Algorithm Analysis Through Proof Complexity

  • Massimo LauriaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10936)


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.


  1. 1.
    Alon, N., Tarsi, M.: Colorings and orientations of graphs. Combinatorica 12(2), 125–134 (1992)MathSciNetCrossRefGoogle Scholar
  2. 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. 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. 4.
    Bayer, D.A.: The division algorithm and the Hilbert scheme. Ph.D. thesis, Harvard University, Cambridge, MA, USA, June 1982.
  5. 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)MathSciNetCrossRefGoogle Scholar
  6. 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 1994Google Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 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. 9.
    Beigel, R., Eppstein, D.: 3-coloring in time \(O(1. 3289^n)\). J. Algorithms 54(2), 168–204 (2005)MathSciNetCrossRefGoogle Scholar
  10. 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 2011MathSciNetCrossRefGoogle Scholar
  11. 11.
    Blake, A.: Canonical expressions in Boolean algebra. Ph.D. thesis, University of Chicago (1938)Google Scholar
  12. 12.
    Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)CrossRefGoogle Scholar
  13. 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 2000MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chvátal, V.: Determining the stability number of a graph. SIAM J. Comput. 6(4), 643–662 (1977)MathSciNetCrossRefGoogle Scholar
  15. 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 1996Google Scholar
  16. 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. 17.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44, 36–50 (1979)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cook, W., Coullard, C.R., Turán, G.: On the complexity of cutting-plane proofs. Discrete Appl. Math. 18(1), 25–38 (1987)MathSciNetCrossRefGoogle Scholar
  19. 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). Scholar
  20. 20.
    De Loera, J.A.: Gröbner bases and graph colorings. Beiträge zur Algebra und Geometrie 36(1), 89–96 (1995). Scholar
  21. 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 2015Google Scholar
  22. 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 2008Google Scholar
  23. 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)MathSciNetCrossRefGoogle Scholar
  24. 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)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Galesi, N., Lauria, M.: On the automatizability of polynomial calculus. Theory Comput. Syst. 47(2), 491–506 (2010)MathSciNetCrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 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. 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. 29.
    Haken, A.: The intractability of resolution. Theoret. Comput. Sci. 39, 297–308 (1985)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hillar, C.J., Windfeldt, T.: Algebraic characterization of uniquely vertex colorable graphs. J. Comb. Theory Ser. B 98(2), 400–414 (2008)MathSciNetCrossRefGoogle Scholar
  31. 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. 13Google Scholar
  32. 32.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and its Applications, vol. 60. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  33. 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. 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). Scholar
  35. 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 2013MathSciNetCrossRefGoogle Scholar
  36. 36.
    Levin, L.A.: Universal sequential search problems. Probl. Peredachi Informatsii 9(3), 115–116 (1973)zbMATHGoogle Scholar
  37. 37.
    Lokshtanov, D., Marx, D., Saurabh, S., et al.: Lower bounds based on the exponential time hypothesis. Bull. EATCS 3(105), 41–72 (2013)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Lovász, L.: Stable sets and polynomials. Discrete Math. 124(1–3), 137–153 (1994)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Matiyasevich, Y.V.: A criterion for vertex colorability of a graph stated in terms of edge orientations. Diskretnyi Analiz 26, 65–71 (1974). English translation of the Russian originalGoogle Scholar
  40. 40.
    Matiyasevich, Y.V.: Some algebraic methods for calculating the number of colorings of a graph. J. Math. Sci. 121(3), 2401–2408 (2004)MathSciNetCrossRefGoogle Scholar
  41. 41.
    McCreesh, C.: Solving hard subgraph problems in parallel. Ph.D. thesis, University of Glasgow (2017)Google Scholar
  42. 42.
    McDiarmid, C.: Colouring random graphs. Ann. Oper. Res. 1(3), 183–200 (1984)MathSciNetCrossRefGoogle Scholar
  43. 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). Scholar
  44. 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). Scholar
  45. 45.
    Nordström, J.: On the interplay between proof complexity and SAT solving. ACM SIGLOG News 2(3), 19–44 (2015)Google Scholar
  46. 46.
    Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120(1), 197–207 (2002)MathSciNetCrossRefGoogle Scholar
  47. 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. 48.
    Pitassi, T., Urquhart, A.: The complexity of the Hajós calculus. SIAM J. Discrete Math. 8(3), 464–483 (1995)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Prosser, P.: Exact algorithms for maximum clique: a computational study. Algorithms 5(4), 545–587 (2012)MathSciNetCrossRefGoogle Scholar
  50. 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 SSSRzbMATHGoogle Scholar
  51. 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. 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. 53.
    Segerlind, N.: The complexity of propositional proofs. Bull. Symb. Log. 13(4), 417–481 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Scienze StatisticheUniversità “La Sapienza” di RomaRomeItaly

Personalised recommendations