At SODA 2006, Martin Grohe and D$\acute{\rm a}$niel Marx's paper "Constraint solving via fractional edge covers" (ACM citation) showed that for the class of hypergraphs $H$ with bounded fractional hypertree width, CSP($H$) $\in PTIME$.
Definitions, etc.
For a great survey of standard tree decompositions and treewidth, see here (Thanks ahead of time, JeffE!).
Let $H$ be a hypergraph.
Then for a hypergraph $H$ and a mapping $\gamma : E(H) \rightarrow [0,\infty)$,
$B(\gamma) = ${$v \in V(H) : \sum_{e \in V(H), v \in e} \gamma(e) \ge 1$}.
Additionally, let weight($\gamma$) = $\sum_{e \in E}\gamma(e)$.
Then a fractional hypertree decomposition of $H$ is a triple $(T, (B_t)_{t \in V(T)}, (\gamma_t)_{t \in V(T)})$, where:
- $(T, (B_t)_{t \in V(T)})$ is a tree decomposition of $H$, and
- $(\gamma_t)_{t \in V(T)}$ is a family of mappings from $E(H)$ to $[0, \infty)$ s.t. for every $t \in V(T), B_t \subseteq B(\gamma_t)$.
Then we say the width of $(T, (B_t)_{t \in V(T)}, (\gamma_t)_{t \in V(T)})$ is $\max${weight$(\gamma_t), t \in V(T)$}.
Finally, the fractional hypertree width of $H$, fhw($H$), is the minimum of the widths over all possible fractional hypertree decompositions of $H$.
Question
As stated above, if the fractional hypertree width of the underlying graph of a CSP is bounded by a constant, then there is a polynomial time algorithm to solve the CSP. However, it was left as an open problem at the end of the linked paper whether there were any polynomial-time solvable families of CSP instances having unbounded hypertree width. (I should also point out, this question is completely resolved in the case of bounded vs. unbounded treewidth (ACM citation) under the assumption that $FPT \ne W[1]$.) Since there's been some time since the first-linked paper, plus I'm relatively unaware of the general state of this sub-field, my question is:
Is anything known about the (in)tractability of CSPs over graphs with unbounded fractional hypertree width?