Communication Complexity of Approximate Nash Equilibria. In Proceedings of the 49th Symposium on Theory of Computing (STOC), pages 878-889. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055407.
  6. Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The Relative Complexity of NP Search Problems. Journal of Computer and System Sciences, 57(1):3-19, 1998. URL: http://dx.doi.org/10.1006/jcss.1998.1575.
  7. Paul Beame, Toniann Pitassi, and Nathan Segerlind. Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity. SIAM Journal on Computing, 37(3):845-869, 2007. URL: http://dx.doi.org/10.1137/060654645.
  8. Paul Beame and Søren Riis. More on the Relative Strength of Counting Principles. In Proceedings of the DIMACS Workshop on Proof Complexity and Feasible Arithmetics, volume 39, pages 13-35, 1998. Google Scholar
  9. Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz. In Proceedings of the 32nd Computational Complexity Conference (CCC), volume 79, pages 30:1-30:24. Schloss Dagstuhl, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.30.
  10. Eli Ben-Sasson and Avi Wigderson. Short Proofs Are Narrow - Resolution Made Simple. Journal of the ACM, 48(2):149-169, 2001. URL: http://dx.doi.org/10.1145/375827.375835.
  11. Andrei Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 319-330, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.37.
  12. Joshua Buresh-Oppenheim and Tsuyoshi Morioka. Relativized NP search problems and propositional proof systems. In Proceedings of the 19th Conference on Computational Complexity (CCC), pages 54-67, 2004. URL: http://dx.doi.org/10.1109/CCC.2004.1313795.
  13. Sam Buss, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. Linear Gaps between Degrees for the Polynomial Calculus Modulo Distinct Primes. Journal of Computer and System Sciences, 62(2):267-289, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1726.
  14. Siu Man Chan. Just a Pebble Game. In Proceedings of the 28th Conference on Computational Complexity (CCC), pages 133-143, 2013. URL: http://dx.doi.org/10.1109/CCC.2013.22.
  15. Siu Man Chan and Aaron Potechin. Tight Bounds for Monotone Switching Networks via Fourier Analysis. Theory of Computing, 10(15):389-419, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a015.
  16. Stephen Cook, Yuval Filmus, and Dai Tri Man Lê. The Complexity of the Comparator Circuit Value Problem. ACM Transactions on Computation Theory, 6(4):15:1-15:44, 2014. URL: http://dx.doi.org/10.1145/2635822.
  17. Constantinos Daskalakis and Christos Papadimitriou. Continuous Local Search. In Proceedings of the 22nd Symposium on Discrete Algorithms (SODA), pages 790-804. SIAM, 2011. URL: http://dl.acm.org/citation.cfm?id=2133036.2133098.
  18. Susanna 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 Symposium on Foundations of Computer Science (FOCS), pages 295-304. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.40.
  19. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. End of Potential Line. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1804.03450.
  20. Tomás Feder and Moshe Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: http://dx.doi.org/10.1137/S0097539794266766.
  21. Anna Gál. A Characterization of Span Program Size and Improved Lower Bounds for Monotone Span Programs. Computational Complexity, 10(4):277-296, 2001. URL: http://dx.doi.org/10.1007/s000370100001.
  22. Anat Ganor and Karthik C. S. Communication Complexity of Correlated Equilibrium with Small Support. In Proceedings of the 22nd International Conference on Randomization and Computation (RANDOM), volume 116, pages 12:1-12:16. Schloss Dagstuhl, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.12.
  23. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone Circuit Lower Bounds from Resolution. In Proceedings of the 50th Symposium on Theory of Computing (STOC), pages 902-911. ACM, 2018. URL: http://dx.doi.org/10.1145/3188745.3188838.
  24. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in Monotone Complexity and TFNP. Technical Report TR18-163, Electronic Colloquium on Computational Complexity (ECCC), 2018. URL: https://eccc.weizmann.ac.il/report/2018/163/.
  25. Mika Göös and Toniann Pitassi. Communication Lower Bounds via Critical Block Sensitivity. In Proceedings of the 46th Symposium on Theory of Computing (STOC), pages 847-856. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591838.
  26. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic Communication vs. Partition Number. In Proceedings of the 56th Symposium on Foundations of Computer Science (FOCS), pages 1077-1088. IEEE, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
  27. Mika Göös, Toniann Pitassi, and Thomas Watson. The Landscape of Communication Complexity Classes. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pages 86:1-86:15. Schloss Dagstuhl, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.86.
  28. Mika Göös and Aviad Rubinstein. Near-Optimal Communication Lower Bounds for Approximate Nash Equilibria. In Proceedings of the 59th Symposium on Foundations of Computer Science (FOCS), 2018. To appear. URL: http://arxiv.org/abs/1805.06387.
  29. Michelangelo Grigni and Michael Sipser. Monotone Complexity. In Proceedings of the London Mathematical Society Symposium on Boolean Function Complexity, pages 57-75. Cambridge University Press, 1992. URL: http://dl.acm.org/citation.cfm?id=167687.167706.
  30. Bernd Halstenberg and Rüdiger Reischuk. Relations Between Communication Complexity Classes. Journal of Computer and System Sciences, 41(3):402-429, 1990. URL: http://dx.doi.org/10.1016/0022-0000(90)90027-I.
  31. Alexandros Hollender and Paul Goldberg. The Complexity of Multi-source Variants of the End-of-Line Problem, and the Concise Mutilated Chessboard. Technical report, Electronic Colloquium on Computational Complexity (ECCC), 2018. URL: https://eccc.weizmann.ac.il/report/2018/120/.
  32. Pavel Hubáček and Eylon Yogev. Hardness of Continuous Local Search: Query Complexity and Cryptographic Lower Bounds. In Proceedings of the 28th Symposium on Discrete Algorithms (SODA), pages 1352-1371, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.88.
  33. Trinh Huynh and Jakob Nordström. On the Virtue of Succinct Proofs: Amplifying Communication Complexity Hardness to Time-Space Trade-Offs in Proof Complexity. In Proceedings of the 44th Symposium on Theory of Computing (STOC), pages 233-248. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2214000.
  34. Russell Impagliazzo, Toniann Pitassi, and Alasdair Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proceedings of the 9th Symposium on Logic in Computer Science (LICS), pages 220-228. IEEE, 1994. URL: http://dx.doi.org/10.1109/LICS.1994.316069.
  35. David Johnson, Christos Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37(1):79-100, 1988. URL: http://dx.doi.org/10.1016/0022-0000(88)90046-3.
  36. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. Google Scholar
  37. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. In Proceedings of the 20th Symposium on Theory of Computing (STOC), pages 539-550. ACM, 1988. URL: http://dx.doi.org/10.1145/62212.62265.
  38. Mauricio Karchmer and Avi Wigderson. On span programs. In Proceedings of the 8th Structure in Complexity Theory Conference, pages 102-111, 1993. URL: http://dx.doi.org/10.1109/SCT.1993.336536.
  39. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  40. László Lovász. On determinants, matchings, and random algorithms. In Proceedings of the 2nd Conference on Fundamentals of Computation Theory (FCT), pages 565-574, 1979. Google Scholar
  41. Ernst Mayr and Ashok Subramanian. The complexity of circuit value and network stability. Journal of Computer and System Sciences, 44(2):302-323, 1992. URL: http://dx.doi.org/10.1016/0022-0000(92)90024-D.
  42. Nimrod Megiddo and Christos Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317-324, 1991. URL: http://dx.doi.org/10.1016/0304-3975(91)90200-L.
  43. Tsuyoshi Morioka. Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic. PhD thesis, University of Toronto, 2005. Google Scholar
  44. Ketan Mulmuley. A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica, 7(1):101-104, 1987. URL: http://dx.doi.org/10.1007/BF02579205.
  45. Igor Oliveira. Unconditional Lower Bounds in Complexity Theory. PhD thesis, Columbia University, 2015. URL: http://dx.doi.org/10.7916/D8ZP45KT.
  46. Christos Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 48(3):498-532, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80063-7.
  47. Toniann Pitassi and Robert Robere. Strongly Exponential Lower Bounds for Monotone Computation. In Proceedings of the 49th Symposium on Theory of Computing (STOC), pages 1246-1255. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055478.
  48. Toniann Pitassi and Robert Robere. Lifting Nullstellensatz to Monotone Span Programs over Any Field. In Proceedings of the 50th Symposium on Theory of Computing (STOC), pages 1207-1219. ACM, 2018. URL: http://dx.doi.org/10.1145/3188745.3188914.
  49. Aaron Potechin. Bounds on Monotone Switching Networks for Directed Connectivity. Journal of the ACM, 64(4):29:1-29:48, 2017. URL: http://dx.doi.org/10.1145/3080520.
  50. Pavel Pudlák. On extracting computations from propositional proofs (a survey). In Proceedings of the 30th Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 8, pages 30-41. Schloss Dagstuhl, 2010. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2010.30.
  51. Pavel Pudlák and Jiří Sgall. Algebraic models of computation and interpolation for algebraic proof systems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 39:279-295, 1998. URL: http://dx.doi.org/10.1090/dimacs/039.
  52. Ran Raz and Pierre McKenzie. Separation of the Monotone NC Hierarchy. Combinatorica, 19(3):403-435, 1999. URL: http://dx.doi.org/10.1007/s004930050062.
  53. Alexander Razborov. Lower bounds on monotone complexity of the logical permanent. Mathematical notes of the Academy of Sciences of the USSR, 37(6):485-493, 1985. URL: http://dx.doi.org/10.1007/BF01157687.
  54. Alexander Razborov. Lower bounds on the monotone complexity of some Boolean functions. Doklady Akademii Nauk USSR, 285:798-801, 1985. Google Scholar
  55. Alexander Razborov. Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izvestiya of the RAN, pages 201-224, 1995. Google Scholar
  56. Robert Robere, Toniann Pitassi, Benjamin Rossman, and Stephen Cook. Exponential Lower Bounds for Monotone Span Programs. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), pages 406-415. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.51.
  57. Tim Roughgarden. Complexity Theory, Game Theory, and Economics. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1801.00734.
  58. Tim Roughgarden and Omri Weinstein. On the Communication Complexity of Approximate Fixed Points. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), pages 229-238. IEEE, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.32.
  59. Thomas Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the 10th Symposium on Theory of Computing (STOC), pages 216-226. ACM, 1978. URL: http://dx.doi.org/10.1145/800133.804350.
  60. Dmitry Sokolov. Dag-Like Communication and Its Applications. In Proceedings of the 12th Computer Science Symposium in Russia (CSR), pages 294-307. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-58747-9_26.
  61. Éva Tardos. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica, 8(1):141-142, 1988. URL: http://dx.doi.org/10.1007/BF02122563.
  62. Alasdair Urquhart. Hard examples for resolution. Journal of the ACM, 34(1):209-219, 1987. URL: http://dx.doi.org/10.1145/7531.8928.
  63. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 331-342, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.38.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail