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Constrained counting is a fundamental problem in artificial intelligence. Apromising new algebraic approach to constrained counting makes use of tensornetworks, following a reduction from constrained counting to the problem oftensor-network contraction. Contracting a tensor network efficiently requiresdetermining an efficient order to contract the tensors inside the network,which is itself a difficult problem.  In this work, we apply graph decompositions to find contraction orders fortensor networks. We prove that finding an efficient contraction order for atensor network is equivalent to the well-known problem of finding an optimalcarving decomposition. Thus memory-optimal contraction orders for planar tensornetworks can be found in cubic time. We show that tree decompositions can beused both to find carving decompositions and to factor tensor networks withhigh-rank, structured tensors.  We implement these algorithms on top of state-of-the-art solvers for treedecompositions and show empirically that the resulting weighted model counteris quite effective and useful as part of a portfolio of counters.

Efficient Contraction of Large Tensor Networks for Weighted Model Counting through Graph Decompositions