AI 中文总结
针对锦标赛加权反馈顶点集问题，提出确定性 $(2+\varepsilon)$-近似算法，结合链分解动态规划与局部比率归约，实现 $n^{2^{O(1/\varepsilon)}}$ 运行时间，并精确求解 $\mathcal B_7$-自由情形。
AI 中文摘要
我们研究锦标赛中的加权反馈顶点集问题。对于每个固定的整数 $k\geq 2$，我们给出一个确定性 $(2+1/k)$-近似算法，其运行时间为 $n^{2^{O(k)}}$，此外还依赖于权重编码长度的多项式。因此，对于每个固定的 $\varepsilon>0$，锦标赛中的加权反馈顶点集具有确定性 $(2+\varepsilon)$-近似，运行时间为 $n^{2^{O(1/\varepsilon)}}$。该算法结合了两个要素。当锦标赛的三角形图有界团数时，其传递补图的链分解产生一个最大权重传递子锦标赛的精确动态规划。当团数较大时，三角形图的结构定理提供一个常数大小的强好代价向量。使用该代价向量的局部比率归约给出所声称的保证。作为副产品，该动态规划在 $\mathcal B_7$-自由锦标赛中精确求解加权反馈顶点集，时间为 $O(n^7)$，其中 $\mathcal B_7$ 是反馈顶点集数至少为三的七顶点锦标赛族。
英文摘要
We study the weighted feedback vertex set problem in tournaments. For every fixed integer $k\geq 2$, we give a deterministic $(2+1/k)$-approximation algorithm with running time $n^{2^{O(k)}}$, apart from polynomial dependence on the encoding length of the weights. Consequently, for every fixed $\varepsilon>0$, weighted feedback vertex set in tournaments has a deterministic $(2+\varepsilon)$-approximation running in time $n^{2^{O(1/\varepsilon)}}$. The algorithm combines two ingredients. When the triangle graph of the tournament has bounded clique number, a chain decomposition of its transitive complement yields an exact dynamic program for a maximum-weight transitive subtournament. When the clique number is large, a structural theorem for triangle graphs supplies a constant-size strongly good cost vector. A local-ratio reduction with this cost vector gives the claimed guarantee. As a by-product, the dynamic program solves weighted feedback vertex set exactly in $\mathcal B_7$-free tournaments in time $O(n^7)$, where $\mathcal B_7$ is the family of seven-vertex tournaments with feedback vertex set number at least three.
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