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 $\omega$, runs in polynomial time and produces a tree decomposition of $H$ of fractional hypertree width $O(\omega \log n \log \omega)$. As an immediate corollary this yields polynomial time $O(\log^2 n \log \omega)$-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 $\omega$ 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 $n^{\omega}m^{O(1)}$ and produces a tree decomposition of $H$ of fractional hypertree width $O(\omega \log^2 \omega)$. This significantly improves over the $(n+m)^{O(\omega^3)}$ time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hypertree width $O(\omega^3)$, 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.

翻译：我们提出了两种新的近似算法来计算输入超图的分数超树宽。第一种算法以分数超树宽至多为$\omega$的$n$顶点$m$边超图$H$作为输入，在多项式时间内运行，并生成分数超树宽为$O(\omega \log n \log \omega)$的$H$的树分解。作为直接推论，这也为（广义）超树宽产生了多项式时间$O(\log^2 n \log \omega)$-近似算法。据我们所知，我们的算法是首个非平凡的多项式时间近似算法，用于分数超树宽和（广义）超树宽，这与仅在$\omega$被视为常数时才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的超图，我们获得了更好的界，与Lanzinger和Razgon最近提出的算法[STACS 2024]相当。第二种算法在$n^{\omega}m^{O(1)}$时间内运行，并生成分数超树宽为$O(\omega \log^2 \omega)$的$H$的树分解。这显著改进了Marx[ACM TALG 2010]的$(n+m)^{O(\omega^3)}$时间算法，该算法生成的树分解分数超树宽为$O(\omega^3)$，无论是在运行时间还是近似比方面。我们的主要技术贡献，以及两种算法背后的关键见解，是图论中经典Menger定理关于团分离器的一个变体：对于每个图$G$、顶点集$A$和$B$、$G$中团的族${\cal F}$以及正有理数$f$，要么存在${\cal F}$中$O(f \cdot \log^2 n)$个团的子族，其并集将$A$与$B$分离，要么存在$f \cdot \log |{\cal F}|$条从$A$到$B$的路径，使得${\cal F}$中没有团与超过$\log |{\cal F}|$条路径相交。