AI 中文总结
研究有向图中最小权重反馈弧集的最优常数，结合顶点剥离与随机排序分析确定最优常数为\((\frac{1}{2}-\frac{\sqrt{2}}{6\sqrt{\Delta}})w(D)\)，并给出更强结果，还得到能达到此界的随机近线性时间算法。
AI 中文摘要
设\(D\)为最大度\(\Delta\geq1\)的有向图（无有向\(2 -\)圈的有向图），配备总权重为\(w(D)\)的非负弧权重，\(\mathrm{fas}_w(D)\)表示\(D\)的反馈弧集的最小权重。Alon（2002）证明了\(\mathrm{fas}_w(D)\leq(\frac{1}{2}-\frac{1}{16\sqrt{2\Delta}})w(D)\)。本文确定了最优常数：\(\mathrm{fas}_w(D)\leq(\frac{1}{2}-\frac{\sqrt{2}}{6\sqrt{\Delta}})w(D)\)。实际上，还展示了更强的结果：\(\mathrm{fas}_w(D)\leq\frac{1}{2}w(D)-\frac{\sqrt{2}}{12}\sum_v w_2(v)\)，其中\(w_2(v)\)是与\(v\)关联的弧权重的\(\ell_2 -\)范数。两个界都由单位权重有向三角形达到，所以常数\(\sqrt{2}/6\)是最优的。证明结合了Berger和Shor的顶点剥离方案与连续随机排序分析，还得到了一个期望达到界的随机近线性时间算法。
英文摘要
Let $D$ be an oriented graph (a digraph with no directed 2-cycles) with maximum degree $Δ\ge 1$, equipped with nonnegative arc weights of total weight $w(D)$, and let $\mathrm{fas}_w(D)$ denote the minimum weight of a feedback arc set of $D$. Alon (2002) proved $\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{1}{16\sqrt{2Δ}})w(D)$. We determine the optimal constant: \[\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{\sqrt{2}}{6\sqrtΔ})w(D).\] In fact, we show a stronger result: $\mathrm{fas}_w(D)\le\frac{1}{2}w(D)-\frac{\sqrt{2}}{12}\sum_v w_2(v)$, where $w_2(v)$ is the $\ell_2$-norm of the weights of the arcs incident with $v$. Both bounds are attained by the unit-weight directed triangle, so the constant $\sqrt{2}/6$ is best possible (already among unweighted oriented graphs). The proof combines the vertex-peeling scheme of Berger and Shor with a continuous random-ordering analysis: realizing the random order by independent uniform labels renders the expected local imbalance at each vertex exactly an integrated Khintchine-type functional, and the theorem reduces to the sharp evaluation \[\inf_{\|a\|_2=1}\int_0^1 \mathbb{E}|\sum_j a_j B_j(q)|\,dq = \frac{\sqrt{2}}{6},\] where the $B_j(q)$ are i.i.d. Bernoulli$(q)$ random variables, which we prove via Fourier analysis. The proof also yields a randomized, near-linear-time algorithm attaining the bounds in expectation.
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