On the complexity of cutting-plane proofs

Cook, William J.; Coullard, Collette R.; Turán, György

doi:10.1016/0166-218x(87)90039-4

Cited by 236 publications

(149 citation statements)

References 10 publications

Supporting

Mentioning

142

Contrasting

Order By: Relevance

“…The "theoretical" approach, i.e. the one found in [12] for instance, translates a cardinality constraint n i=1 l i ≤ k using n k+1 negative clauses of size k + 1. Such encoding is called binomial because of the number of generated clauses.…”

Section: Introductionmentioning

confidence: 99%

Detecting Cardinality Constraints in CNF

Biere

Berre²,

Lonca³

et al. 2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We present novel approaches to detect cardinality constraints expressed in CNF. The first approach is based on a syntactic analysis of specific data structures used in SAT solvers to represent binary and ternary clauses, whereas the second approach is based on a semantic analysis by unit propagation. The syntactic approach computes an approximation of the cardinality constraints AtMost-1 and AtMost-2 constraints very fast, whereas the semantic approach has the property to be generic, i.e. it can detect cardinality constraints AtMost-k for any k, at a higher computation cost. Our experimental results suggest that both approaches are efficient at recovering AtMost-1 and AtMost-2 cardinality constraints.

show abstract

Section: Introductionmentioning

confidence: 99%

Detecting Cardinality Constraints in CNF

Biere

Berre²,

Lonca³

et al. 2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…[9,4,7]. This system operates with linear inequalities with integer coefficients by rules of addition and rounding.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies

Kojevnikov

Itsykson

2006

Automata, Languages and Programming

View full text Add to dashboard Cite

We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS + proof into Positivstellensatz proof of Grigoriev et al., extracting a "good" expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.

show abstract

“…In other words, (1) can potentially be a strict inclusion such that M(P ∩ {x ∈ R n | cx ≥ d + 1}) = ∅ while M(P ) ∩ {x ∈ R n | cx ≥ d + 1} = ∅. This is equivalent to saying that we can verify the validity of cx ≤ d, however we are not able to compute cx ≤ d. To the best of our knowledge, the only paper discussing a related idea is [4], but theoretical and computational potential of this approach has not been further investigated.…”

Section: Introductionmentioning

confidence: 89%

Design and Verify: A New Scheme for Generating Cutting-Planes

Dey

Pokutta

2011

Integer Programming and Combinatoral Optimization

View full text Add to dashboard Cite

A cutting-plane procedure for integer programming (IP) problems usually involves invoking a blackbox procedure (such as the Gomory-Chvátal (GC) procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality cx ≤ d where c and d are integral, independent of the cutting-plane black-box. In the second step, we verify that the designed inequality is a valid inequality by verifying that the set P ∩ {x ∈ R n | cx ≥ d + 1} ∩ Z n is empty using cutting-planes from the black-box. Here P is the feasible region of the linear-programming relaxation of the IP. We refer to the closure of all cutting-planes that can be verified to be valid using a specific cutting-plane black-box as the verification closure of the considered cutting-plane black-box. This paper undertakes a systematic study of properties of verification closures of various cutting-plane black-box procedures.

show abstract

On the complexity of cutting-plane proofs

Cited by 236 publications

References 10 publications

Detecting Cardinality Constraints in CNF

Detecting Cardinality Constraints in CNF

Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies

Design and Verify: A New Scheme for Generating Cutting-Planes

Contact Info

Product

Resources

About