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Lower bounds for cutting planes proofs with small coefficients

Maria Bonet, Toniann Pitassi, Ran Raz
1995 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing - STOC '95  
Acknowledgements The authors would like to thank Russell Impagliazzo, Jan Krajicek and Sam Buss for very helpful conversations.  ...  in small-weights Cutting Planes.  ...  In Section 4, we will prove lower bounds for both of the above generalizations of Cutting Planes.  ... 
doi:10.1145/225058.225275 dblp:conf/stoc/BonetPR95 fatcat:zqdlacylzvcxpiq4yw7w77j4r4

The Space Complexity of Cutting Planes Refutations

Nicola Galesi, Pavel Pudlák, Neil Thapen, Marc Herbstritt
2015 Computational Complexity Conference  
cutting planes proofs of the pigeonhole principle with coefficients bounded by two.  ...  This is in contrast to the weaker resolution proof system, for which the analogous space measure has been well-studied and many optimal lower bounds are known.  ...  Results for cutting planes with small coefficients The constant space refutations in Theorem 5 use coefficients as big as 2 n , and these seem to be necessary for our proof technique to work.  ... 
doi:10.4230/lipics.ccc.2015.433 dblp:conf/coco/GalesiPT15 fatcat:n736vrqq25d5pkys57umbmhpj4

Semantic Versus Syntactic Cutting Planes

Yuval Filmus, Pavel Hrubeš, Massimo Lauria, Marc Herbstritt
2016 Symposium on Theoretical Aspects of Computer Science  
First, we show that the lower bound technique of [22] applies also to semantic cutting planes: the proof system has feasible interpolation via monotone real circuits, which gives an exponential lower bound  ...  In particular, we give a formula for which any refutation in syntactic cutting planes requires exponential length, while there is a polynomial length refutation in semantic cutting planes.  ...  A lower bound for cutting planes with small coefficients was obtained by [4] and [18] , and [14] gave a lower bound for the tree-like version of the system.  ... 
doi:10.4230/lipics.stacs.2016.35 dblp:conf/stacs/FilmusHL16 fatcat:pp6k6laesfapjnpbqzx6g3p2oe

Stabbing Planes [article]

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere
2022 arXiv   pre-print
Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot  ...  We develop a new semi-algebraic proof system called Stabbing Planes which formalizes modern branch-and-cut algorithms for integer programming and is in the style of DPLL-based modern SAT solvers.  ...  This allowed them to lift the exponential lower bounds on Cutting Planes proofs [36, 39, 48, 67] to SP * , and even to SP proofs with coefficients of size exp(n δ ) for some constant δ < 1.  ... 
arXiv:1710.03219v2 fatcat:4dedtl7iordyvcmquixcpwmd6y

On the Power and Limitations of Branch and Cut [article]

Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, Avi Wigderson
2021 arXiv   pre-print
First, we show that any Stabbing Planes proof with bounded coefficients SP* can be translated into Cutting Planes.  ...  As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*.  ...  Our second lower bound is a new linear depth lower bound for semantic Cutting Planes proofs.  ... 
arXiv:2102.05019v2 fatcat:2kls6apafjbzxlsw2pfpkyissm

A Rank Lower Bound for Cutting Planes Proofs of Ramsey's Theorem

Massimo Lauria
2016 ACM Transactions on Computation Theory  
In particular we focus on the propositional proof system cutting planes; we prove that the upper bound "r(k, k) ≤ 4 k " requires cutting planes proof of high rank.  ...  We study the complexity of proving upper bounds for the number r(k, k).  ...  Lower bounds for more natural formulas exist for cutting plane proofs of restricted forms (e.g. when the numeric coefficients are small [6] or the proof is tree-like [19] ).  ... 
doi:10.1145/2903266 fatcat:aikdisyslzflboe65jqgi6p4l4

A finitely converging cutting plane technique

James B Orlin
1985 Operations Research Letters  
The primary 'theoretical' contribution is the simplicity of the proof of convergence. programming * integer * cutting planes  ...  We consider a new finitely convergent cutting plane algorithm for mixed integer linear programs in which the optimal objective value is assumed to be integral.  ...  Acknowledgement I wish to thank David Bell for his helpful comments on an earlier version of this note.  ... 
doi:10.1016/0167-6377(85)90041-0 fatcat:boo5hptlbzdkxdhcxwq3qc5ola

Page 7539 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
Summary: “We consider small-weight cutting planes (CP" ) proofs; that is, cutting planes (CP) proofs with coefficients up to Poly(n).  ...  proofs with small coefficients.  ... 

Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity [article]

Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, Marc Vinyals
2020 arXiv   pre-print
We apply our generalized theorem to solve two open problems: * We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients  ...  Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential  ...  Part of this work was carried out while several of the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational  ... 
arXiv:2001.02144v1 fatcat:urgi4cd5vvczdnsqcnqloxsj6q

Cuts from proofs: a complete and practical technique for solving linear inequalities over integers

Isil Dillig, Thomas Dillig, Alex Aiken
2011 Formal methods in system design  
Our main insight is that by focusing on the defining constraints of a vertex, we can compute a proof of unsatisfiability for the intersection of the defining constraints and use this proof to systematically  ...  exclude subspaces of the feasible region with no integer points.  ...  We would also like to thank David Dill for his useful suggestions and Suhabe Bugrara for his comments on a draft of this paper.  ... 
doi:10.1007/s10703-011-0127-z fatcat:4y5meofrvfe67l7vaatv4oa7yy

Cuts from Proofs: A Complete and Practical Technique for Solving Linear Inequalities over Integers [chapter]

Isil Dillig, Thomas Dillig, Alex Aiken
2009 Lecture Notes in Computer Science  
Our main insight is that by focusing on the defining constraints of a vertex, we can compute a proof of unsatisfiability for the intersection of the defining constraints and use this proof to systematically  ...  exclude subspaces of the feasible region with no integer points.  ...  We would also like to thank David Dill for his useful suggestions and Suhabe Bugrara for his comments on a draft of this paper.  ... 
doi:10.1007/978-3-642-02658-4_20 fatcat:yn6f3645ijbihpdqqahps5u6ie

A Rank Lower Bound for Cutting Planes Proofs of Ramsey's Theorem [chapter]

Massimo Lauria
2013 Lecture Notes in Computer Science  
In particular we focus on the propositional proof system cutting planes; we prove that the upper bound "r(k, k) ≤ 4 k " requires cutting planes proof of high rank.  ...  We study the complexity of proving upper bounds for the number r(k, k).  ...  We are now ready to prove the lower bound on rank of cutting planes proof of the Ramsey number upper bound. Theorem 7. For all even k ≥ 4, cutting planes rank of formula Ram k is at least 2 k/2−1 .  ... 
doi:10.1007/978-3-642-39071-5_26 fatcat:tb624rsot5dtbi56vwgstndnku

Optimized Risk Scores

Berk Ustun, Cynthia Rudin
2017 Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD '17  
We formulate the risk score problem as a mixed integer nonlinear program, and present a cutting plane algorithm to recover its optimal solution.  ...  These models are widely used in medicine and criminal justice, but are difficult to learn from data because they need to be calibrated, sparse, use small integer coefficients, and obey application-specific  ...  We would like to thank Paul Rubin for helpful discussions. We would also like to thank our collaborators Aaron Struck and Brandon Westover for their guidance on the seizure prediction application.  ... 
doi:10.1145/3097983.3098161 dblp:conf/kdd/UstunR17 fatcat:vcfyutt3jfa2lfbagzdeakhsbu

Depth lower bounds in Stabbing Planes for combinatorial principles [article]

Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martin
2021 arXiv   pre-print
The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments  ...  By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.  ...  In this paper we further develop the study of proof depth for SP. Rank lower bound techniques for Cutting Planes are essentially of two types.  ... 
arXiv:2102.07622v2 fatcat:gjrnams6ynhf3kjj7ih67uegam

Bounds on the size of branch-and-bound proofs for integer knapsacks

Bala Krishnamoorthy
2008 Operations Research Letters  
Using a direct counting argument, we derive lower and upper bounds for the number of nodes enumerated by linear programming-based branch-and-bound (B&B) method to prove the infeasibility of an integer  ...  Acknowledgments The author would like to thank Gábor Pataki from the University of North Carolina at Chapel Hill for having constructive discussions and providing helpful suggestions on the material presented  ...  Most commercial solvers use cutting planes in conjunction with B&B (termed branch-and-cut methods).  ... 
doi:10.1016/j.orl.2007.04.011 fatcat:dzcckx3lxfetvoxbpbqf3hm6ey
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