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Lauria, Massimoorcid.org/0000-0003-4003-3168

Open this publication in new window or tab >>The complexity of proving that a graph is Ramsey### Lauria, Massimo

### Pudlak, Pavel

### Rodl, Vojtch

### Thapen, Neil

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_some",{id:"formSmash:j_idt204:0:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_otherAuthors",{id:"formSmash:j_idt204:0:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Combinatorica, ISSN 0209-9683, E-ISSN 1439-6912, Vol. 37, no 2, p. 253-268Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

SPRINGER HEIDELBERG, 2017
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-207698 (URN)10.1007/s00493-015-3193-9 (DOI)000399890000008 ()2-s2.0-85018519537 (Scopus ID)
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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every c-Ramsey graph must contain a large subgraph with some properties typical for random graphs.

QC 20170531

Available from: 2017-05-31 Created: 2017-05-31 Last updated: 2022-06-27Bibliographically approvedOpen this publication in new window or tab >>Tight Size-Degree Bounds for Sums-of-Squares Proofs### Lauria, Massimo

### Nordström, Jakob

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Computational Complexity, ISSN 1016-3328, E-ISSN 1420-8954, Vol. 26, no 4, p. 911-948Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

SPRINGER BASEL AG, 2017
##### Keywords

Proof complexity, resolution, Lasserre, Positivstellensatz, sums-of-squares, SOS, semidefinite programming, size, degree, rank, clique, lower bound
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-272426 (URN)10.1007/s00037-017-0152-4 (DOI)000415306000004 ()2-s2.0-85018513275 (Scopus ID)
#####

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##### Note

KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Theoretical Computer Science, TCS.

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size for values of d = d(n) from constant all the way up to for some universal constant . This shows that the running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajiek (Arch Math Log 43(4):427-441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL '03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450-476, 2015; ACM Trans Comput Log 17:19:1-19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.

QC 20200422

Available from: 2020-04-22 Created: 2020-04-22 Last updated: 2024-03-18Bibliographically approvedOpen this publication in new window or tab >>From small space to small width in resolution### Filmus, Y.

### Lauria, Massimo

### Mikša, Mladen

### Nordström, Jakob

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_some",{id:"formSmash:j_idt204:2:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_some",multiple:true}); ### Vinyals, M.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_otherAuthors",{id:"formSmash:j_idt204:2:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt204_2_j_idt208_j_idt222",{id:"formSmash:j_idt204:2:j_idt208:j_idt222",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_j_idt222",onLabel:"Hide others...",offLabel:"Show others..."}); 2015 (English)In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 16, no 4, article id 28Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

PCR, Polynomial calculus, Proof complexity, Resolution, Space, Width
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-174699 (URN)10.1145/2746339 (DOI)000365216500001 ()2-s2.0-84941557324 (Scopus ID)
#####

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##### Funder

EU, FP7, Seventh Framework ProgrammeEU, European Research Council, 621-2010-4797, 621-2012-5645Swedish Research Council, 621-2010-4797, 621-2012-5645
##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexitymeasure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similarmethods.

QC 20151111

Available from: 2015-11-11 Created: 2015-10-07 Last updated: 2022-06-23Bibliographically approvedOpen this publication in new window or tab >>Hardness of Approximation in PSPACE and Separation Results for Pebble Games### Chan, S. M.

### Lauria, Massimo

### Nordstrom, Jakob

### Vinyals, Marc

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_some",{id:"formSmash:j_idt204:3:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_otherAuthors",{id:"formSmash:j_idt204:3:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_otherAuthors",multiple:true}); 2015 (English)In: Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, Institute of Electrical and Electronics Engineers (IEEE), 2015, p. 466-485Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Institute of Electrical and Electronics Engineers (IEEE), 2015
##### Keywords

Dymond-Tompa game, pebbling, PSPACE-complete, PSPACE-hardness of approximation, Raz-Mc Kenzie game, resolution depth, reversible pebbling, separation
##### National Category

Electrical Engineering, Electronic Engineering, Information Engineering
##### Identifiers

urn:nbn:se:kth:diva-186800 (URN)10.1109/FOCS.2015.36 (DOI)000379204700027 ()2-s2.0-84960371365 (Scopus ID)9781467381918 (ISBN)
##### Conference

56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, 17 October 2015 through 20 October 2015
#####

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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of deciding whether s pebbles suffice to reversibly pebble a DAG G is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond - Tompa and Raz - McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time.

QC 20160516

Available from: 2016-05-16 Created: 2016-05-13 Last updated: 2024-01-10Bibliographically approvedOpen this publication in new window or tab >>Space complexity in polynomial calculus### Filmus, Yuval

### Lauria, Massimo

### Nordstrom, Jakob

### Ron-Zewi, Noga

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_some",{id:"formSmash:j_idt204:4:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_some",multiple:true}); ### Thapen, Neil

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_otherAuthors",{id:"formSmash:j_idt204:4:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt204_4_j_idt208_j_idt222",{id:"formSmash:j_idt204:4:j_idt208:j_idt222",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_j_idt222",onLabel:"Hide others...",offLabel:"Show others..."}); 2015 (English)In: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 44, no 4, p. 1119-1153Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

proof complexity, space, polynomial calculus, polynomial calculus resolution, PCR, resolution, lower bounds
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:kth:diva-174007 (URN)10.1137/120895950 (DOI)000360654100008 ()2-s2.0-84940668727 (Scopus ID)
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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

During the last 10 to 15 years, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important concern in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well-understood from this point of view. For other proof systems of interest, however, such as polynomial calculus or cutting planes, progress has been more limited. Essentially nothing has been known about space complexity in cutting planes, and for polynomial calculus the only lower bound has been for conjunctive normal form (CNF) formulas of unbounded width in [Alekhnovich et al., SIAM J. Comput., 31 (2002), pp. 1184-1211], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any k-CNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied by Alekhnovich et al.: (1) We prove an omega(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHPmn with m pigeons and n holes, and show that this is tight. (2) For PCR, we prove an omega(n) space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over Alekhnovich et al. measured in the width of the formulas. (3) We then present another encoding of the pigeonhole principle that has constant width, and prove an omega(n) space lower bound in PCR for these formulas as well. (4) Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.

QC 20150930

Available from: 2015-09-30 Created: 2015-09-24 Last updated: 2022-06-23Bibliographically approvedOpen this publication in new window or tab >>Tight size-degree bounds for sums-of-squares proofs### Lauria, Massimo

### Nordström, Jakob

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_some",{id:"formSmash:j_idt204:5:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_otherAuthors",{id:"formSmash:j_idt204:5:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_otherAuthors",multiple:true}); 2015 (English)In: Leibniz International Proceedings in Informatics, LIPIcs, Dagstuhl Publishing , 2015, Vol. 33, p. 448-466Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Dagstuhl Publishing, 2015
##### Keywords

Clique, Degree, Lasserre, Lower bound, Positivstellensatz, Proof complexity, Rank, Resolution, Semidefinite programming, Size, SOS, Sums-ofsquares
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-187388 (URN)10.4230/LIPIcs.CCC.2015.448 (DOI)000494655400023 ()2-s2.0-84958245402 (Scopus ID)978-393989781-1 (ISBN)
##### Conference

30th Conference on Computational Complexity, CCC 2015; Portland; United States
#####

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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size nΩ(d) for values of d = d(n) from constant all the way up to nδ for some universal constant δ. This shows that the nO(d) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Krajícek’04] and [Dantchev and Riis’03], and then applying a restriction argument as in [Atserias, Müller, and Oliva’13] and [Atserias, Lauria, and Nordström’14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.

QC 20160527

Available from: 2016-05-27 Created: 2016-05-23 Last updated: 2022-06-22Bibliographically approvedOpen this publication in new window or tab >>From small space to small width in resolution### Filmus, Y.

### Lauria, Massimo

### Mikša, Mladen

### Nordström, Jakob

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_some",{id:"formSmash:j_idt204:6:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_some",multiple:true}); ### Vinyals, Marc

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_otherAuthors",{id:"formSmash:j_idt204:6:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt204_6_j_idt208_j_idt222",{id:"formSmash:j_idt204:6:j_idt208:j_idt222",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_j_idt222",onLabel:"Hide others...",offLabel:"Show others..."}); 2014 (English)In: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2014, Vol. 25, p. 300-311Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2014
##### Series

Leibniz International Proceedings in Informatics, LIPIcs, ISSN 1868-8969 ; 25
##### Keywords

PCR, Polynomial calculus, Proof complexity, Resolution, Space, Width
##### National Category

Computer and Information Sciences
##### Identifiers

urn:nbn:se:kth:diva-158079 (URN)10.4230/LIPIcs.STACS.2014.300 (DOI)000521069500027 ()2-s2.0-84907818998 (Scopus ID)978-393989765-1 (ISBN)
##### Conference

31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014, Lyon, France, 5 March 2014 through 8 March 2014
#####

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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexity measure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

QC 20141222

Available from: 2014-12-22 Created: 2014-12-22 Last updated: 2022-06-23Bibliographically approvedOpen this publication in new window or tab >>Narrow proofs may be maximally long### Atserias, A.

### Lauria, Massimo

### Nordström, Jakob

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_some",{id:"formSmash:j_idt204:7:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_otherAuthors",{id:"formSmash:j_idt204:7:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_otherAuthors",multiple:true}); 2014 (English)In: Proceedings of the Annual IEEE Conference on Computational Complexity, 2014, p. 286-297Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Keywords

degree, Lasserre, length, PCR, polynomial calculus, proof complexity, rank, resolution, Sherali-Adams, size, width, Calculations, Computational complexity, Optical resolving power, Polynomials
##### National Category

Computer and Information Sciences
##### Identifiers

urn:nbn:se:kth:diva-168868 (URN)10.1109/CCC.2014.36 (DOI)000345759400028 ()2-s2.0-84906667234 (Scopus ID)9781479936267 (ISBN)
##### Conference

29th Annual IEEE Conference on Computational Complexity, CCC 2014, 11 June 2014 through 13 June 2014, Vancouver, BC
#####

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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size nω(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(ω) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however-the formulas we study have Lasserre proofs of constant rank and size polynomial in both n and w.

QC 20150611

Available from: 2015-06-11 Created: 2015-06-09 Last updated: 2022-06-23Bibliographically approvedOpen this publication in new window or tab >>A characterization of tree-like Resolution size### Beyersdorff, Olaf

### Galesi, Nicola

### Lauria, Massimo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_some",{id:"formSmash:j_idt204:8:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_otherAuthors",{id:"formSmash:j_idt204:8:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_otherAuthors",multiple:true}); 2013 (English)In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 113, no 18, p. 666-671Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Computational complexity, Proof complexity, Prover-Delayer games, Resolution
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:kth:diva-127469 (URN)10.1016/j.ipl.2013.06.002 (DOI)000322610700005 ()2-s2.0-84879377601 (Scopus ID)
#####

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##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas (Beyersdorff et al. (2013) [2]) and for the classical pigeonhole principle (Beyersdorff et al. (2010) [1]). The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i.e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.

Not duplicate with DiVA 587524

QC 20230202

Available from: 2013-09-05 Created: 2013-08-30 Last updated: 2023-02-02Bibliographically approvedOpen this publication in new window or tab >>A rank lower bound for cutting planes proofs of Ramsey's theorem### Lauria, Massimo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_some",{id:"formSmash:j_idt204:9:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_otherAuthors",{id:"formSmash:j_idt204:9:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_otherAuthors",multiple:true}); 2013 (English)In: Theory and Applications of Satisfiability Testing – SAT 2013: 16th International Conference, Helsinki, Finland, July 8-12, 2013. Proceedings, Springer, 2013, p. 351-364Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2013
##### Series

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), ISSN 0302-9743 ; 7962 LNCS
##### Keywords

Combinatorics, Cutting planes, Independent set, Lower bounds, Propositional proof systems, Ramsey's theorem, Upper Bound, Artificial intelligence, Computer science, Formal logic
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:kth:diva-134088 (URN)10.1007/978-3-642-39071-5_26 (DOI)2-s2.0-84879905485 (Scopus ID)978-364239070-8 (ISBN)
##### Conference

16th International Conference on Theory and Applications of Satisfiability Testing, SAT 2013; Helsinki; Finland; 8 July 2013 through 12 July 2013
#####

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##### Funder

EU, European Research Council
##### Note

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Ramsey's Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that there is a function r such that any simple graph with r(k,s) vertices contains either a clique of size k or an independent set of size s. We study the complexity of proving upper bounds for the number r(k,k). In particular we focus on the propositional proof system cutting planes; we prove that the upper bound "r(k,k) ≤ 4k" requires cutting planes proof of high rank. In order to do that we show a protection lemma which could be of independent interest.

QC 20131115

Available from: 2013-11-15 Created: 2013-11-15 Last updated: 2022-06-23Bibliographically approved