Abstract
We study the performance of a best reply algorithm for online resource allocation problems with a diseconomy of scale. In an online resource allocation problem, we are given a set of resources and a set of requests that arrive in an online manner. Each request consists of a set of feasible allocations and an allocation is a set of resources. The total cost of an allocation vector is given by the sum of the resources’ costs, where each resource’s cost depends on the total load on the resource under the allocation vector. We analyze the natural online procedure where each request is allocated greedily to a feasible set of resources that minimizes the individual cost of that particular request. In the literature, this algorithm is also known as a one-round walk in congestion games starting from the empty state. For unweighted resource allocation problems with polynomial cost functions with maximum degree d, upper bounds on the competitive ratio of this greedy algorithm were known only for the special cases \(d\in \{1, 2, 3\}\). In this paper, we show a general upper bound on the competitive ratio of \(d(d / W(\frac{1.2d-1}{d+1}))^{d+1}\) for the unweighted case where W denotes the Lambert-W function on \(\mathbb {R}_{\ge 0}\). For the weighted case, we show that the competitive ratio of the greedy algorithm is bounded from above by \((d/W(\frac{d}{d+1}))^{d+1}\).
A. Tönnis—Partially supported by CONICYT grant PCI PII 20150140 and ERC Starting Grant 306465 (BeyondWorstCase).
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References
Aland, S., Dumrauf, D., Gairing, M., Monien, B., Schoppmann, F.: Exact price of anarchy for polynomial congestion games. SIAM J. Comput. 40, 1211–1233 (2011)
Albers, S.: Energy-efficient algorithms. Commun. ACM 53(5), 86–96 (2010)
Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. SIAM J. Comput. 42(1), 160–177 (2013)
Awerbuch, B., Azar, Y., Epstein, A., Mirrokni, V.S., Skopalik, A.: Fast convergence to nearly optimal solutions in potential games. In: Proceedings of the 9th ACM Conference on Electronic Commerce (EC), pp. 264–273 (2008)
Awerbuch, B., Azar, Y., Grove, E.F., Kao, M., Krishnan, P., Vitter, J.S.: Load balancing in the l\(_{\rm p}\) norm. In: Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 383–391 (1995)
Bilò, V.: A unifying tool for bounding the quality of non-cooperative solutions in weighted congestion games. Theory Comput. Syst. 62(5), 1288–1317 (2018). https://doi.org/10.1007/s00224-017-9826-1
Bilò, V., Fanelli, A., Flammini, M., Moscardelli, L.: Performance of one-round walks in linear congestion games. Theory Comput. Syst. 49(1), 24–45 (2011)
Bilò, V., Vinci, C.: On the impact of singleton strategies in congestion games. In: Proceedings of the 25th Annual European Symposium on Algorithms (ESA), pp. 17:1–17:14 (2017)
Bjelde, A., Klimm, M., Schmand, D.: Brief announcement: approximation algorithms for unsplittable resource allocation problems with diseconomies of scale. In: Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 227–229 (2017)
Caragiannis, I.: Better bounds for online load balancing on unrelated machines. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 972–981 (2008)
Caragiannis, I., Flammini, M., Kaklamanis, C., Kanellopoulos, P., Moscardelli, L.: Tight bounds for selfish and greedy load balancing. Algorithmica 61(3), 606–637 (2011)
Christodoulou, G., Gairing, M.: Price of stability in polynomial congestion games. ACM Trans. Econ. Comput. 4(2), 10:1–10:17 (2016)
Christodoulou, G., Gairing, M., Giannakopoulos, Y., Spirakis, P.G.: The price of stability of weighted congestion games. In: Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP), pp. 150:1–150:16 (2018)
Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pp. 67–73 (2005)
Christodoulou, G., Mirrokni, V.S., Sidiropoulos, A.: Convergence and approximation in potential games. Theor. Comput. Sci. 438, 13–27 (2012)
Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 604–612 (2004)
Fanelli, A., Flammini, M., Moscardelli, L.: The speed of convergence in congestion games under best-response dynamics. ACM Trans. Algorithms 8(3), 25:1–25:15 (2012)
Farzad, B., Olver, N., Vetta, A.: A priority-based model of routing. Chicago J. Theor. Comput. Sci. 2008 (2008). Article 1
Goemans, M.X., Mirrokni, V.S., Vetta, A.: Sink equilibria and convergence. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 142–154 (2005)
Harks, T., Heinz, S., Pfetsch, M.E.: Competitive online multicommodity routing. Theory Comput. Syst. 45(3), 533–554 (2009)
Harks, T., Heinz, S., Pfetsch, M.E., Vredeveld, T.: Online multicommodity routing with time windows. ZIB Report 07-22, Zuse Institute Berlin (2007)
Harks, T., Klimm, M.: On the existence of pure nash equilibria in weighted congestion games. Math. Oper. Res. 37, 419–436 (2012)
Klimm, M., Schmand, D., Tönnis, A.: The online best reply algorithm for resource allocation problems. CoRR abs/1805.02526. https://arxiv.org/abs/1805.02526
Makarychev, K., Sviridenko, M.: Solving optimization problems with diseconomies of scale via decoupling. In: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 571–580 (2014)
Meyers, C.A., Schulz, A.S.: The complexity of welfare maximization in congestion games. Networks 59(2), 252–260 (2012)
Mirrokni, V.S., Vetta, A.: Convergence issues in competitive games. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) APPROX/RANDOM 2004. LNCS, vol. 3122, pp. 183–194. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27821-4_17
Orlin, J.B., Punnen, A.P., Schulz, A.S.: Approximate local search in combinatorial optimization. SIAM J. Comput. 33(5), 1201–1214 (2004)
Roughgarden, T.: Barriers to near-optimal equilibria. In: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 71–80 (2014)
Roughgarden, T.: Intrinsic robustness of the price of anarchy. J. ACM 62(5), 32:1–32:42 (2015)
Suri, S., Tóth, C.D., Zhou, Y.: Selfish load balancing and atomic congestion games. Algorithmica 47(1), 79–96 (2007)
U.S. Bureau of Public Roads: Traffic assignment manual. U.S. Department of Commerce, Urban Planning Division, Washington, DC (1964)
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Klimm, M., Schmand, D., Tönnis, A. (2019). The Online Best Reply Algorithm for Resource Allocation Problems. In: Fotakis, D., Markakis, E. (eds) Algorithmic Game Theory. SAGT 2019. Lecture Notes in Computer Science(), vol 11801. Springer, Cham. https://doi.org/10.1007/978-3-030-30473-7_14
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