分数超树宽的近似计算
Efficient Approximation of Fractional Hypertree Width
AI总结：
本文提出两个新的多项式时间近似算法计算分数超树宽，分别达到O(ω log n log ω)和O(ω log² ω)的近似比，并基于Menger定理的团分离子变体给出关键理论支撑。
摘要：
我们给出了两个新的近似算法，用于计算输入超图的分数超树宽。第一个算法以具有至多ω分数超树宽的n个顶点、m条边的超图H作为输入，在多项式时间内运行，并生成H的一个分数超树宽为O(ω log n log ω)的树分解。作为一个直接推论，这也为（广义）超树宽给出了多项式时间的O(log² n log ω)近似算法。据我们所知，我们的算法是分数超树宽和（广义）超树宽的第一个非平凡多项式时间近似算法，与此前仅在ω被视为常数时才能在多项式时间内运行的算法形成对比。对于具有有界交性质（bounded intersection property）的超图，我们得到更好的界，可与Lanzinger和Razgon [STACS 2024]最近的算法相比。第二个算法在n^ω m^{O(1)}时间内运行，并生成H的一个分数超树宽为O(ω log² ω)的树分解。这显著改进了Marx [ACM TALG 2010]的(n+m)^{O(ω³)}时间算法，该算法生成分数超树宽为O(ω³)的树分解，无论从运行时间还是近似比来看都是如此。我们的主要技术贡献，也是这两个算法背后的关键洞察，是经典Menger定理针对图中团分离子（clique separators）的一个变体：对于每个图G、顶点集A和B、G中团的族F以及正有理数f，要么存在F中O(f · log² n)个团构成的子族，其并集将A与B分离，要么存在从A到B的f · log |F|条路径，使得F中没有任何团与超过log |F|条路径相交。
英文摘要：
We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input $n$-vertex $m$-edge hypergraph $H$ of fractional hypertree width at most $ω$, runs in polynomial time and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log n \log ω)$. As an immediate corollary this yields polynomial time $O(\log^2 n \log ω)$-approximation algorithms for (generalized) hypertree width as well. To the best of our knowledge our algorithm is the first non-trivial polynomial-time approximation algorithm for fractional hypertree width and (generalized) hypertree width, as opposed to algorithms that run in polynomial time only when $ω$ is considered a constant. For hypergraphs with the bounded intersection property we get better bounds, comparable with that recent algorithm of Lanzinger and Razgon [STACS 2024]. The second algorithm runs in time $n^ωm^{O(1)}$ and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log^2 ω)$. This significantly improves over the $(n+m)^{O(ω^3)}$ time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hypertree width $O(ω^3)$, both in terms of running time and the approximation ratio. Our main technical contribution, and the key insight behind both algorithms, is a variant of the classic Menger's Theorem for clique separators in graphs: For every graph $G$, vertex sets $A$ and $B$, family ${\cal F}$ of cliques in $G$, and positive rational $f$, either there exists a sub-family of $O(f \cdot \log^2 n)$ cliques in ${\cal F}$ whose union separates $A$ from $B$, or there exist $f \cdot \log |{\cal F}|$ paths from $A$ to $B$ such that no clique in ${\cal F}$ intersects more than $\log |{\cal F}|$ paths.
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