AI 中文总结
该研究分析带预测的在线度量匹配中近完美一致性的代价，给出确定性算法的鲁棒性下界与上界，确定双服务器权衡等，推导随机算法的比率界及最优指数。
AI 中文摘要
我们研究带每请求动作预测的在线度量匹配问题。在实线上，每个确定性(1+ε)-一致性算法的鲁棒性至少为1+∑_{j=1}^{k-1}2^{j+1}/ε^j，且我们为任意度量给出了具有相同主导项的确定性算法。因此，对每个固定的k，当ε→0时，ε^{k-1}R_k^R(1+ε)=lim_{ε→0}ε^{k-1}R_k(1+ε)=2^k。对于0<ε≤1/(k-1)，该比较在绝对常数范围内一致。我们确定了两种场景下的双服务器权衡，以及当0<ε≤√13-3时实线上三服务器的值为1+4/ε+8/ε²。当预测标签互异时，算法的代价最多为预测匹配代价的(1+ε)倍。对于随机算法，固定k的依赖仍为Θ_k(1/ε^{k-1})。对k一致而言，鲁棒性最多为M_0(ε)ρ_k^0，其中ρ_k^0是实线上最优的无严格预测的随机比率，M_0(ε)=(2e+o(1))e^{2/ε}。对每个η>0，当k≥C_η/ε时，存在下界exp((2-η)/ε)。随机上界源于两个在线算法的比较定理，其状态可通过累积代价有界耦合。对每个固定c>1，这些假设下的最小比较因子M^*(c,ε)满足lim_{ε→0}εlog M^*(c,ε)=2。该保证为严格乘法形式，无依赖直径的加性项。不可撤销度量匹配和度量任务系统满足这些假设，且指数2在这些假设下是最优的。
英文摘要
We study online metric matching with per-request action predictions. On the real line, every deterministic $(1+\varepsilon)$-consistent algorithm has robustness at least $1+\sum_{j=1}^{k-1}2^{j+1}/\varepsilon^j$, and we give a deterministic algorithm for arbitrary metrics with the same leading term. Thus, for every fixed $k$, $\lim_{\varepsilon\downarrow 0}\varepsilon^{k-1}R_k^{\mathbb{R}}(1+\varepsilon)=\lim_{\varepsilon\downarrow 0}\varepsilon^{k-1}R_k(1+\varepsilon)=2^k$. The comparison is uniform up to an absolute constant for $0<\varepsilon\le 1/(k-1)$. We determine the two-server trade-off in both settings and the real-line three-server value $1+4/\varepsilon+8/\varepsilon^2$ for $0<\varepsilon\le\sqrt{13}-3$. When the predicted labels are distinct, the algorithm pays at most $(1+\varepsilon)$ times the cost of the predicted matching. For randomised algorithms, the fixed-$k$ dependence remains $Θ_k(1/\varepsilon^{k-1})$. Uniformly in $k$, robustness is at most $M_0(\varepsilon)ρ_k^0$, where $ρ_k^0$ is the optimal strict prediction-free randomised ratio on the real line and $M_0(\varepsilon)=(2e+o(1))e^{2/\varepsilon}$. For every $η>0$, a lower bound $\exp((2-η)/\varepsilon)$ holds once $k\ge C_η/\varepsilon$. The randomised upper bound follows from a comparison theorem for two online algorithms whose states can be coupled at a cost bounded by their cumulative costs. For every fixed $c>1$, the least comparison factor $M^*(c,\varepsilon)$ under these assumptions satisfies $\lim_{\varepsilon\downarrow 0}\varepsilon\log M^*(c,\varepsilon)=2$. The guarantee is strictly multiplicative and has no diameter-dependent additive term. Irrevocable metric matching and metrical task systems satisfy the assumptions, and the exponent $2$ is optimal under them.
Comments33 pages
