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Lower bounds for cutting planes proofs with small coefficients

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

doi:10.1145/225058.225275
dblp:conf/stoc/BonetPR95
fatcat:zqdlacylzvcxpiq4yw7w77j4r4
*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*. ...##
###
The Space Complexity of Cutting Planes Refutations

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. ...

##
###
Semantic Versus Syntactic Cutting Planes

2016
*
Symposium on Theoretical Aspects of Computer Science
*

First, we show that the

doi:10.4230/lipics.stacs.2016.35
dblp:conf/stacs/FilmusHL16
fatcat:pp6k6laesfapjnpbqzx6g3p2oe
*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. ...##
###
Stabbing Planes
[article]

2022
*
arXiv
*
pre-print

Finally, we prove linear

arXiv:1710.03219v2
fatcat:4dedtl7iordyvcmquixcpwmd6y
*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. ...##
###
On the Power and Limitations of Branch and Cut
[article]

2021
*
arXiv
*
pre-print

First, we show that any Stabbing

arXiv:2102.05019v2
fatcat:2kls6apafjbzxlsw2pfpkyissm
*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*. ...##
###
A Rank Lower Bound for Cutting Planes Proofs of Ramsey's Theorem

2016
*
ACM Transactions on Computation Theory
*

In particular we focus on the propositional

doi:10.1145/2903266
fatcat:aikdisyslzflboe65jqgi6p4l4
*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] ). ...##
###
A finitely converging cutting plane technique

1985
*
Operations Research Letters
*

The primary 'theoretical' contribution is the simplicity of the

doi:10.1016/0167-6377(85)90041-0
fatcat:boo5hptlbzdkxdhcxwq3qc5ola
*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. ...##
###
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]

2020
*
arXiv
*
pre-print

We apply our generalized theorem to solve two open problems: * We present the first result that demonstrates a separation in

arXiv:2001.02144v1
fatcat:urgi4cd5vvczdnsqcnqloxsj6q
*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 ...##
###
Cuts from proofs: a complete and practical technique for solving linear inequalities over integers

2011
*
Formal methods in system design
*

Our main insight is that by focusing on the defining constraints of a vertex, we can compute a

doi:10.1007/s10703-011-0127-z
fatcat:4y5meofrvfe67l7vaatv4oa7yy
*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. ...##
###
Cuts from Proofs: A Complete and Practical Technique for Solving Linear Inequalities over Integers
[chapter]

2009
*
Lecture Notes in Computer Science
*

Our main insight is that by focusing on the defining constraints of a vertex, we can compute a

doi:10.1007/978-3-642-02658-4_20
fatcat:yn6f3645ijbihpdqqahps5u6ie
*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. ...##
###
A Rank Lower Bound for Cutting Planes Proofs of Ramsey's Theorem
[chapter]

2013
*
Lecture Notes in Computer Science
*

In particular we focus on the propositional

doi:10.1007/978-3-642-39071-5_26
fatcat:tb624rsot5dtbi56vwgstndnku
*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 . ...##
###
Optimized Risk Scores

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

doi:10.1145/3097983.3098161
dblp:conf/kdd/UstunR17
fatcat:vcfyutt3jfa2lfbagzdeakhsbu
*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. ...##
###
Depth lower bounds in Stabbing Planes for combinatorial principles
[article]

2021
*
arXiv
*
pre-print

The techniques known so far to prove size and depth

arXiv:2102.07622v2
fatcat:gjrnams6ynhf3kjj7ih67uegam
*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. ...##
###
Bounds on the size of branch-and-bound proofs for integer knapsacks

2008
*
Operations Research Letters
*

Using a direct counting argument, we derive

doi:10.1016/j.orl.2007.04.011
fatcat:dzcckx3lxfetvoxbpbqf3hm6ey
*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). ...
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