Sherali-Adams and the binary encoding of combinatorial principles

We consider the Sherali-Adams (SA) refutation system together with the unusual binary encoding of certain combinatorial principles. For the unary encoding of the Pigeonhole Principle and the Least Number Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the Pigeonhole Principle requires exponentially-sized SA refutations, whereas the binary encoding of the Least Number Principle admits logarithmic rank, polynomially-sized SA refutations. We continue by considering a refutation system between SA and Lasserre (Sum-of-Squares). In this system, the Least Number Principle requires linear rank while the Pigeonhole Principle becomes constant rank.

[1]  Monique Laurent,et al.  A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming , 2003, Math. Oper. Res..

[2]  Søren Riis.  A complexity gap for tree resolution , 2001, computational complexity.

[3]  Aaron Potechin.  Sum of squares bounds for the total ordering principle , 2018, ArXiv.

[4]  Albert Atserias,et al.  Narrow Proofs May Be Maximally Long , 2014, Computational Complexity Conference.

[5]  Vojtech Rödl,et al.  The complexity of proving that a graph is Ramsey , 2017 .

[6]  David Steurer,et al.  Sum-of-squares proofs and the quest toward optimal algorithms , 2014, Electron. Colloquium Comput. Complex..

[7]  Nicola Galesi,et al.  A Framework for Space Complexity in Algebraic Proof Systems , 2015, J. ACM.

[8]  Barnaby Martin,et al.  Resolution and the binary encoding of combinatorial principles , 2018, Electron. Colloquium Comput. Complex..

[9]  Yuval Filmus,et al.  Space Complexity in Polynomial Calculus , 2015, SIAM J. Comput..

[10]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[11]  Massimo Lauria,et al.  Tight Size-Degree Bounds for Sums-of-Squares Proofs , 2017, computational complexity.

[12]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..

[13]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[14]  Barnaby Martin,et al.  Tight rank lower bounds for the Sherali-Adams proof system , 2009, Theor. Comput. Sci..

[15]  Jean B. Lasserre,et al.  An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs , 2001, IPCO.

[16]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[17]  Stefan S. Dantchev.  Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems , 2007, STOC '07.

[18]  Mark Nicholas Charles Rhodes.  Rank Lower Bounds for the Sherali-Adams Operator , 2007, CiE.

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

[20]  Ralph E. Gomory,et al.  Solving linear programming problems in integers , 1960 .

[21]  Dima Grigoriev,et al.  Complexity of Semi-algebraic Proofs , 2002, STACS.

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

[23]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[24]  Neil Thapen,et al.  Total Space in Resolution , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.