Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs

  • Erik D. Demaine
  • Quanquan C. LiuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


Pebble games are single-player games on DAGs involving placing and moving pebbles on nodes of the graph according to a certain set of rules. The goal is to pebble a set of target nodes using a minimum number of pebbles. In this paper, we present a possibly simpler proof of the result in [4] and strengthen the result to show that it is PSPACE-hard to determine the minimum number of pebbles to an additive \(n^{1/3-\epsilon }\) term for all \(\epsilon > 0\), which improves upon the currently known additive constant hardness of approximation [4] in the standard pebble game. We also introduce a family of explicit, constant indegree graphs with n nodes where there exists a graph in the family such that using \(0< k < \sqrt{n}\) pebbles requires \(\varOmega ((n/k)^k)\) moves to pebble in both the standard and black-white pebble games. This independently answers an open question summarized in [14] of whether a family of DAGs exists that meets the upper bound of \(O(n^k)\) moves using constant k pebbles with a different construction than that presented in [1].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alwen, J., de Rezende, S.F., Nordström, J., Vinyals, M.: Cumulative space in black-white pebbling and resolution. In: Innovations in Theoretical Computer Science, ITCS 2017, Berkeley, CA, USA, pp. 9–11, January 2017Google Scholar
  2. 2.
    Alwen, J., Serbinenko, V.: High parallel complexity graphs and memory-hard functions. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14–17, pp. 595–603 (2015).
  3. 3.
    Bennett, C.H.: Time/space trade-offs for reversible computation. SIAM J. Comput. 18(4), 766–776 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chan, S.M., Lauria, M., Nordström, J., Vinyals, M.: Hardness of approximation in PSPACE and separation results for pebble games. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, October 17–20, 2015, pp. 466–485 (2015).
  5. 5.
    Cook, S., Sethi, R.: Storage requirements for deterministic / polynomial time recognizable languages. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, STOC 1974, pp. 33–39. ACM, New York (1974).
  6. 6.
    Demaine, E.D., Liu, Q.C.: Inapproximability of the standard pebble game and hard to pebble graphs. CoRR (2017)Google Scholar
  7. 7.
    van Emde Boas, P., van Leeuwen, J.: Move rules and trade-offs in the pebble game. In: Weihrauch, K. (ed.) GI-TCS 1979. LNCS, vol. 67, pp. 101–112. Springer, Heidelberg (1979). doi: 10.1007/3-540-09118-1_12 CrossRefGoogle Scholar
  8. 8.
    Gilbert, J.R., Lengauer, T., Tarjan, R.E.: The pebbling problem is complete in polynomial space. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC 1979, pp. 237–248. ACM, New York (1979).
  9. 9.
    Gilbert, J.R., Tarjan, R.E.: Variations of a pebble game on graphs. Technical report, Stanford, CA, USA (1978)Google Scholar
  10. 10.
    Hertel, P., Pitassi, T.: The PSPACE-completeness of black-white pebbling. SIAM J. Comput. 39(6), 2622–2682 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hopcroft, J., Paul, W., Valiant, L.: On time versus space. J. ACM 24(2), 332–337 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jia-Wei, H., Kung, H.T.: I/O complexity: the red-blue pebble game. In: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, STOC 1981, pp. 326–333. ACM, New York (1981).
  13. 13.
    Lengauer, T., Tarjan, R.E.: Upper and lower bounds on time-space tradeoffs. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC 1979, pp. 262–277. ACM, New York (1979).
  14. 14.
    Nordstrom, J.: New wine into old wineskins: A survey of some pebbling classics with supplemental results (2015)Google Scholar
  15. 15.
    Paul, W.J., Tarjan, R.E., Celoni, J.R.: Space bounds for a game on graphs. In: Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, STOC 1976, pp. 149–160. ACM, New York (1976).
  16. 16.
    Sethi, R.: Complete register allocation problems. SIAM J. Comput. 4(3), 226–248 (1975). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MIT CSAILCambridgeUSA

Personalised recommendations