计数超图染色的不可逼近性
Inapproximability of counting hypergraph colourings
摘要：
最近的发展，在近似计数取得了惊人的进展，在发展快速算法的方法来近似的解决方案的数量约束满足问题（CSP）与大的arities，使用连接到Lovasz局部引理。然而，这些方法的边界与非布尔域的CSP没有很好地理解。我们在本文中的目标是填补这一空白，并通过研究这类CSP，超图着色的原型问题，获得强不可逼近的结果。更准确地说，我们主要研究了度有界的$Δ$的$K$ -一致超图的$q$ -着色的近似计数问题.如果$Δ\lesssim \frac{q^{K/3-1}}{4^KK^2}$（Jain，Pham和Vuong，2021; He，Sun和Wu，2021），则存在有效算法。然而，令人惊讶的是，硬度界限甚至不知道更容易的问题，找到着色。对于计数问题，情况甚至更不清楚，没有证据表明正确的常数控制$K$的指数增长。为此，我们首先建立，对于一般的$q$计算硬度找到一个简单的着色/线性超图发生在$Δ\gtrsim Kq^K$，几乎匹配的算法从Lovasz局部引理。我们的第二个和主要的贡献是获得一个更精细的计数问题，远远超出了硬度找到一个着色，我们猜想是渐近紧（常数因子）。我们特别表明，对于所有甚至$q\geq 4$它是NP-难近似的着色数时，$Δ\gtrsim q^{K/2}$。
英文摘要：
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovasz Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this paper is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting $q$-colourings on $K$-uniform hypergraphs with bounded degree $Δ$. An efficient algorithm exists if $Δ\lesssim \frac{q^{K/3-1}}{4^KK^2}$ (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021). Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of $K$. To this end, we first establish that for general $q$ computational hardness for finding a colouring on simple/linear hypergraphs occurs at $Δ\gtrsim Kq^K$, almost matching the algorithm from the Lovasz Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture that is asymptotically tight (up to constant factors). We show in particular that for all even $q\geq 4$ it is NP-hard to approximate the number of colourings when $Δ\gtrsim q^{K/2}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
