Skip to main content
SpringerLink
Log in

Cache Me if You Can: Capacitated Selfish Replication Games in Networks

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

In Peer-to-Peer (P2P) network systems, content (object) delivery between nodes is often required. One way to study such a distributed system is by defining games, which involve selfish nodes that make strategic choices on replicating content in their local limited memory (cache) or accessing content from other nodes for a cost. These Selfish Replication games have been introduced in Chun et al. (2004) for nodes that do not have any capacity limits, leaving the capacitated problem, i.e. Capacitated Selfish Replication (CSR) games, open. In this work, we first form the model of the CSR games, for which we perform a Nash equilibria analysis. In particular, we focus on hierarchical networks, given their extensive use to model communication costs of content delivery in P2P systems. We present an exact polynomial-time algorithm for any hierarchical network, under two constraints on the utility functions: 1) “Nearer is better”, i.e. the closest the content is to the node the less its access cost is, and 2) “Independence of irrelevant alternatives”, i.e. aggregation of individual node preferences. This generalization represents a vast class of utilities and more interestingly allows each of the nodes to have simultaneously completely different functional forms of utility functions. In this general framework, we present CSR games results on arbitrary networks and outline the boundary between intractability and effective computability in terms of the network structure, object preferences, and the total number of objects. Moreover, we prove that the problem of equilibria existence becomes NP-hard for general CSR games. By adding some constraints in the number of objects and their preferences, we show that the equilibrium can be found in polynomial time. Finally, we introduce the fractional version of CSR games (F-CSR) to represent content distribution. We prove that equilibrium exists for every F-CSR game, but it is PPAD-complete.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. not to be confused with “fictitious play” [24] which involves learning

  2. We let each node be both descendant and ancestor of itself.

  3. We define a total preorder as a binary relation that satisfies reflexivity, transitivity, and totality. By totality we mean that for any i, j, k, either j ≥ik or k ≥ij.

  4. A strict weak order is a strict partial order >, i.e. a transitive relation that is irreflexive, in which the “neither a > b nor b > a” relation is transitive. Strict weak orders and total preorders are widely used in the field of microeconomics.

References

  1. Ahmadyan, S.N., Etesami, S.R., Poor, H.V.: A random tree search algorithm for Nash equilibrium in capacitated selfish replication games. In: IEEE 55th conference on decision and control (CDC), pp 4439–4444. https://doi.org/10.1109/CDC.2016.7798943 (2016)

  2. Angel, E., Bampis, E., Pollatos, G.G., Zissimopoulos, V.: Optimal data placement on networks with constant number of clients. Theoretical Computer Science (2013)

  3. Arrow, K.: Social choice and individual values. Yale University Press (1951)

  4. Baev, I.D., Rajaraman, R., Swamy, C.: Approximation Algorithms for Data Placement Problems. SIAM J. Comput. 38(4), 1411–1429 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byers, J.W., Luby, M., Mitzenmacher, M., Rege, A.: A digital fountain approach to reliable distribution of bulk data. In: SIGCOMM ’98, pp 56–67 (1998)

    Article  Google Scholar 

  6. Chen, X., Deng, X., Teng, S.H.: Settling the complexity of computing two-player Nash equilibria. Journal of the ACM (JACM), 56(3) (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Y., Katz, R.H., Kubiatowicz, J.D.: Scan: A dynamic, scalable, and efficient content distribution network. In: Mattern, F., Naghshineh, M. (eds.) Pervasive computing, pp 282–296. Springer, Berlin (2002)

    Google Scholar 

  8. Chun, B.G., Chaudhuri, K., Wee, H., Barreno, M., Papadimitriou, C.H., Kubiatowicz, J.: Selfish caching in distributed systems: A game-theoretic analysis. In: ACM symposium on principles of distributed computing (PODC), pp 21–30 (2004)

  9. Dabek, F., Kaashoek, M.F., Karger, D., Morris, R., Stoica, I.: Wide-area cooperative storage with cfs. In: Proceedings of the 8th ACM symposium on operating systems principles, SOSP ’01. https://doi.org/10.1145/502034.502054, pp 202–215. ACM, New York (2001)

  10. Danzig, P.: Netcache architecture and deployment. Comput. Netw. ISDN Syst. 30(22-23), 2081–2091 (1998). https://doi.org/10.1016/S0169-7552(98)00250-5

    Article  Google Scholar 

  11. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. STOC ACM, 71–78 (2006)

  12. Devanur, N.R., Garg, N., Khandekar, R., Pandit, V., Saberi, A., Vazirani, V.V.: Price of anarchy, locality gap, and a network service provider game. In: WINE, pp 1046–1055 (2005)

  13. Douceur, J.R., Wattenhofer, R.P.: Large-scale simulation of replica placement algorithms for a serverless distributed file system. In: MASCOTS 2001 proceedings 9th international symposium on modeling, analysis and simulation of computer and telecommunication systems, pp. 311–319. https://doi.org/10.1109/MASCOT.2001.948882 (2001)

  14. Etesami, S.R., Basar, T.: Pure Nash equilibrium in capacitated selfish replication (CSR) game. arXiv:1404.3442 (2014)

  15. Etesami, S.R., Basar, T.: Approximation algorithm for the binary-preference capacitated selfish replication game and a tight bound on its price of anarchy. arXiv:1506.04047v2 (2016)

  16. Etesami, S.R., Basar, T.: Pure Nash equilibrium in a capacitated resource allocation game with binary preferences. arXiv:1404.3442v3(2016)

  17. Etesami, S.R., Basar, T.: Pure Nash equilibrium in a capacitated selfish resource allocation game. IEEE Transactions on Control of Network Systems PP(99), 1–1 (2016)

    MATH  Google Scholar 

  18. Etesami, S.R., Başar, T.: Network games, pp 1–46. Springer International Publishing, Cham (2017). https://doi.org/10.1007/978-3-319-27335-8_10-1

    Google Scholar 

  19. Etesami, S.R., Başar, T.: An approximation algorithm and price of anarchy for the binary-preference capacitated selfish replication game. In: 54th IEEE conference on decision and control (CDC), pp. 3568–3573. https://doi.org/10.1109/CDC.2015.7402771 (2015)

  20. Etesami, S.R., Başar, T.: Price of anarchy and an approximation algorithm for the binary-preference capacitated selfish replication game. Automatica 76(Supplement C), 153–163 (2017). https://doi.org/10.1016/j.automatica.2016.10.002

    Article  MathSciNet  MATH  Google Scholar 

  21. Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.: On a Network Creation Game. In: PODC ’03: Proceedings of the 22nd annual symposium on principles of distributed computing. https://doi.org/10.1145/872035.872088, pp 347–351. ACM Press, New York (2003)

  22. Fan, L., Cao, P., Almeida, J., Broder, A.Z.: Summary cache: A scalable wide-area web cache sharing protocol. In: Proceedings of the ACM SIGCOMM ’98 conference on applications, technologies, architectures, and protocols for computer communication, SIGCOMM ’98. https://doi.org/10.1145/285237.285287, pp 254–265. ACM, New York (1998)

  23. Feldman, M., Chuang, J.: Overcoming free-riding behavior in peer-to-peer systems. ACM Sigecom Exchanges 5(4), 41–50 (2005)

    Article  Google Scholar 

  24. Fudenberg, D., Levine, D.: The theory of learning in games. MIT Press (1998)

  25. Garces-Erice, L., Biersack, E.W., Felber, P.A., Ross, K.W., Urvoy-Keller, G.: Hierarchical peer-to-peer systems. Parallel Process. Lett. 13 (04), 643–657 (2003). https://doi.org/10.1142/S0129626403001574

    Article  MathSciNet  Google Scholar 

  26. Garey, M., Johnson, D.: Computers and intractability. Freeman Press (1979)

  27. Goemans, M.X., Li, L., Mirrokni, V.S., Thottan, M.: Market sharing games applied to content distribution in ad hoc networks. IEEE J. Sel. Areas Commun. 24 (5), 1020–1033 (2006). https://doi.org/10.1109/JSAC.2006.872884

    Article  Google Scholar 

  28. Goemans, M.X., Li, L., Mirrokni, V.S., Thottan, M.: Market sharing games applied to content distribution in Ad Hoc networks. IEEE J. Sel. Areas Commun. 24(5), 1020–1033 (2006)

    Article  Google Scholar 

  29. Gopalakrishnan, R., Kanoulas, D., Karuturi, N., Pandu Rangan, C., Rajaraman, R., Sundaram, R.: Cache me if you can: Capacitated selfish replication games. In: Latin American symposium on theoretical informatics (LATIN), vol. 7256, pp. 420-432 (2012)

    Chapter  Google Scholar 

  30. Gribble, S.D., Halevy, A.Y., Ives, Z.G., Rodrig, M., Suciu, D.: What can database do for peer-to-peer?. In: WebDB workshop on databases and the web (2001)

  31. Hu, X., Gong, J.: PhD forum: Not so cooperative caching. In: 21st IEEE international conference on network protocols (ICNP), pp. 1–3. https://doi.org/10.1109/ICNP.2013.6733656 (2013)

  32. Hu, X.Y., Gong, J.: Study on the theoretical framework of not so cooperative caching. J. Internet Technol. 15(3), 351–362 (2014). https://doi.org/10.6138/JIT.2014.15.3.04

    Google Scholar 

  33. Iyer, S., Rowstron, A.I.T., Druschel, P.: Squirrel: a decentralized peer-to-peer web cache. In: PODC (2002)

  34. Jain, K., Vazirani, V.V.: Primal-dual approximation algorithms for metric facility location and k-median problems. In: 40th annual symposium on foundations of computer science (Cat. No.99CB37039), pp. 2–13. https://doi.org/10.1109/SFFCS.1999.814571(1999)

  35. Jamin, S., Jin, C., Jin, Y., Raz, D., Shavitt, Y., Zhang, L.: On the placement of internet instrumentation. In: Proceedings IEEE INFOCOM 2000. Conference on computer communications. 19th annual joint conference of the ieee computer and communications societies (Cat. No.00CH37064), vol. 1, pp. 295-304. https://doi.org/10.1109/INFCOM.2000.832199 (2000)

  36. Jamin, S., Jin, C., Kurc, A.R., Raz, D., Shavitt, Y.: Constrained mirror placement on the internet. In: Proceedings IEEE INFOCOM 2001. Conference on computer communications. 20th annual joint conference of the ieee computer and communications society (Cat. No.01CH37213), vol. 1, pp. 31–40. https://doi.org/10.1109/INFCOM.2001.916684 (2001)

  37. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How Easy is Local Search? J. Comput. Syst. Sci. 37(1), 79–100 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  38. Karger, D., Lehman, E., Leighton, T., Panigrahy, R., Levine, M., Lewin, D.: Consistent hashing and random trees: Distributed caching protocols for relieving hot spots on the world wide web. In: Proceedings of the 29th annual ACM symposium on theory of computing (STOC), pp. 654–663 (1997)

  39. Kintali, S., Poplawski, L.J., Rajaraman, R., Sundaram, R., Teng, S.H.: Reducibility among fractional stability problems. Proceedings of the 50th annual IEEE symposium on foundations of computer science (FOCS) (2009)

  40. Ko, B.J., Rubenstein, D.: Distributed self-stabilizing placement of replicated resources in emerging networks. IEEE/ACM Trans. Netw. 13(3), 476–487 (2005). https://doi.org/10.1109/TNET.2005.850196

    Article  Google Scholar 

  41. Korupolu, M., Plaxton, C.G., Rajaraman, R.: Placement algorithms for hierarchical cooperative caching. J. Algorithms 38, 260–302 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Korupolu, M.R., Dahlin, M.: Coordinated placement and replacement for large-scale distributed caches. IEEE Trans. Knowl. Data Eng. 14(6), 1317–1329 (2002)

    Article  Google Scholar 

  43. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: STACS, pp. 404–413 (1999)

    Chapter  Google Scholar 

  44. Kubiatowicz, J., Bindel, D., Chen, Y., Czerwinski, S., Eaton, P., Geels, D., Gummadi, R., Rhea, S., Weatherspoon, H., Weimer, W., Wells, C., Zhao, B.: Oceanstore: An architecture for global-scale persistent storage. ACM SIGPLAN Not. 35(11), 190–201 (2000). https://doi.org/10.1145/356989.357007

    Article  Google Scholar 

  45. Laoutaris, N., Smaragdakis, G., Bestavros, A., Stavrakakis, I.: Mistreatment in distributed caching groups: Causes and implications. In: INFOCOM (2006)

  46. Laoutaris, N., Telelis, O., Zissimopoulos, V., Stavrakakis, I.: Distributed selfish replication. IEEE Trans. Parallel Distrib. Syst. 17(12), 1401–1413 (2006)

    Article  Google Scholar 

  47. Laoutaris, N., Smaragdakis, G., Oikonomou, K., Stavrakakis, I., Bestavros, A.: Distributed placement of service facilities in large-scale networks. In: INFOCOM, pp. 2144–2152 (2007)

  48. Leff, A., Wolf, J.L., Yu, P.S.: Replication algorithms in a remote caching architecture. IEEE Trans. Parallel Distrib. Syst. 4(11), 1185–1204 (1993)

    Article  MATH  Google Scholar 

  49. Li, B., Golin, M.J., Italiano, G.F., Deng, X., Sohraby, K.: On the optimal placement of web proxies in the internet. In: IEEE INFOCOM ’99. Conference on computer communications. Proceedings. 18th annual joint conference of the IEEE computer and communications societies. The future is now (Cat. No.99CH36320), vol. 3, pp. 1282–1290. https://doi.org/10.1109/INFCOM.1999.752146 (1999)

  50. Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Jansen, K, Leonardi, S, Vazirani, V (eds.) Approximation algorithms for combinatorial optimization, pp 229–242. Springer, Berlin (2002)

    Chapter  MATH  Google Scholar 

  51. McCuaig, W., Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. In: STOC, pp. 402–405 (1997)

  52. Mettu, R.R., Plaxton, C.G.: The online median problem. In: Proceedings of the 41st annual symposium on foundations of computer science. FOCS ’00, p 339. IEEE Computer Society, Washington (2000)

  53. Nisan, N., Roughgarden, T., Tardos, É., Vazirani VV: Algorithmic game theory. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  54. Osborne, M.J., Rubinstein, A.: A course in game theory. MIT Press (1994)

  55. Pacifici, V.: Resource allocation in operator-owned content delivery systems. KTH School of Electrical Engineering, PhD thesis (2016)

    Google Scholar 

  56. Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. JCSS 48(3), 498–532 (1994)

    MathSciNet  MATH  Google Scholar 

  57. Pollatos, G.G., Telelis, O., Zissimopoulos, V.: On the social cost of distributed selfish content replication. In: Networking, pp. 195–206 (2008)

  58. Qiu, L., Padmanabhan, V.N., Voelker, G.M.: On the placement of web server replicas. In: Proceedings IEEE INFOCOM 2001. Conference on computer communications. 20th annual joint conference of the IEEE computer and communications society (Cat. No.01CH37213), vol. 3, pp. 1587-1596. https://doi.org/10.1109/INFCOM.2001.916655 (2001)

  59. Rabin, M.: Efficient dispersal of information for security, load balancing and fault tolerance. J. ACM 36, 335–348 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  60. Rabinovich, M., Rabinovich, I., Rajaraman, R., Aggarwal, A.: A dynamic object replication and migration protocol for an internet hosting service. In: Proceedings. 19th IEEE international conference on distributed computing systems (Cat. No.99CB37003), pp. 101–113. https://doi.org/10.1109/ICDCS.1999.776511 (1999)

  61. Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Annals of Mathematics, pp. 929–975 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  62. Rosenwein, M.B.: Discrete location theory. In: Mirchandani, P.B., Francis, R.L. (eds.) . Networks 24(2):124–125 https://doi.org/10.1002/net.3230240212, p 555. Wiley, New York (1990)

    Article  Google Scholar 

  63. Rowstron, A., Druschel, P.: Storage management and caching in past, a large-scale, persistent peer-to-peer storage utility. In: Proceedings of the 18th ACM symposium on operating systems principles, SOSP ’01. https://doi.org/10.1145/502034.502053, pp 188–201. ACM, New York (2001)

  64. Saito, Y., Karamanolis, C., Karlsson, M., Mahalingam, M.: Taming aggressive replication in the pangaea wide-area file system. SIGOPS Oper. Syst. Rev. 36(SI), 15–30 (2002). https://doi.org/10.1145/844128.844131

    Article  Google Scholar 

  65. Schrijver, A.: Combinatorial optimization (3 Vol.) Springer-Verlag, Berlin (2003)

    MATH  Google Scholar 

  66. Shokrollahi, A.: Raptor codes. In: IEEE Trans Inf Theory, pp. 2551–2567 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  67. Tang, X., Chanson, S.T.: Coordinated en-route web caching. IEEE Trans. Comput. 51(6), 595–607 (2002). https://doi.org/10.1109/TC.2002.1009146

    Article  Google Scholar 

  68. Tewari, R., Dahlin, M., Vin, H.M., Kay, J.S.: Design Considerations for Distributed Caching on the Internet. In: ICDCS, pp 273–284 (1999)

  69. Tsaknakis, H., Spirakis, P.G., Kanoulas, D.: Performance evaluation of a descent algorithm for bi-matrix games. In: Internet and network economics, 4th international workshop, WINE 2008, Shanghai, China, December 17-20, 2008. Proceedings, pp 222–230. https://doi.org/10.1007/978-3-540-92185-1_29 (2008)

    Google Scholar 

  70. Vetta, A.: Nash equilibria in competitive societies, with applications to facility location traffic routing and auctions. In: FOCS (2002)

  71. Wolfson, O., Jajodia, S., Huang, Y.: An Adaptive Data Replication Algorithm. ACM Trans. Database Syst. 22, 255–314 (1997)

    Article  Google Scholar 

  72. Younger, D.H.: Graphs with interlinked directed circuits. In: Proceedings of midwestern symposium on circuit theory, vol. 2, pp. XVI2.1–XVI2.7 (1973)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering Chennai, Indian Institute of Technology Madras, Play Field Avenue, Indian Institute Of Technology, Chennai, TN, 600036, India

    Ragavendran Gopalakrishnan, Naga Naresh Karuturi & C. Pandu Rangan

  2. Humanoids and Human-Centered Mechatronics Lab, Istituto Italiano di Tecnologia (IIT), Via Morego, 30, Genoa, 16163, Italy

    Dimitrios Kanoulas

  3. School of Computer and Information Science, Northeastern University, 440 Huntington Avenue 240 West Village H, Boston, MA, 02115, USA

    Rajmohan Rajaraman & Ravi Sundaram

Authors
  1. Ragavendran Gopalakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitrios Kanoulas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Naga Naresh Karuturi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Pandu Rangan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Rajmohan Rajaraman
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ravi Sundaram
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dimitrios Kanoulas.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Gopalakrishnan and Karuturi were partially supported by a generous gift from Northeastern University alumnus Madhav Anand. This work was also partially supported by NSF grants CCF-0635119 and CNS-0915985. A preliminary version of this work appeared in LATIN 2012 [29].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gopalakrishnan, R., Kanoulas, D., Karuturi, N.N. et al. Cache Me if You Can: Capacitated Selfish Replication Games in Networks. Theory Comput Syst 64, 272–310 (2020). https://doi.org/10.1007/s00224-019-09939-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-019-09939-7

Keywords

Search

Navigation