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The k-server conjecture, first posed by Manasse, McGeoch and Sleator in 1988, states that a k-competitive deterministic algorithm for the k-server problem exists. It is conjectured that the work function algorithm (WFA) achieves this guarantee, a multi-purpose algorithm with applications to various online problems. This has been shown for several special cases: k = 2, (k + 1)-point metrics, (k + 2)-point metrics, the line metric, weighted star metrics, and k = 3 in the Manhattan plane. The known proofs of these results are based on potential functions tied to each particular special case, thus requiring six different potential functions for the six cases. We present a single potential function proving k-competitiveness of WFA for all these cases. We also use this potential to show k-competitiveness of WFA on multiray spaces and for k = 3 on trees. While the Double Coverage algorithm was known to be k-competitive for these latter cases, it has been open for WFA. Our potential captures a type of lazy adversary and thus shows that in all settled cases, the worst-case adversary is lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the lazy adversary would resolve the k-server conjecture. To our major surprise, this is not the case, as we show (using connections to the k-taxi problem) that our potential fails for three servers on the circle. Thus, our potential highlights laziness of the adversary as a fundamental property that is shared by all settled cases but violated in general. On the one hand, this weakens our confidence in the validity of the k-server conjecture. On the other hand, if the k-server conjecture holds, then we believe it can be proved by a variant of our potential.

We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k-server problem, also known as the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k)-server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

The k-server conjecture, first posed by Manasse, McGeoch and Sleator in 1988, states that a k-competitive deterministic algorithm for the k-server problem exists. It is conjectured that the work function algorithm (WFA) achieves this guarantee, a multi-purpose algorithm with applications to various online problems. This has been shown for several special cases: k = 2, (k + 1)-point metrics, (k + 2)-point metrics, the line metric, weighted star metrics, and k = 3 in the Manhattan plane. The known proofs of these results are based on potential functions tied to each particular special case, thus requiring six different potential functions for the six cases. We present a single potential function proving k-competitiveness of WFA for all these cases. We also use this potential to show k-competitiveness of WFA on multiray spaces and for k = 3 on trees. While the Double Coverage algorithm was known to be k-competitive for these latter cases, it has been open for WFA. Our potential captures a type of lazy adversary and thus shows that in all settled cases, the worst-case adversary is lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the lazy adversary would resolve the k-server conjecture. To our major surprise, this is not the case, as we show (using connections to the k-taxi problem) that our potential fails for three servers on the circle. Thus, our potential highlights laziness of the adversary as a fundamental property that is shared by all settled cases but violated in general. On the one hand, this weakens our confidence in the validity of the k-server conjecture. On the other hand, if the k-server conjecture holds, then we believe it can be proved by a variant of our potential.

We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k-server problem, also known as the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k)-server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

We study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We derive upper and (asymptotically) matching lower bounds on the (strong) price of anarchy of linear bottleneck congestion games for a natural load balancing social cost objective (i.e., minimize the maximum latency of a facility). We restrict our studies to linear latency functions. Linear bottleneck congestion games still constitute a rich class of games and generalize, for example, load balancing games with identical or uniformly related machines with or without restricted assignments.

We study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We derive upper and (asymptotically) matching lower bounds on the (strong) price of anarchy of linear bottleneck congestion games for a natural load balancing social cost objective (i.e., minimize the maximum latency of a facility). We restrict our studies to linear latency functions. Linear bottleneck congestion games still constitute a rich class of games and generalize, for example, load balancing games with identical or uniformly related machines with or without restricted assignments.