走向k服务器猜想：一个统一的潜力，推动边界到圆
Towards the k-server conjecture: A unifying potential, pushing the frontier to the circle
摘要：
$k$ -服务器猜想，首先由Manasse，McGeoch和Sleator在1988年提出，指出$k$ -服务器问题存在一个$k$ -竞争确定性算法.据证实，功函数算法（WFA）实现了这一保证，一个多用途的算法与应用程序的各种在线问题。这已经在几种特殊情况下得到了证明：$k=2$、$(k+1)$ -点度量、$(k+2)$ -点度量、线度量、加权星度量和曼哈顿平面中的$k=3$。这些结果的已知证明是基于与每个特定特殊情况相关联的势函数，因此需要六种不同的势函数用于六种情况。我们提出了一个单一的潜在功能证明$k$ -竞争力的WFA的所有这些情况下。我们还使用这种潜力，以显示$k$ -竞争力的WFA的多射线空间和$k=3$的树木。虽然已知DoubleCoverage算法在后一种情况下具有$k$竞争力，但它对WFA是开放的。我们的潜力捕捉一种类型的懒惰的对手，从而表明，在所有解决的情况下，最坏的情况下的对手是懒惰的。Chrobak和Larmore在1992年提出，一个潜在的捕获懒惰的对手将解决$k$ -服务器猜想。令我们非常惊讶的是，事实并非如此，因为我们（使用与$k$ -taxi问题的连接）表明，我们的潜力对于圆上的三个服务器失败。因此，我们的潜力突出了对手的懒惰，这是所有已解决案件所共有的基本属性，但一般都受到侵犯。一方面，这削弱了我们对$k$ -服务器猜想有效性的信心。另一方面，如果$k$ -服务器猜想成立，那么我们相信它可以通过我们的势的变体来证明。
英文摘要：
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 DoubleCoverage 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.
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