Exponential resolution lower bounds for weak pigeonhole principle and perfect matching formulas over sparse graphs

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

[1]  Michael Alekhnovich,et al.  Pseudorandom Generators in Propositional Proof Complexity , 2004, SIAM J. Comput..

[2]  Alexander A. Razborov,et al.  Clique is hard on average for regular resolution , 2018, STOC.

[3]  Alasdair Urquhart.  Resolution Proofs of Matching Principles , 2004, Annals of Mathematics and Artificial Intelligence.

[4]  Russell Impagliazzo,et al.  The Resolution Complexity of Independent Sets and Vertex Covers in Random Graphs , 2007, computational complexity.

[5]  Ran Raz,et al.  Regular resolution lower bounds for the weak pigeonhole principle , 2001, STOC '01.

[6]  Dmitry Itsykson,et al.  Tight Lower Bounds on the Resolution Complexity of Perfect Matching Principles , 2016, Fundam. Informaticae.

[7]  Alexander A. Razborov.  Resolution lower bounds for perfect matching principles , 2004, J. Comput. Syst. Sci..

[8]  Søren Riis,et al.  "Planar" tautologies hard for resolution , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[9]  Søren Riis,et al.  Tree resolution proofs of the weak pigeon-hole principle , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[10]  Shuo Pang,et al.  Large Clique is Hard on Average for Resolution , 2021, Electron. Colloquium Comput. Complex..

[11]  Michael Alekhnovich,et al.  Lower bounds for polynomial calculus: non-binomial case , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

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

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

[14]  Alexander A. Razborov,et al.  Read-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus , 1997, STOC '97.

[15]  Samuel R. Buss,et al.  Resolution Proofs of Generalized Pigeonhole Principles , 1988, Theor. Comput. Sci..

[16]  Olaf Beyersdorff,et al.  A Lower Bound for the Pigeonhole Principle in Tree-like Resolution by Asymmetric Prover-Delayer Games , 2010, Electron. Colloquium Comput. Complex..

[17]  Michael Alekhnovich,et al.  Mutilated chessboard problem is exponentially hard for resolution , 2004, Theor. Comput. Sci..

[18]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

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

[20]  Stefan S. Dantchev.  Resolution width-size trade-offs for the Pigeon-Hole Principle , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[21]  Alasdair Urquhart.  Width versus size in resolution proofs , 2007, Theor. Comput. Sci..

[22]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[23]  Toniann Pitassi,et al.  Simplified and improved resolution lower bounds , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[24]  Archie Blake.  Canonical expressions in Boolean algebra , 1938 .

[25]  Enkatesan G Uruswami.  Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes , 2008 .

[26]  Alexander A. Razborov,et al.  Lower bounds for the polynomial calculus , 1998, computational complexity.

[27]  Samuel R. Buss,et al.  Resolution and the Weak Pigeonhole Principle , 1997, CSL.

[28]  Alexander A. Razborov,et al.  Pseudorandom generators hard for $k$-DNF resolution and polynomial calculus resolution , 2015 .

[29]  Alexander A. Razborov,et al.  Proof Complexity of Pigeonhole Principles , 2001, Developments in Language Theory.

[30]  Alexander A. Razborov,et al.  Improved Resolution Lower Bounds for the Weak Pigeonhole Principle , 2001, Electron. Colloquium Comput. Complex..

[31]  Alexander A. Razborov,et al.  Electronic Colloquium on Computational Complexity, Report No. 75 (2001) Resolution Lower Bounds for the Weak Functional Pigeonhole Principle , 2001 .

[32]  Michael Alekhnovich,et al.  Space complexity in propositional calculus , 2000, STOC '00.