The Online Best Reply Algorithm for Resource Allocation Problems

Abstract

We study online resource allocation problems with a diseconomy of scale. In these problems, there are certain requests, each demanding a set of resources, that arrive in an online manner. The cost of each resource is semi-convex and grows superlinearly in the total load on the resource. An irrevocable allocation decision has to be made directly after the arrival of each request with the goal to minimize the total cost on the resources. We focus on two simple greedy online policies that provide very fast and easy approximation algorithms.The first policy is to minimize the individual cost of the current online request with respect to all previous requests that have been allocated before. The second policy is to minimize the marginal total cost over all requests that have arrived up to this point. In the literature, these type of algorithms is also considered as one-round walks in congestion games starting from the empty state.We consider the weighted and unweighted version of the problem. In the weighted variant, and for cost functions that are polynomials with maximal degree d and positive coefficients, we proof a tight competitive ratio of d √ 2 − 1 −(d+1) for the marginal total cost policy. This interestingly exactly matches the approximation factor for the corresponding multiple-round walk algorithm. Our work indicates that one-round walks that start in an empty starting state are exactly as efficient as multiple-round walks. We also show that this does not carry over to the unweighted version of the problem. For unweighted instances, we provide lower bounds for both policies that are significantly larger than the corresponding multiple-round walks. We complement our results with an upper and lower bound on the solution quality of the personal cost policy for weighted and unweighted instances.

The Impact of Selfish Behavior in Load Balancing Games

“…γ W (PoA , W(P(d))) has been already evaluated in [3,23]. The bounds related to the -approximate one-walks generated by selfish players generalize the results obtained in [24], in which the same bounds have been shown for affine latency functions and = 0, only; for more general polynomial latency functions and = 0, the bounds related to the -approximate one-walks have been re-obtained and shown in more detail in [38], subsequently to the preliminary version of our work.…”

“…Row 2 of Figure 4 generalizes row 1 of Figure 2, which holds only with respect to exact one-round walks generated by cooperative players in games with affine latency functions. Row 3 of Figure 4 implies that the upper bounds provided in [24] and [38] for the competitive ratio of exact one-round walks in general congestion games with affine and polynomial latency functions, respectively, are tight even for load balancing games. We stress out that, even for general congestion games, no lower bound for the competitive ratio of exact one-round walks involving selfish players was known prior to this work, except for the case of unweighted games with affine latencies [9].…”

“…For the case of selfish players and still under affine latency functions, [9,24] show that the competitive ratio is 2 + √ 5 for unweighted congestion games, while, for weighted players, [24] gives an upper bound of 4 + 2 √ 3. For the more general case of polynomial latency functions, [38] determines explicit and good upper bounds on the competitive ratio in unweighted and weighted congestion games; prior to [38], analogue results for weighted games with polynomial latency functions have been obtained in the preliminary version of our work [12].…”

“…For more general polynomial latency functions with maximum degree d > 3 and for any ≥ 0, an upper bound on γ U (CR s , U(P(d))) can be trivially obtained by reusing the upper bound shown in Subsection 3.3, that holds for more general weighted games; for exact one-round walks, a better upper bound has been recently provided in [38]. By Remark 5, we can get good lower bounds on the performance of -approximate one-round walks, having a similar representation as in the cases of EM ∈ {PoA , CR c }.…”

Nash Social Welfare in Selfish and Online Load Balancing

“…[14,28] analyse a different online algorithm (usually termed one-round walk starting from the empty state) for load balancing and prove that its competitive ratio is 2 + √ 5 under affine latency functions. Bounds for the case of polynomial latencies are given in [13,16,46], while [15,60] address more general latency functions with respect to atomic and non-atomic congestion games, respectively.…”