We study the problem of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted a lot of attention. For list packing the setup is similar but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. In particular, the existence of a list packing implies the existence of a list coloring. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting. In graphs of maximum degree $\Delta$ and when the number of colors is at least $\Omega(\Delta^2)$, we give an FPRAS based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree.

翻译：我们研究图列表打包数量的近似计数问题。常规顶点着色和列表着色的类似问题已引起广泛关注。列表打包的设置与之类似，但我们寻求将颜色列表完全分解为两两不相交的适当列表着色。特别地，列表打包的存在性意味着列表着色的存在性。近期关于列表打包的研究聚焦于存在性或其数量的极值结果，但本文转向计数相关的算法层面。当图的最大度为$\Delta$且颜色数至少为$\Omega(\Delta^2)$时，我们基于自然马尔可夫链（格劳伯动力学）的快速混合性给出一个全多项式随机近似方案（FPRAS），并采用路径耦合法进行分析。本研究的动机之一是探究一种非典型自旋系统——其中每个顶点的自旋数远大于图的度数。