In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP($\mathcal{C}$,-), in which the goal is, given a relational structure $\mathbf{A}$ from a class $\mathcal{C}$ of structures and an arbitrary structure $\mathbf{B}$, to find the number of homomorphisms from $\mathbf{A}$ to $\mathbf{B}$. Flum and Grohe showed that #CSP($\mathcal{C}$,-) is solvable in polynomial time if $\mathcal{C}$ has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if $\mathcal{C}$ is a recursively enumerable class of relational structures of bounded arity, then assuming FPT $\neq$ #W[1], there are no other cases of #CSP($\mathcal{C}$,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04]. We show that, assuming FPT $\neq$ W[1] (under randomised parametrised reductions) and for $\mathcal{C}$ satisfying certain general conditions, #CSP($\mathcal{C}$,-) is not solvable even approximately for $\mathcal{C}$ of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP($\mathcal{C}$,-). In particular, our condition generalises the case when $\mathcal{C}$ is closed under taking minors.

中文翻译：

---

从另一侧看到的近似计数 CSP

在本文中，我们研究了计算#CSP($\mathcal{C}$,-) 形式的约束满足问题 (CSP) 的复杂性，其中目标是，给定一个关系结构 $\mathbf{A}$一类 $\mathcal{C}$ 结构和一个任意结构 $\mathbf{B}$，找到从 $\mathbf{A}$ 到 $\mathbf{B}$ 的同态数。Flum 和 Grohe 表明，如果 $\mathcal{C}$ 具有有界树宽 [FOCS'02]，则 #CSP($\mathcal{C}$,-) 在多项式时间内是可解的。基于 Grohe [JACM'07] 在决策 CSP 上的工作，Dalmau 和 Jonsson 然后表明，如果 $\mathcal{C}$ 是一个递归可枚举的有界元关系结构类，那么假设 FPT $\neq$ # W[1]，#CSP($\mathcal{C}$,-) 没有其他情况可以在多项式时间（甚至固定参数时间）[TCS'04] 中精确求解。我们表明，假设 FPT $\neq$ W[1]（在随机参数化约简下）并且 $\mathcal{C}$ 满足某些一般条件，#CSP($\mathcal{C}$,-) 甚至对于 $ \mathcal{C}$ 无界树宽；也就是说，#CSP($\mathcal{C}$,-) 没有固定参数可处理（因此也不是完全多项式）随机近似方案。特别是，我们的条件概括了 $\mathcal{C}$ 在接受未成年人的情况下关闭的情况。