Skip to main content
SpringerLink
Log in

Algorithm Analysis Through Proof Complexity

  • Conference paper
  • First Online:
Sailing Routes in the World of Computation (CiE 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10936))

Included in the following conference series:

Abstract

Proof complexity can be a tool for studying the efficiency of algorithms. By proving a single lower bound on the length of certain proofs, we can get running time lower bounds for a wide category of algorithms. We survey the proof complexity literature that adopts this approach relative to two \(\mathsf {NP}\)-problems: k-clique and 3-coloring.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The possibility of short proofs for all tautologies was already mentioned by Gödel in a letter to von Neumann [27].

  2. 2.

    A SAT solver is a software that decides satisfiability. While nowadays solvers go beyond resolution, their main component still builds resolution proofs.

  3. 3.

    The graph has n vertices and the edges are independent \(\{0,1\}\)-valued random variables with expected value p.

References

  1. Alon, N., Tarsi, M.: Colorings and orientations of graphs. Combinatorica 12(2), 125–134 (1992)

    Article  MathSciNet  Google Scholar 

  2. Atserias, A., Bonacina, I., de Rezende, S.F., Lauria, M., Nordström, J., Razborov, A.A.: Clique is hard on average for regular resolution. In: Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC 2008) (2018, to appear)

    Google Scholar 

  3. Atserias, A., Ochremiak, J.: Proof complexity meets algebra. In: ICALP 2017. Leibniz International Proceedings in Informatics (LIPIcs), vol. 80, pp. 110:1–110:14. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2017)

    Google Scholar 

  4. Bayer, D.A.: The division algorithm and the Hilbert scheme. Ph.D. thesis, Harvard University, Cambridge, MA, USA, June 1982. https://www.math.columbia.edu/~bayer/papers/Bayer-thesis.pdf

  5. Beame, P., Culberson, J.C., Mitchell, D.G., Moore, C.: The resolution complexity of random graph \(k\)-colorability. Discrete Appl. Math. 153(1–3), 25–47 (2005)

    Article  MathSciNet  Google Scholar 

  6. Beame, P., Impagliazzo, R., Krajíček, J., Pitassi, T., Pudlák, P.: Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. In: Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1994), pp. 794–806, November 1994

    Google Scholar 

  7. Beame, P., Impagliazzo, R., Sabharwal, A.: The resolution complexity of independent sets and vertex covers in random graphs. Comput. Complex. 16(3), 245–297 (2007)

    Article  MathSciNet  Google Scholar 

  8. Beame, P., Pitassi, T.: Propositional proof complexity: past, present, and future. In: Current Trends in Theoretical Computer Science, pp. 42–70. World Scientific Publishing (2001)

    Google Scholar 

  9. Beigel, R., Eppstein, D.: 3-coloring in time \(O(1. 3289^n)\). J. Algorithms 54(2), 168–204 (2005)

    Article  MathSciNet  Google Scholar 

  10. Beyersdorff, O., Galesi, N., Lauria, M.: Parameterized complexity of DPLL search procedures. ACM Trans. Comput. Log. 14(3), 20:1–20:21 (2013). Preliminary version in SAT 2011

    Article  MathSciNet  Google Scholar 

  11. Blake, A.: Canonical expressions in Boolean algebra. Ph.D. thesis, University of Chicago (1938)

    Google Scholar 

  12. Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)

    Article  Google Scholar 

  13. Buresh-Oppenheim, J., Clegg, M., Impagliazzo, R., Pitassi, T.: Homogenization and the polynomial calculus. Comput. Complex. 11(3–4), 91–108 (2002). Preliminary version in ICALP 2000

    Article  MathSciNet  Google Scholar 

  14. Chvátal, V.: Determining the stability number of a graph. SIAM J. Comput. 6(4), 643–662 (1977)

    Article  MathSciNet  Google Scholar 

  15. Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Groebner basis algorithm to find proofs of unsatisfiability. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996), pp. 174–183, May 1996

    Google Scholar 

  16. Cook, S.A.: The complexity of theorem proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (STOC 1971), pp. 151–158 (1971)

    Google Scholar 

  17. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44, 36–50 (1979)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, New York (2007). https://doi.org/10.1007/978-0-387-35651-8

    Book  MATH  Google Scholar 

  20. De Loera, J.A.: Gröbner bases and graph colorings. Beiträge zur Algebra und Geometrie 36(1), 89–96 (1995). https://www.emis.de/journals/BAG/vol.36/no.1/

    MathSciNet  MATH  Google Scholar 

  21. De Loera, J.A., Margulies, S., Pernpeintner, M., Riedl, E., Rolnick, D., Spencer, G., Stasi, D., Swenson, J.: Graph-coloring ideals: Nullstellensatz certificates, Gröbner bases for chordal graphs, and hardness of Gröbner bases. In: Proceedings of the 40th International Symposium on Symbolic and Algebraic Computation (ISSAC 2015), pp. 133–140, July 2015

    Google Scholar 

  22. De Loera, J.A., Lee, J., Malkin, P.N., Margulies, S.: Hilbert’s Nullstellensatz and an algorithm for proving combinatorial infeasibility. In: Proceedings of the 21st International Symposium on Symbolic and Algebraic Computation (ISSAC 2008), pp. 197–206, July 2008

    Google Scholar 

  23. De Loera, J.A., Lee, J., Malkin, P.N., Margulies, S.: Computing infeasibility certificates for combinatorial problems through Hilbert’s Nullstellensatz. J. Symb. Comput. 46(11), 1260–1283 (2011)

    Article  MathSciNet  Google Scholar 

  24. De Loera, J.A., Lee, J., Margulies, S., Onn, S.: Expressing combinatorial problems by systems of polynomial equations and Hilbert’s Nullstellensatz. Comb. Probab. Comput. 18(04), 551–582 (2009)

    Article  MathSciNet  Google Scholar 

  25. Galesi, N., Lauria, M.: On the automatizability of polynomial calculus. Theory Comput. Syst. 47(2), 491–506 (2010)

    Article  MathSciNet  Google Scholar 

  26. Galesi, N., Lauria, M.: Optimality of size-degree trade-offs for polynomial calculus. ACM Trans. Comput. Log. 12(1), 4:1–4:22 (2010)

    Article  Google Scholar 

  27. Gödel, K.: Ein brief an Johann von Neumann, 20. März, 1956. In: Clote, P., Krajíček, J. (eds.) Arithmetic, Proof Theory, and Computational Complexity, pp. 7–9. Oxford University Press, Oxford (1993)

    Google Scholar 

  28. Hajós, G.: Üver eine konstruktion nicht n-farbbarer graphen. Wissenschaftliche Zeitschrift der Martin-Luther-Universitat Halle-Wittenberg, A 10, 116–117 (1961)

    Google Scholar 

  29. Haken, A.: The intractability of resolution. Theoret. Comput. Sci. 39, 297–308 (1985)

    Article  MathSciNet  Google Scholar 

  30. Hillar, C.J., Windfeldt, T.: Algebraic characterization of uniquely vertex colorable graphs. J. Comb. Theory Ser. B 98(2), 400–414 (2008)

    Article  MathSciNet  Google Scholar 

  31. Husfeldt, T.: Graph colouring algorithms. In: Beineke, L.W., Wilson, R.J. (eds.) Topics in Chromatic Graph Theory, Encyclopedia of Mathematics and its Applications, pp. 277–303. Cambridge University Press, May 2015. Chap. 13

    Google Scholar 

  32. Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and its Applications, vol. 60. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  33. Lauria, M., Nordström, J.: Graph colouring is hard for algorithms based on Hilbert’s Nullstellensatz and Gröbner bases. In: O’Donnell, R. (ed.) 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), vol. 79, pp. 2:1–2:20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2017)

    Google Scholar 

  34. Lauria, M., Pudlák, P., Rödl, V., Thapen, N.: The complexity of proving that a graph is Ramsey. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 684–695. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39206-1_58

    Chapter  Google Scholar 

  35. Lauria, M., Pudlák, P., Rödl, V., Thapen, N.: The complexity of proving that a graph is Ramsey. Combinatorica 37(2), 253–268 (2017). Preliminary version in ICALP 2013

    Article  MathSciNet  Google Scholar 

  36. Levin, L.A.: Universal sequential search problems. Probl. Peredachi Informatsii 9(3), 115–116 (1973)

    MATH  Google Scholar 

  37. Lokshtanov, D., Marx, D., Saurabh, S., et al.: Lower bounds based on the exponential time hypothesis. Bull. EATCS 3(105), 41–72 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Lovász, L.: Stable sets and polynomials. Discrete Math. 124(1–3), 137–153 (1994)

    Article  MathSciNet  Google Scholar 

  39. Matiyasevich, Y.V.: A criterion for vertex colorability of a graph stated in terms of edge orientations. Diskretnyi Analiz 26, 65–71 (1974). http://logic.pdmi.ras.ru/~yumat/papers/22paper/. English translation of the Russian original

    Google Scholar 

  40. Matiyasevich, Y.V.: Some algebraic methods for calculating the number of colorings of a graph. J. Math. Sci. 121(3), 2401–2408 (2004)

    Article  MathSciNet  Google Scholar 

  41. McCreesh, C.: Solving hard subgraph problems in parallel. Ph.D. thesis, University of Glasgow (2017)

    Google Scholar 

  42. McDiarmid, C.: Colouring random graphs. Ann. Oper. Res. 1(3), 183–200 (1984)

    Article  MathSciNet  Google Scholar 

  43. Mnuk, M.: Representing graph properties by polynomial ideals. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2001, pp. 431–444. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-642-56666-0_33

    Chapter  Google Scholar 

  44. Nordström, J.: A (biased) proof complexity survey for SAT practitioners. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 1–6. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09284-3_1

    Chapter  Google Scholar 

  45. Nordström, J.: On the interplay between proof complexity and SAT solving. ACM SIGLOG News 2(3), 19–44 (2015)

    Google Scholar 

  46. Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120(1), 197–207 (2002)

    Article  MathSciNet  Google Scholar 

  47. Pevzner, P.A., Sze, S.-H., et al.: Combinatorial approaches to finding subtle signals in DNA sequences. In: ISMB, vol. 8, pp. 269–278 (2000)

    Google Scholar 

  48. Pitassi, T., Urquhart, A.: The complexity of the Hajós calculus. SIAM J. Discrete Math. 8(3), 464–483 (1995)

    Article  MathSciNet  Google Scholar 

  49. Prosser, P.: Exact algorithms for maximum clique: a computational study. Algorithms 5(4), 545–587 (2012)

    Article  MathSciNet  Google Scholar 

  50. Razborov, A.A.: Lower bounds for the monotone complexity of some Boolean functions. Soviet Math. Dokl. 31(2), 354–357 (1985). English translation of a paper in Doklady Akademii Nauk SSSR

    MATH  Google Scholar 

  51. Rossman, B.: On the constant-depth complexity of \(k\)-clique. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 721–730. ACM (2008)

    Google Scholar 

  52. Rossman, B.: The monotone complexity of \(k\)-clique on random graphs. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, 23–26 October 2010, Las Vegas, Nevada, USA, pp. 193–201. IEEE Computer Society (2010)

    Google Scholar 

  53. Segerlind, N.: The complexity of propositional proofs. Bull. Symb. Log. 13(4), 417–481 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Statistiche, Università “La Sapienza” di Roma, Rome, Italy

    Massimo Lauria

Authors
  1. Massimo Lauria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Massimo Lauria .

Editor information

Editors and Affiliations

  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lauria, M. (2018). Algorithm Analysis Through Proof Complexity. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-94418-0_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94417-3

  • Online ISBN: 978-3-319-94418-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation