发表机构
University of Bremen（不来梅大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对排除小图类，提出确定性常数轮 LOCAL 算法，实现支配集近似比接近最优，并解决平面图下确界为 7 的开放问题。
AI 中文摘要
对于每个固定的真小闭类 $\mathscr C$ 和每个 $\epsilon>0$，我们给出一个确定性 LOCAL 算法，该算法在 $\mathscr C$ 中的每个图 $G$ 上返回一个大小至多为 $(2a(\mathscr C)+1+\epsilon)\gamma_f(G)$ 的支配集。这里 $a(\mathscr C)$ 是 $\mathscr C$ 中边顶点比的上确界，$\gamma_f(G)$ 是分数支配数。该类还承认分数支配集的确定性 $(1+\epsilon)$-近似，以及一个总是返回支配集且期望大小至多为 $(1+\epsilon)\gamma(G)$ 的随机算法。在每种情况下，轮数仅依赖于 $\mathscr C$ 和 $\epsilon$。这些算法都不需要顶点数或最大度作为输入的一部分。对于平面图，这给出了确定性保证 $(7+\epsilon)\gamma_f(G)$。结合 Hilke、Lenzen 和 Suomela 的下界，这确定了平面最小支配集的确定性常数轮近似比的下确界为 $7$，解决了自他们工作以来一直悬而未决的问题。相应的下确界，相对于整数最优值，对于欧拉亏格至多任意固定 $g\ge0$ 的图为 $7$，对于 $K_t$-小自由图（$3\le t\le9$）为 $2t-3$，对于树宽或路径宽至多任意固定 $r\ge1$ 的图为 $2r+1$。我们还证明，对于每个整数 $r\ge1$，没有确定性常数轮 LOCAL 算法在路径的 $r$ 次幂上达到低于 $2r+1$ 的近似比，即使每个顶点知道顶点数。这给出了一个新的证明，即对于平面图、有界树宽或路径宽的图以及 $K_t$-小自由图（$3\le t\le9$），极限常数是最优的。对于无三角形平面图，相应的下确界为 $5$。
英文摘要
For every fixed proper minor-closed class $\mathscr C$ and every $ε>0$, we give a deterministic LOCAL algorithm that returns a dominating set of size at most $(2a(\mathscr C)+1+ε)γ_f(G)$ on every $G\in\mathscr C$. Here $a(\mathscr C)$ is the supremum edge-to-vertex ratio in $\mathscr C$, and $γ_f(G)$ is the fractional domination number. The class also admits a deterministic $(1+ε)$-approximation for fractional dominating set and a randomized algorithm that always returns a dominating set and has expected size at most $(1+ε)γ(G)$. In each case, the number of rounds depends only on $\mathscr C$ and $ε$. None of these algorithms requires the number of vertices or the maximum degree as part of the input. For planar graphs, this gives the deterministic guarantee $(7+ε)γ_f(G)$. Together with the lower bound of Hilke, Lenzen and Suomela, it determines the infimum of the deterministic constant-round approximation ratios for planar minimum dominating set as $7$, settling a question that had remained open since their work. The corresponding infima, measured against the integral optimum, are $7$ for graphs of Euler genus at most any fixed $g\ge0$, $2t-3$ for $K_t$-minor-free graphs with $3\le t\le9$, and $2r+1$ for graphs of treewidth or pathwidth at most any fixed $r\ge1$. We also prove that, for every integer $r\ge1$, no deterministic constant-round LOCAL algorithm achieves an approximation ratio below $2r+1$ on the $r$-th powers of paths, even when every vertex knows the number of vertices. This gives a new proof that the limiting constants are optimal for planar graphs, graphs of bounded treewidth or pathwidth, and $K_t$-minor-free graphs with $3\le t\le9$. For triangle-free planar graphs, the corresponding infimum is $5$.
