Susanna F. de Rezende
Jakob Nordström
ABSTRACT
We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali–Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Müller (JACM 2020) that established an analogous result for Resolution.
- Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, and Avi Wigderson. 2002. Space Complexity in Propositional Calculus. SIAM J. Comput., 31, 4, 2002. Pages 1184–1211. https://doi.org/10.1137/S0097539700366735 Google Scholar
Digital Library
- Michael Alekhnovich, Sam Buss, Shlomo Moran, and Toniann Pitassi. 2001. Minimum propositional proof length is NP-hard to linearly approximate. Journal of Symbolic Logic, 66, 1, 2001. Pages 171–191. https://doi.org/10.2307/2694916 Google ScholarCross Ref
- Michael Alekhnovich and Alexander Razborov. 2008. Resolution Is Not Automatizable Unless W[P] Is Tractable. SIAM J. Comput., 38, 4, 2008. Pages 1347–1363. https://doi.org/10.1137/06066850X Google ScholarDigital Library
- Albert Atserias and Maria Luisa Bonet. 2004. On the automatizability of resolution and related propositional proof systems. Information and Computation, 189, 2, 2004. Pages 182–201. https://doi.org/10.1016/j.ic.2003.10.004 Google ScholarDigital Library
- Albert Atserias and Víctor Dalmau. 2008. A combinatorial characterization of resolution width. J. Comput. System Sci., 74, 3, 2008. Pages 323–334. https://doi.org/10.1016/j.jcss.2007.06.025 Google ScholarDigital Library
- Albert Atserias and Tuomas Hakoniemi. 2019. Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs. In Proceedings of the 34th Computational Complexity Conference (CCC). 137, Pages 24:1–24:20. https://doi.org/10.4230/LIPIcs.CCC.2019.24 Google ScholarDigital Library
- Albert Atserias and Moritz Müller. 2020. Automating Resolution is NP-Hard. J. ACM, 67, 5, 2020. https://doi.org/10.1145/3409472 Google ScholarDigital Library
- Boaz Barak, Jonathan Kelner, and David Steurer. 2015. Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method. In Proceedings of the 47th Symposium on Theory of Computing (STOC). Pages 143–151. https://doi.org/10.1145/2746539.2746605 Google ScholarDigital Library
- Roberto J. Bayardo Jr. and Robert Schrag. 1997. Using CSP Look-Back Techniques to Solve Real-World SAT Instances. In Proceedings of the 14th National Conference on Artificial Intelligence (AAAI). Pages 203–208.Google Scholar
- Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. 1998. The Relative Complexity of NP Search Problems. J. Comput. System Sci., 57, 1, 1998. Pages 3–19. https://doi.org/10.1006/jcss.1998.1575 Google ScholarDigital Library
- Paul Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi, and Pavel Pudlák. 1994. Lower bounds on Hilbert's Nullstellensatz and propositional proofs. In Proceedings of the 35th Symposium on Foundations of Computer Science (FOCS). Pages 794–806. https://doi.org/10.1109/SFCS.1994.365714 Google ScholarDigital Library
- Paul Beame, Henry Kautz, and Ashish Sabharwal. 2004. Towards Understanding and Harnessing the Potential of Clause Learning. Journal of Artificial Intelligence Research, 22, 2004. Pages 319–351. https://doi.org/10.1613/jair.1410 Google ScholarCross Ref
- Zoë Bell. 2020. Automating Regular or Ordered Resolution is NP-Hard. https://eccc.weizmann.ac.il/report/2020/105/Google Scholar
- Eli Ben-Sasson and Avi Wigderson. 2001. Short Proofs Are Narrow–-Resolution Made Simple. J. ACM, 48, 2, 2001. Pages 149–169. issn:0004-5411 https://doi.org/10.1145/375827.375835 Google ScholarDigital Library
- Christoph Berkholz. 2018. The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs. In Proceedings of the 35th Symposium on Theoretical Aspects of Computer Science (STACS). 96, Pages 11:1–11:14. https://doi.org/10.4230/LIPIcs.STACS.2018.11 Google ScholarCross Ref
- Maria Luisa Bonet, Carlos Domingo, Ricard Gavald\`a, Alexis Maciel, and Toniann Pitassi. 2004. Non-Automatizability of Bounded-Depth Frege Proofs. Computational Complexity, 13, 1-2, 2004. Pages 47–68. https://doi.org/10.1007/s00037-004-0183-5 Google ScholarDigital Library
- Maria Luisa Bonet and Nicola Galesi. 2001. Optimality of size-width tradeoffs for resolution. Computational Complexity, 10, 4, 2001. Pages 261–276. https://doi.org/10.1007/s000370100000 Google ScholarDigital Library
- Maria Luisa Bonet, Toniann Pitassi, and Ran Raz. 1997. No Feasible Interpolation for TC^0-Frege Proofs. In Proceedings of the 38th Symposium on Foundations of Computer Science (FOCS). Pages 254–263. https://doi.org/10.1109/SFCS.1997.646114 Google ScholarCross Ref
- Mar\'ia Luisa Bonet, Toniann Pitassi, and Ran Raz. 2000. On Interpolation and Automatization for Frege Systems. SIAM J. Comput., 29, 6, 2000. Pages 1939–1967. https://doi.org/10.1137/S0097539798353230 Google ScholarDigital Library
- Joshua Buresh-Oppenheim, Matthew Clegg, Russell Impagliazzo, and Toniann Pitassi. 2002. Homogenization and the Polynomial Calculus. Computational Complexity, 11, 3-4, 2002. Pages 91–108. https://doi.org/10.1007/s00037-002-0171-6 Google ScholarDigital Library
- Sam Buss, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. 2001. Linear Gaps between Degrees for the Polynomial Calculus Modulo Distinct Primes. J. Comput. System Sci., 62, 2, 2001. Pages 267–289. https://doi.org/0.1006/jcss.2000.1726Google ScholarDigital Library
- Matthew Clegg, Jeff Edmonds, and Russell Impagliazzo. 1996. Using the Groebner Basis Algorithm to Find Proofs of Unsatisfiability. In Proceedings of the 28th Symposium on Theory of Computing (STOC). Pages 174–183. https://doi.org/10.1145/237814.237860 Google ScholarDigital Library
- Stefan Dantchev and Sø ren Riis. 2003. On Relativisation and Complexity Gap for Resolution-Based Proof Systems. In Computer Science Logic. Springer. Pages 142–154. https://doi.org/10.1007/978-3-540-45220-1_14 Google ScholarCross Ref
- Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, and Marc Vinyals. 2020. Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity. In Proceedings of the 61st Symposium on Foundations of Computer Science (FOCS). Pages 24–30. https://doi.org/10.1109/focs46700.2020.00011 Google ScholarCross Ref
- Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals. 2016. 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. https://doi.org/10.1109/FOCS.2016.40 Google ScholarCross Ref
- Noah Fleming, Pravesh Kothari, and Toniann Pitassi. 2019. Semialgebraic Proofs and Efficient Algorithm Design. Foundations and Trends in Theoretical Computer Science, 14, 1-2, 2019. Pages 1–221. https://doi.org/10.1561/0400000086 Google ScholarDigital Library
- Nicola Galesi and Massimo Lauria. 2010. On the Automatizability of Polynomial Calculus. Theory of Computing Systems, 47, 2, 2010. Pages 491–506. https://doi.org/10.1007/s00224-009-9195-5 Google ScholarDigital Library
- Nicola Galesi and Massimo Lauria. 2010. Optimality of Size-Degree Tradeoffs for Polynomial Calculus. ACM Transactions on Computational Logic, 12, 1, 2010. https://doi.org/10.1145/1838552.1838556 Google ScholarDigital Library
- Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. 2018. Monotone Circuit Lower Bounds from Resolution. In Proceedings of the 50th Symposium on Theory of Computing (STOC). Pages 902–911. https://doi.org/10.1145/3188745.3188838 Google ScholarDigital Library
- Michal Garlík. 2019. Resolution Lower Bounds for Refutation Statements. In Proceedings of the 44th Mathematical Foundations of Computer Science (MFCS). 138, Pages 37:1–37:13. https://doi.org/10.4230/LIPIcs.MFCS.2019.37 Google ScholarCross Ref
- Michal Garlík. 2020. Failure of Feasible Disjunction Property for k-DNF Resolution and NP-hardness of Automating It. https://eccc.weizmann.ac.il/report/2020/037/Google Scholar
- Konstantinos Georgiou and Avner Magen. 2008. Limitations of the Sherali-Adams lift and project system: Compromising local and global arguments. http://www.cs.utoronto.ca/pub/reports/csrg/587/CSRG-587.pdfGoogle Scholar
- Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. 2019. Adventures in Monotone Complexity and TFNP. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS). Pages 38:1–38:19. https://doi.org/10.4230/LIPIcs.ITCS.2019.38 Google ScholarCross Ref
- Mika Göös, Sajin Koroth, Ian Mertz, and Toniann Pitassi. 2020. Automating Cutting Planes is NP-Hard. In Proceedings of the 52nd Symposium on Theory of Computing (STOC). Pages 68–77. https://doi.org/10.1145/3357713.3384248 Google ScholarDigital Library
- Mika Göös and Toniann Pitassi. 2018. Communication Lower Bounds via Critical Block Sensitivity. SIAM J. Comput., 47, 5, 2018. Pages 1778–1806. https://doi.org/10.1137/16M1082007 Google ScholarDigital Library
- Samuel Hopkins, Pravesh Kothari, Aaron Potechin, Prasad Raghavendra, Tselil Schramm, and David Steurer. 2017. The Power of Sum-of-Squares for Detecting Hidden Structures. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS). Pages 720–731. https://doi.org/10.1109/FOCS.2017.72 Google ScholarCross Ref
- Trinh Huynh and Jakob Nordström. 2012. 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. isbn:978-1-4503-1245-5 https://doi.org/10.1145/2213977.2214000 Google ScholarDigital Library
- Russell Impagliazzo, Pavel Pudlák, and Ji\v r\'i Sgall. 1999. Lower Bounds for the Polynomial Calculus and the Gröbner Basis Algorithm. Computational Complexity, 8, 2, 1999. Pages 127–144. https://doi.org/10.1007/s000370050024 Google ScholarDigital Library
- Kazuo Iwama. 1997. Complexity of finding short resolution proofs. In Mathematical Foundations of Computer Science (MFCS). Pages 309–318. https://doi.org/10.1007/BFb0029974 Google ScholarCross Ref
- Emil Je\v rábek. 2007. On Independence of Variants of the Weak Pigeonhole Principle. Journal of Logic and Computation, 17, 3, 2007. Pages 587–604. https://doi.org/10.1093/logcom/exm017 Google ScholarCross Ref
- Stasys Jukna. 2012. Boolean Function Complexity: Advances and Frontiers. Algorithms and Combinatorics. 27, Springer.Google Scholar
- Pravesh Kothari, Jacob Steinhardt, and David Steurer. 2018. Robust moment estimation and improved clustering via sum of squares. In Proceedings of the 50th Symposium on Theory of Computing (STOC). Pages 1035–1046. https://doi.org/10.1145/3188745.3188970 Google ScholarDigital Library
- Jan Krajíček and Pavel Pudlák. 1998. Some Consequences of Cryptographical Conjectures for S^1_2 and EF. Information and Computation, 140, 1, 1998. Pages 82–94. https://doi.org/10.1006/inco.1997.2674 Google ScholarDigital Library
- Jean Lasserre. 2001. An Explicit Exact SDP Relaxation for Nonlinear 0–1 Programs. In Proceedings of the 8th International Conference on Integer Programming and Combinatorial Optimization (IPCO). Pages 293–303. https://doi.org/10.1007/3-540-45535-3_23 Google ScholarCross Ref
- Massimo Lauria and Jakob Nordström. 2017. Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gröbner Bases. In Proceedings of the 32nd Computational Complexity Conference (CCC). Pages 2:1–2:20. https://doi.org/10.4230/LIPIcs.CCC.2017.2 Google ScholarCross Ref
- Massimo Lauria and Jakob Nordström. 2017. Tight Size-Degree Bounds for Sums-of-Squares Proofs. Computational Complexity, 26, 4, 2017. Pages 911–948. https://doi.org/10.1007/s00037-017-0152-4 Google ScholarDigital Library
- Tengyu Ma, Jonathan Shi, and David Steurer. 2016. Polynomial-Time Tensor Decompositions with Sum-of-Squares. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS). Pages 438–446. https://doi.org/10.1109/FOCS.2016.54 Google ScholarCross Ref
- Jo\~ao P. Marques-Silva and Karem A. Sakallah. 1999. GRASP: A Search Algorithm for Propositional Satisfiability. IEEE Trans. Comput., 48, 5, May, 1999. Pages 506–521. https://doi.org/10.1109/12.769433 Google ScholarDigital Library
- Ian Mertz, Toniann Pitassi, and Yuanhao Wei. 2019. Short Proofs Are Hard to Find. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP). 132, Pages 84:1–84:16. https://doi.org/10.4230/LIPIcs.ICALP.2019.84 Google ScholarCross Ref
- Matthew W. Moskewicz, Conor F. Madigan, Ying Zhao, Lintao Zhang, and Sharad Malik. 2001. Chaff: Engineering an Efficient SAT Solver. In Proceedings of the 38th Design Automation Conference (DAC). Pages 530–535. https://doi.org/10.1145/378239.379017 Google ScholarDigital Library
- Ryan O'Donnell. 2017. SOS Is Not Obviously Automatizable, Even Approximately. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS). 67, Schloss Dagstuhl. Pages 59:1–59:10. isbn:978-3-95977-029-3 https://doi.org/10.4230/LIPIcs.ITCS.2017.59 Google ScholarCross Ref
- Ryan O'Donnell and Tselil Schramm. 2019. Sherali–Adams Strikes Back. In Proceedings of the 34th Computational Complexity Conference (CCC). Pages 8:1–8:30. https://doi.org/10.4230/LIPIcs.CCC.2019.8 Google ScholarDigital Library
- Ryan O'Donnell and Yuan Zhou. 2013. Approximability and proof complexity. In Proceedings of the 24th Symposium on Discrete Algorithms (SODA). Pages 1537–1556. https://doi.org/10.1137/1.9781611973105.111 Google ScholarCross Ref
- Christos Papadimitriou. 1994. On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. J. Comput. System Sci., 48, 3, 1994. Pages 498–532. https://doi.org/10.1016/S0022-0000(05)80063-7 Google ScholarDigital Library
- Pablo Parrilo. 2000. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization.Google Scholar
- Toniann Pitassi and Nathan Segerlind. 2012. Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász–Schrijver Procedures. SIAM J. Comput., 41, 1, 2012. Pages 128–159. https://doi.org/10.1137/100816833 Google ScholarDigital Library
- Aaron Potechin. 2020. Sum of Squares Bounds for the Ordering Principle. In Proceedings of the 35th Computational Complexity Conference (CCC). 169, Pages 38:1–38:37. issn:1868-8969 https://doi.org/10.4230/LIPIcs.CCC.2020.38 Google ScholarDigital Library
- Pavel Pudlák. 2000. Proofs as Games. The American Mathematical Monthly, 107, 6, 2000. Pages 541–550. issn:00029890, 19300972 https://doi.org/10.2307/2589349 Google ScholarCross Ref
- Pavel Pudlák. 2003. On reducibility and symmetry of disjoint NP pairs. Theoretical Computer Science, 295, 2003. Pages 323–339. https://doi.org/10.1016/S0304-3975(02)00411-5 Google ScholarCross Ref
- Pavel Pudlák and Neil Thapen. 2019. Random resolution refutations. Computational Complexity, 28, 2, 2019. Pages 185–239. https://doi.org/10.1007/s00037-019-00182-7 Google ScholarDigital Library
- Prasad Raghavendra and Benjamin Weitz. 2017. On the Bit Complexity of Sum-of-Squares Proofs. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP). Pages 80:1–80:13. https://doi.org/10.4230/LIPIcs.ICALP.2017.80 Google ScholarCross Ref
- Alexander Razborov. 1998. Lower bounds for the polynomial calculus. Computational Complexity, 7, 1998. Pages 291–324. https://doi.org/10.1007/s000370050013 Google ScholarDigital Library
- Hanif Sherali and Warren Adams. 1994. A hierarchy of relaxations and convex hull characterizations for mixed-integer zero–one programming problems. Discrete Applied Mathematics, 52, 1, 1994. Pages 83–106. https://doi.org/10.1016/0166-218X(92)00190-W Google ScholarDigital Library
- Naum Shor. 1987. An Approach to Obtaining Global Extremums in Polynomial Mathematical Programming Problems. Cybernetics, 23, 5, 1987. Pages 695–700. https://doi.org/10.1007/BF01074929 Google ScholarCross Ref
Index Terms
- Automating algebraic proof systems is NP-hard
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