Automating cutting planes is NP-hard

We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula F, It is -hard to find a CP refutation of F in time polynomial in the length of the shortest such refutation; and unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a tree-like CP refutation of F in time polynomial in the length of the shortest such refutation. The first result extends the recent breakthrough of Atserias and M'uller (FOCS 2019) that established an analogous result for Resolution. Our proofs rely on two new lifting theorems: (1) Dag-like lifting for gadgets with many output bits. (2) Tree-like lifting that simulates an r-round protocol with gadgets of query complexity O(r).

[1]  Jakob Nordström,et al.  On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity , 2012, STOC '12.

[2]  Stephen A. Cook,et al.  An Exponential Lower Bound for the Size of Monotone Real Circuits , 1999, J. Comput. Syst. Sci..

[3]  Michael Alekhnovich,et al.  Resolution Is Not Automatizable Unless W[P] Is Tractable , 2008, SIAM J. Comput..

[4]  Moni Naor,et al.  Search problems in the decision tree model , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[5]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

[6]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, STOC '95.

[7]  Or Meir,et al.  Query-to-Communication Lifting Using Low-Discrepancy Gadgets , 2019, Electron. Colloquium Comput. Complex..

[8]  Pritish Kamath,et al.  Adventures in Monotone Complexity and TFNP , 2018, Electron. Colloquium Comput. Complex..

[9]  Toniann Pitassi,et al.  Random CNFs are Hard for Cutting Planes , 2017, Electron. Colloquium Comput. Complex..

[10]  Jan Krajícek.  Interpolation by a Game , 1998, Math. Log. Q..

[11]  Jan Kraíček,et al.  Interpolation by a Game , 1998 .

[12]  Pavel Pudlák.  On extracting computations from propositional proofs (a survey) , 2010, FSTTCS.

[13]  Ran Raz,et al.  No feasible interpolation for TC/sup 0/-Frege proofs , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[14]  Eli Ben-Sasson,et al.  Short proofs are narrow—resolution made simple , 2001, JACM.

[15]  William J. Cook,et al.  Chvátal closures for mixed integer programming problems , 1990, Math. Program..

[16]  Alasdair Urquhart,et al.  Formal Languages]: Mathematical Logic--mechanical theorem proving , 2022 .

[17]  Pavel Pudlák.  Proofs as Games , 2000, Am. Math. Mon..

[18]  Albert Atserias,et al.  Automating Resolution is NP-Hard , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[19]  Ran Raz,et al.  Separation of the Monotone NC Hierarchy , 1999, Comb..

[20]  Michal Garlík.  Resolution Lower Bounds for Refutation Statements , 2019, Electron. Colloquium Comput. Complex..

[21]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[22]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[23]  Toniann Pitassi,et al.  Communication lower bounds via critical block sensitivity , 2013, STOC.

[24]  Massimo Lauria,et al.  On the Automatizability of Polynomial Calculus , 2010, Theory of Computing Systems.

[25]  Ryan O'Donnell,et al.  SOS Is Not Obviously Automatizable, Even Approximately , 2016, ITCS.

[26]  Toniann Pitassi,et al.  Non-Automatizability of Bounded-Depth Frege Proofs , 2004, computational complexity.

[27]  Czech Republickrajicek.  Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic , 2007 .

[28]  Dmitry Sokolov.  Dag-Like Communication and Its Applications , 2016, CSR.

[29]  Toniann Pitassi,et al.  Short Proofs Are Hard to Find , 2019, ICALP.

[30]  Ankit Garg,et al.  Monotone circuit lower bounds from resolution , 2018, Electron. Colloquium Comput. Complex..

[31]  Michael Alekhnovich,et al.  Minimum propositional proof length is NP-hard to linearly approximate , 1998, Journal of Symbolic Logic.

[32]  Pavel Pudlák,et al.  Random Formulas, Monotone Circuits, and Interpolation , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[33]  Or Meir,et al.  Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[34]  Bingkai Lin,et al.  A Simple Gap-producing Reduction for the Parameterized Set Cover Problem , 2019, ICALP.

[35]  Toniann Pitassi,et al.  Deterministic Communication vs. Partition Number , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[36]  Marc Vinyals,et al.  How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity) , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[37]  Russell Impagliazzo,et al.  Using the Groebner basis algorithm to find proofs of unsatisfiability , 1996, STOC '96.

[38]  Toniann Pitassi,et al.  Query-to-Communication Lifting for BPP , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[39]  William J. Cook,et al.  On the complexity of cutting-plane proofs , 1987, Discret. Appl. Math..

[40]  Jan Krajícek,et al.  Some Consequences of Cryptographical Conjectures for S12 and EF , 1998, Inf. Comput..

[41]  Kazuo Iwama.  Complexity of Finding Short Resolution Proofs , 1997, MFCS.

[42]  Pravesh Kothari,et al.  Semialgebraic Proofs and Efficient Algorithm Design , 2019, Electron. Colloquium Comput. Complex..

[43]  A. Razborov.  Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic , 1995 .

[44]  Prasad Raghavendra,et al.  On the Bit Complexity of Sum-of-Squares Proofs , 2017, ICALP.

[45]  Ran Raz,et al.  On Interpolation and Automatization for Frege Systems , 2000, SIAM J. Comput..

[46]  Pavel Pudlák,et al.  A note on monotone real circuits , 2018, Inf. Process. Lett..

[47]  HierarchyRan Raz,et al.  Separation of the Monotone NC , 1999 .

[48]  Carlos Gershenson,et al.  Information and Computation , 2013, Handbook of Human Computation.

[49]  Maria Luisa Bonet,et al.  On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems , 2000, SIAM J. Comput..

[50]  Albert Atserias,et al.  A combinatorial characterization of resolution width , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..