图平衡问题的Wang-Sitters舍入方案的11/6上确界
The 11/6 supremum of the Wang-Sitters rounding scheme for graph balancing
- Hunter College and the Graduate Center, CUNY（亨特学院及纽约市立大学研究生中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究确定了Wang-Sitters图平衡舍入方案的11/6上确界，推导了不同β范围的最坏情况保证，发现2/3为最优阈值，保证在1/2处跳变。
AI中文摘要：
Wang和Sitters提出的图平衡问题的11/6近似算法并非单一算法，而是一组允许的执行方式：步骤1可返回松弛问题的任意可行解，步骤3可将剩余作业匹配到舍入打开的时隙的任意方式。我们精确确定了该自由度允许的情况：比值可任意接近11/6，但无法达到该值，因此11/6是约束所有允许执行的最小常数，且无任何执行能达到该值。随后我们确定了最坏情况保证作为大作业阈值β的函数（相对于最优解），在Wang和Sitters给出的范围1/2 < β < 1内，该保证恰好为max{3/2 + β/2, 5/2 - β}。我们还将相同的资格规则扩展到β∈(0,1/2]（超出其声明范围，此时大作业的两个份额均可达到阈值，因此步骤2获得第三种选择），并确定了该范围内的保证：恰好为3/2 + (1-β)floor(1/β)，因此当β减小时该值无界。因此，在(0,1)内的每个阈值处的最坏情况比值均已知，且无任何阈值能达到该值。因此，2/3是唯一的最优阈值，且保证在1/2处发生跳变，而非平滑退化。一篇配套说明探讨了什么无法确定该常数。
英文摘要：
Wang and Sitters' 11/6-approximation for graph balancing is not one algorithm but a set of permitted executions: Step 1 may return any feasible solution of the relaxation and Step 3 any of the many ways to match the remaining jobs into the slots the rounding opens. We determine exactly what that latitude permits: ratios arbitrarily close to 11/6, and none reaching it, so 11/6 is the least constant that bounds every permitted run, and no run attains it. We then determine the worst-case guarantee as a function of the big-job threshold beta, measured against the optimum itself. On Wang and Sitters' own range 1/2 < beta < 1 the guarantee is exactly max{3/2 + beta/2, 5/2 - beta}. We then extend the same eligibility rule to 0 < beta <= 1/2 -- outside the range they state, and where a big job's two shares can both reach the threshold, so Step 2 acquires a third choice -- and determine the guarantee there as well: exactly 3/2 + (1-beta)floor(1/beta), hence unbounded as beta falls. The worst-case ratio is therefore known at every threshold in (0,1), and attained at none. Consequently 2/3 is the unique optimal threshold, and the guarantee jumps at one half rather than degrading smoothly. A companion note asks what does not fix the constant.
