# Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue

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4 Lower Bound Upper Bound Lower Bound Upper Bound d = d = d = d = d = d Fig. 1. Summary of upper and lower bounds on the competitive ratio for certain values of d. As a first step, we resolve this question positively in a slightly simpler setting, which we call the allocation problem. In the allocation problem, the seller introduces the products one-by-one and sets a fixed price b(j) for each product j. Upon arrival of a product, each buyer announces whether it is interested in buying it for the set price and the seller decides (instantly) to which of the interested buyers to sell the product. We have indications that solving the more general ad-auctions problem requires overcoming a few additional obstacles. Nevertheless, achieving better competitive factors for the allocation problem is a necessary non-trivial step. We design an online algorithm with competitive ratio C(d) = 1 d 1 d(1+ 1 d 1) d 1. This factor is strictly better than 1 1/e for any value of d, and approaches (1 1/e) from above as d goes to infinity. We also prove lower bounds for the problem that indicate that the competitive factor of our online algorithm is quite tight. Our improved bounds for certain values of d are shown in Figure 1. Our improved competitive factors are obtained via a new approach. Our algorithm is composed of two conceptually separate phases that run simultaneously. The first phase generates online a fractional solution for the problem. A fractional solution for the problem allows the algorithm to sell each product in fractions to several buyers. This problem has a motivation of its own in case products can be divided between buyers. An example of a divisible product is the allocation of bandwidth in a communication network. This part of our algorithm that generates a fractional solution in an online fashion is somewhat counter-intuitive. In particular, a newly arrived product is not split equally between buyers who have spent the least fraction of their budget. Such an algorithm is referred to as a water level algorithm and it is not hard to verify that it does not improve upon the (1 1/e) worst case ratio, even for small values of d. Rather, the idea is to split the product between several buyers that have approximately spent the same fraction of their total budget. The analysis is performed through (online) linear programming dual fitting: we maintain during each step of the online algorithm a dual fractional solution that bounds the optimum solution from above. We also remark that this part of the algorithm yields a competitive solution even when the prices of the products are large compared with the budgets of the buyers. As a special case, the first phase implies a C(d)-competitive algorithm for the online maximum fractional matching problem in bounded degree bipartite graphs [13]. The second phase consists of rounding the fractional solution (obtained in the first phase) in an online fashion. We note again that this is only a conceptual phase which is simultaneously implemented with the previous phase. This step can be easily done by using randomized rounding. However, we show how to perform the rounding deterministically by constructing a suitable potential function. The potential function is inspired by the pessimistic estimator used to derandomize the offline problem. We show that if

7 primal cost is at most 1 + 1/() the change in the dual profit. The value of c is then maximized such that both the primal and the dual solutions remain feasible. Theorem 1. The allocation algorithm is (1 1/c) (1 R max )-competitive, where c = 1 (1 + R max ) Rmax. When Rmax 0 the competitive ratio tends to (1 1/e). Proof. We prove three simple claims: 1. The algorithm produces a primal feasible solution. 2. In each iteration: (change in primal objective function)/ (change in dual objective function) c The algorithm produces an almost feasible dual solution. Proof of (1): Consider a primal constraint corresponding to buyer i and product j. If x(i) 1 then the primal constraint is satisfied. Otherwise, the algorithm allocates the product to the buyer i for which b(i, j)(1 x(i )) is maximized. Setting z(j) = b(i, j)(1 x(i )) guarantees that the constraint is satisfied for all (i, j). Subsequent increases of the variables x(i) s cannot make the solution infeasible. Proof of (2): Whenever the algorithm updates the primal and dual solutions, the change in the dual profit is b(i, j). (Note that even if the remaining budget of buyer i to which product j is allocated is less than its bid b(i, j), variable y(i, j) is still set to 1.) The change in the primal cost is: x(i) + z(j) = ( b(i, j)x(i) + ) ( b(i, j) + b(i, j)(1 x(i)) = b(i, j) ). Proof of (3): The algorithm never updates the dual solution for buyers satisfying x(i) 1. We prove that for any buyer i, when j M b(i, j)y(i, j), then x(i) 1. This is done by proving that: x(i) 1 ( ) j M c b(i,j)y(i,j) 1. (1) Thus, whenever j M b(i, j)y(i, j), we get that x(i) 1. We prove (1) by induction on the (relevant) iterations of the algorithm. Initially, this assumption is trivially true. We are only concerned with iterations in which a product, say k, is sold to buyer i. In such an iteration we get that: ( ) b(i, k) x(i) end = x(i)start [ = 1 1 c b(i, k) j M\{k} b(i,j)y(i,j) 1 [ ( j M\{k} b(i,j)y(i,j) c 1 + [ c j M\{k} b(i,j)y(i,j) c () ] ( b(i, k) 1 + ) ] b(i, k) 1 ( ) ] b(i,k) 1 ) + b(i, k) () = 1 [ j M b(i,j)y(i,j) c (2) ] 1 (3) where Inequality (2) follows from the induction hypothesis, and Inequality (3) follows since, for any 0 x y 1, ln(1+x) x ln(1+y) y. Note that when b(i,k) = R max

12 that is strictly more than zero. For instance, a buyer is always willing to buy an item if he only needs to pay 10% of its value (as estimated by the buyer s bid). Our modified algorithm for this more general setting takes advantage of this fact to improve the worst case competitive ratio. In particular, let a min = min n i=1{a(i, r i )} be the minimum aggression level of the buyers, then the competitive factor of the algorithm e 1 is e a min. If the minimum level is only 10% (0.1), for example, the competitive ratio is 0.656, compared with /e of the basic ad-auctions algorithm. Let R max = max i I,j M,1 r ri 1{ a(i,r)b(i,j) B(i,r) } be the maximum ratio between a charge to a budget and the total budget. The modified linear program for the more general risk management setting along with our modified algorithm are deferred to the full version. ( ) c 1 Theorem 5. The algorithm is c a min (1 R max )-competitive, where c tends to e when R max 0. Acknowledgements: We thank Allan Borodin for pointing to us the mutiple slot setting. References 1. N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, and J. Naor. The online set cover problem. In Proceedings of the 35th STOC, pp , N. Andelman and Y. Mansour. Auctions with budget constraints. In Proc. of the 9th Scandinavian Workshop on Algorithm Theory, pp , J. Aspnes, Y. Azar, A. Fiat, S. Plotkin, and O. Waarts. On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM, 44(3): , B. Awerbuch, Y. Azar, and S. Plotkin. Throughput-competitive online routing. In Proc. of the 34th Annual Symposium on Foundations of Computer Science, pp , A. Blum and J. Hartline. Near-optimal online auctions C. Borgs, J. Chayes, N. Immorlica, M. Mahdian, and A. Saberi. Multi-unit auctions with budget-constrained bidders. In Proc. of the 6th EC, pp , N. Buchbinder and J. Naor. Online primal-dual algorithms for covering and packing problems. In Proc. of the 13th Annual European Symposium on Algorithms, pp , N. Buchbinder and J. Naor. A primal-dual approach to online routing and packing. In Proc. of the 47th Annual Symposium on Foundations of Computer Science, , D. R. Dooly, S. Goldman, and S. D. Scott. On-line analysis of the TCP acknowledgement delay problem. Journal of the ACM, 48: , A. Goel, A. Meyerson, and S. A. Plotkin. Combining fairness with throughput: Online routing with multiple objectives. J. Comput. Syst. Sci., 63(1):62-79, B. Kalyanasundaram and K. R. Pruhs. An optimal deterministic algorithm for online b - matching. Theoretical Computer Science, 233(1-2): , A. R. Karlin, C. Kenyon, and D. Randall. Dynamic TCP acknowledgement and other stories about e/(e-1). In Proc. of the 33rd Symposium on Theory of Computing, pp , R. Karp, U. Vazirani, and V. Vazirani. An optimal algorithm for online bipartite matching. In In Proceedings of the 22nd Annual Symposium on Theory of Computing, pp , M. Mahdian and A. Saberi. Multi-unit auctions with unknown supply. In EC 06: Proceedings of the 7th ACM conference on Electronic commerce, pp , A. Mehta, A. Saberi, U. Vazirani, and V. Vazirani. Adwords and generalized on-line matching. In Proc. of the 46th IEEE Symp. on Foundations of Computer Science, , S. Robinson. Computer scientists optimize innovative ad auction. SIAM News, 38: , 2005.

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