W. Cook, C. R. Coullard, and Gy. Tur' an, On the complexity of cutting plane proofs, Discrete Appl. Math., (1987), 25--38.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....namely polyhedral combinatorics (see, e.g. 28, 39, 41] On the other hand, Balas et al. 2] successfully incorporated Gomory s mixed integer cuts within a Branch and Cut framework. Third, cutting planes are of interest to mathematical logic and complexity theory. Cook, Coullard, and Turn [15] were the first to consider cutting plane proofs as a propositional proof system. In particular, they pointed out that the cutting plane proof system is a strengthening of resolution proofs. Since the work of Haken [30] exponential lower bounds are known for the latter. Results of Chvtal, Cook, ....

W. Cook, C. R. Coullard, and Gy. Turn. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

.... form the basis of popular algorithms such as Branch and Bound, or Gomory s Cutting Planes algorithm (see [Sch86] From the combinatorial optimization perspective, they are a powerful tool for discovering the structure of integer hulls (see [Sch86] again, and [CCH89] Cook, Coullard, and Tur an [CCT87] introduced a new perspective. They saw Chv atal Gomory cuts as a rule for inferring valid inequalities, and de ned a proof system for proving the unsatis ability of formulae in propositional logic. The resulting proof system is called cutting planes (CP) It is well known that sets of Boolean ....

....inequalities by deriving 1 0. Next we give a method to translate clauses into inequalities. A clause x j 1 x j k :x l 1 :x l m is translated into the inequality x j 1 x j k (1 x l 1 ) 1 x l m ) 1: It was proved by Cook, Coullard, and Tur an [CCT87] that CP polynomially simulates Resolution when clauses are presented as linear inequalities according to the previous translation. Therefore, the system is complete. Next we de ne a few complexity measures of the proof system. De nition 2 The length of a CP proof is the number of ....

W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18:25-38, 1987.

.... form the basis of popular algorithms such as Branch and Bound, or Gomory s Cutting Planes algorithm (see [Sch86] From the combinatorial optimization perspective, they are a powerful tool for discovering the structure of integer hulls (see [Sch86] again, and [CCH89] Cook, Coullard, and Turan [CCT87] introduced a new perspective. They saw Chvatal Gomory cuts as a rule for inferring valid inequalities, and defined a proof system for proving the unsatisfiability of formulae in propositional logic. The resulting proof system is called cutting planes (CP) It is well known that sets of Boolean ....

....clauses into inequalities. A clause x j 1 Delta Delta Delta x j k :x l 1 Delta Delta Delta :x l m is translated into the inequality x j 1 Delta Delta Delta x j k (1 Gamma x l 1 ) Delta Delta Delta (1 Gamma x l m ) 1: It was proved by Cook, Coullard, and Turan [CCT87] that CP polynomially simulates Resolution when clauses are presented as linear inequalities according to the previous translation. Therefore, the system is complete. Next we define a few complexity measures of the proof system. Definition 2 The length of a CP proof is the number of ....

W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

.... languagues are by no means worse (note that there is a polynomialtime reduction between any two co NP complete languages) For example, recently there was an increased interest in proof systems for systems of polynomial equations [BIK 96, CEI96] linear inequalities [Gom63, Chv73, CCT87, CCH89] and polynomial inequalities [Lov94, LS91, Pud99, Das01, Das02, GHP02] It is more natural to regard these systems as refutation systems , because the theorems here are exactly the systems of (in)equalities that have no appropriate (e.g. 0 1 or integer) solutions. The most part of ....

W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18(1):25-38, 1987.

....for general integer programming created by Gomory [19] The algorithm was rarely used in practice because it converged slowly, but it was recognized by Chvatal [6] that the method could function as a proof system. There are many studies examining the complexity and strength of the CP proof system [5, 9, 18, 33], and it was shown early on by Cook that the CP system is properly stronger than resolution in that existence of a polynomial length resolution proof implies the existence of a polynomial length CP proof, but the reverse need not hold [9] Constraints in CP are expressed as linear ....

.... complexity and strength of the CP proof system [5, 9, 18, 33] and it was shown early on by Cook that the CP system is properly stronger than resolution in that existence of a polynomial length resolution proof implies the existence of a polynomial length CP proof, but the reverse need not hold [9]. Constraints in CP are expressed as linear inequalities where x 1 , x 2 , x n are non negative integer variables and a 1 , a 2 , a n and k are integers. The system has two rules of inference: i) derive a new inequality by taking a linear combination of a set of ....

[Article contains additional citation context not shown here]

W. Cook, C. Coullard, and G. Turan. On the complexity of cutting plane proofs. Journal of Discrete Applied Math, 18:25--38, 1987.

....for general integer programming created by Gomory [19] The algorithm was rarely used in practice because it converged slowly, but it was recognized by Chv atal [6] that the method could function as a proof system. There are many studies examining the complexity and strength of the CP proof system [5, 9, 18, 33], and it was shown early on by Cook that the CP system is properly stronger than resolution in that existence of a polynomial length resolution proof implies the existence of a polynomial length CP proof, but the reverse need not hold [9] Constraints in CP are expressed as linear inequalities ....

.... complexity and strength of the CP proof system [5, 9, 18, 33] and it was shown early on by Cook that the CP system is properly stronger than resolution in that existence of a polynomial length resolution proof implies the existence of a polynomial length CP proof, but the reverse need not hold [9]. Constraints in CP are expressed as linear inequalities a j x j k where x 1 ; x 2 ; xn are non negative integer variables and a 1 ; a 2 ; an and k are integers. The system has two rules of inference: i) derive a new inequality by taking a linear combination of a set of ....

[Article contains additional citation context not shown here]

W. Cook, C. Coullard, and G. Turan. On the complexity of cutting plane proofs. Journal of Discrete Applied Math, 18:25-38, 1987.

....signi cantly. Several gaps between these two kinds of proof systems were demonstrated in [GH01] Systems of polynomial inequalities yield much more powerful proof systems than these operating with equations only, such as NS or PC. Historically rst such a proof system is Cutting Planes (CP) [Gom63, Chv73, CCT87, CCH89], see also Subsection 2.3. This system uses linear inequalities (with integer coecients) Exponential lower bounds on proof size were established for CP with polynomially bounded coecients [BPR95] as well as for the general case [Pud97] Another family of well studied proof systems are so called ....

....the depth of multiplications invoked in a derivation. A series of lower bounds for various versions of the LS rank were obtained in the context of optimization theory [ST99, CD01, Das01, GT01] For a counterpart notion in CP, the so called Chv atal rank [Chv73] lower bounds were established in [CCT87, CCH89]. To the best of our knowledge, the connection between the Chv atal rank and CP proof complexity is not very well understood, despite a number of interesting recent results [BEHS99, ES99] As a rule, however, diverse versions of the rank grow at most linear, while we are looking for non linear ....

[Article contains additional citation context not shown here]

W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25-38, 1987.

....a set of inequalities has a CP refutation i# it has no 0, 1 solution. Any assignment satisfying the original clauses is actually a 0, 1 solution of the corresponding inequalities, provided that we assign the numerical value 1 to True and the value 0 to False. It is easy to translate, see [11], resolution refutations into CP refutations similar in size to the original resolution refutations. Moreover if the resolution refutation is tree like, the resulting CP refutation is also tree like. 2.2 Monotone Real Circuits An important part of this paper is concerned with monotone real ....

W. Cook, C. Coullard, and G. Turan. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

....Several gaps between these two kinds of proof systems were demonstrated in [GH01] Systems of polynomial inequalities yield much more powerful proof systems than these operating with equations only, such as NS or PC. Historically rst such a proof system is Cutting Planes (CP) Gom63, Chv73, CCT87, CCH89] see also Subsection 2.3. This system uses linear inequalities (with integer coecients) Exponential lower bounds on proof size were established for CP with polynomially bounded coecients [BPR95] as well as for the general case [Pud97] Another family of well studied proof systems are ....

....the depth of multiplications invoked in a derivation. A series of lower bounds for various versions of the LS rank were obtained in the context of optimization theory [ST99, CD01, Das01, GT01] For a counterpart notion in CP, the so called Chv atal rank [Chv73] lower bounds were established in [CCT87, CCH89] To the best of our knowledge, the connection between the Chv atal rank and CP proof complexity is not very well understood, despite a number of interesting recent results [BEHS99, ES99] As a rule, however, diverse versions of the rank grow at most linear, while we are looking for ....

[Article contains additional citation context not shown here]

W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25-38, 1987.

....Several gaps between these two kinds of proof systems were demonstrated in [GH01] Systems of polynomial inequalities yield much more powerful proof systems than these operating with equations only, such as NS or PC. The first proof system working with inequalities was Cutting Planes (CP) [Gom63, Chv73, CCT87, CCH89], see also Subsection 2.3. This system uses linear inequalities (with integer coefficients) Exponential lower bounds on proof size were established for CP with polynomially bounded coefficients [BPR95] as well as for the general case [Pud97] Another family of well studied proof systems are ....

....the depth of multiplications invoked in a derivation. A series of lower bounds for various versions of the LS rank were obtained in the context of optimization theory [ST99, CD01, Das01, GT01] For a counterpart notion in CP, the so called Chv atal rank [Chv73] lower bounds were established in [CCT87, CCH89]. To the best of our knowledge, the connection between the Chv atal rank and CP proof complexity is not very well understood, despite a number of interesting recent results [BEHS99, ES99] As a rule, however, diverse versions of the rank grow at most linear, while we are looking for non linear ....

[Article contains additional citation context not shown here]

W. Cook, C. R. Coullard, and G. Tur'an. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25--38, 1987.

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W. Cook, C. R. Coullard, and Gy. Tur' an, On the complexity of cutting plane proofs, Discrete Appl. Math., (1987), 25--38.

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W. Cook, C. R. Coullard, and Gy. Tur'an, On the complexity of cutting plane proofs. Disc. Appl. Math., (1987), 25--38.

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W. Cook, R. Coullard, and G. Turan. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25-- 38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25-38, 1987.

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W. Cook, C.R. Coullard, and G. Turan. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25-38, 1987.

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W. Cook, C. R. Coullard, and G. Tur an, On the complexity of cutting plane proofs, Discrete Applied Mathematics, 18 (1987), pp. 25--38.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25-38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25-38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18:25--38, 1987.

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W. Cook, C. R. Coullard, and G. Turan. On the complexity of cutting-plane proofs. Discrete Appl. Math., 18(1):25-38, 1987.

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W. Cook,C.R. Coullard, and Gy. Turan, On the Complexity of Cutting Plane Proofs, Discrete Applied Mathematics, 18, (1987), 25-38.

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Cook, W., Coullard, C.R., Turan, G.: On the complexity of cutting-plane proofs. Discrete Appl. Math. 18 (1987) 25-38

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