Lifting with Sunflowers

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Whereas previous proofs used sophisticated Fourier analytic techniques, our proof uses elementary counting together with a novel connection to the sunflower lemma. In addition to a simplified proof, our approach also gives quantitative improvements in terms of gadget size. Focusing on one of the most widely used gadgets—the index gadget—existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to poly logarithmic, similar to the known bounds for the corresponding robust sunflower lemmas.

[1]  Toniann Pitassi,et al.  Automating cutting planes is NP-hard , 2020, Electron. Colloquium Comput. Complex..

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

[3]  Prasad Raghavendra,et al.  Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of CSPs , 2016, STOC.

[4]  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.

[5]  Shachar Lovett,et al.  Rectangles Are Nonnegative Juntas , 2015, SIAM J. Comput..

[6]  Prasad Raghavendra,et al.  Approximate Constraint Satisfaction Requires Large LP Relaxations , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[7]  Toniann Pitassi,et al.  Strongly exponential lower bounds for monotone computation , 2017, Electron. Colloquium Comput. Complex..

[8]  Xiaodi Wu,et al.  Raz-McKenzie simulation with the inner product gadget , 2017, Electron. Colloquium Comput. Complex..

[9]  Arkadev Chattopadhyay,et al.  Simulation Theorems via Pseudo-random Properties , 2017, computational complexity.

[10]  Shachar Lovett,et al.  Improved bounds for the sunflower lemma , 2019, Electron. Colloquium Comput. Complex..

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

[12]  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).

[13]  A. Razborov.  Communication Complexity , 2011 .

[14]  Rahul Jain,et al.  Extension Complexity of Independent Set Polytopes , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

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

[16]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[17]  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).

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

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

[20]  Mika Göös,et al.  Lower Bounds for Clique vs. Independent Set , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[21]  Toniann Pitassi,et al.  Exponential Lower Bounds for Monotone Span Programs , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[22]  Ran Raz,et al.  Higher lower bounds on monotone size , 2000, STOC '00.

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

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

[25]  Alexander A. Sherstov.  Communication Lower Bounds Using Directional Derivatives , 2014, JACM.

[26]  Anup Rao.  Coding for Sunflowers , 2019, ArXiv.

[27]  Alexander A. Sherstov.  The Pattern Matrix Method , 2009, SIAM J. Comput..

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

[29]  Xin Li,et al.  Sunflowers and Quasi-sunflowers from Randomness Extractors , 2018, Electron. Colloquium Comput. Complex..

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

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

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

[33]  Toniann Pitassi,et al.  Lifting Nullstellensatz to monotone span programs over any field , 2018, Electron. Colloquium Comput. Complex..

[34]  Prasad Raghavendra,et al.  Lower Bounds on the Size of Semidefinite Programming Relaxations , 2014, STOC.

[35]  H. L. Abbott,et al.  Intersection Theorems for Systems of Sets , 1972, J. Comb. Theory, Ser. A.