最长队列丢弃竞争比的新界：1.46929591 <= CR(LQD) <= 1.683652
New Bounds on the Competitive Ratio of Longest Queue Drop: 1.46929591 <= CR(LQD) <= 1.683652
- St. Petersburg Department of the Steklov Institute of Mathematics（圣彼得堡列别捷夫数学研究所圣彼得堡分部）
- St. Petersburg State University（圣彼得堡国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究改进最长队列丢弃（LQD）策略的竞争比界，通过前置负载族下界和连续包络松弛上界，将界从[1.44546086, 1.6918]收紧至[1.46929591, 1.683652]，并修复了先前证明中的漏洞。
AI中文摘要：
我们研究了最长队列丢弃（LQD）的竞争比，这是共享内存交换机中经典的缓冲区管理策略，此前发表的界为 CR(LQD) 属于 [1.44546086, 1.6918]。我们改进了上下界。对于下界，我们在一个新的实例族——前置负载族上提供了一个精确整数证书，该证书给出 CR(LQD) >= 184815365566285 / 125784985866185 = 1.46929591...，这是在特定有限实例上针对精确最优离线策略评估得到的精确比率。该实例在一种固定平局规则下评估，但该界不依赖于平局规则：一个自适应耦合将该证书值传递到每个非预知确定性平局规则，以及自适应对手意义下的每个随机平局规则。对于上界，我们证明 CR(LQD) <= 1.683652，方法是将 Antoniadis 等人（2024）终局中的逐数据包端点松弛替换为相同支付表达式的连续包络松弛，我们以闭式精确求解了该松弛；与下界不同，此上界对每种平局规则均成立。在此过程中，我们发现了已发表证明的聚合步骤（其引理18）推导中的一个漏洞。我们用一个摊销引理和精确的有限头部分析修复了聚合；该修复恢复了已发表的 1.6918，进而恢复了 Antoniadis 等人（ICALP 2021）较弱的会议保证 1.707，并且也支持了我们进一步的改进。
英文摘要：
We study the competitive ratio of Longest Queue Drop (LQD), the canonical buffer management policy for shared memory switches, for which the previously published bounds were CR(LQD) in [1.44546086, 1.6918]. We improve both ends. For the lower bound, we provide an exact-integer certificate on a new instance family, the front-loaded family, which gives $\mathrm{CR}(\mathrm{LQD}) >= 184815365566285 / 125784985866185 = 1.46929591...$, the exact ratio on a specific finite instance evaluated against the exactly optimal offline policy. The instance is evaluated under one fixed tie rule, but the bound does not depend on the tie rule: an adaptive coupling transfers the certified value to every non-clairvoyant deterministic tie rule, and to every randomized tie rule in the adaptive adversary sense. For the upper bound, we prove that CR(LQD) <= 1.683652 by replacing the per-packet endpoint relaxation in the endgame of Antoniadis et al. (2024) with the continuum envelope relaxation of the same payment expression, which we solve exactly in closed form; unlike the lower bound, this upper bound holds for every tie rule. In the process, we find a gap in the derivation of the published proof's aggregation step (their Lemma 18). We repair the aggregation with one amortized lemma and an exact finite-head analysis; the repair restores the published 1.6918, which in turn restores the weaker conference guarantee 1.707 of Antoniadis et al. (ICALP 2021), and it also supports our further improvement.
