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We initiate the study of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted substantial 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. The existence of a list packing implies the existence of a list coloring, but the converse is false. 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 and sampling. In graphs of maximum degree Δ and when the number of colors is at least Ω(Δ²), we give a fully polynomial-time randomized approximation scheme (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. 

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently.