Dividing a Boolean formula into smaller independent sub-formulae can be a useful technique for accelerating the solution of Boolean problems, including SAT and #SAT. Nevertheless, and despite promising early results, formula partitioning is hardly used in state-of-the-art solvers. In this paper, we show that this is rooted in a lack of consistency of the usefulness of formula partitioning techniques. In particular, we evaluate two existing and a novel partitioning model, coupled with two existing and two novel partitioning algorithms, on a wide range of benchmark instances. Our results show that there is no one-size-fits-all solution: for different formula types, different partitioning models and algorithms are the most suitable. While these results might seem negative, they help to improve our understanding about formula partitioning; moreover, the findings also give guidance as to which method to use for what kinds of formulae.

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We study financial networks where banks are connected by debt contracts. We consider the operation of debt swapping when two creditor banks decide to exchange an incoming payment obligation, thus leading to a locally different network structure. We say that a swap is positive if it is beneficial for both of the banks involved; we can interpret this notion either with respect to the amount of assets received by the banks, or their exposure to different shocks that might hit the system.We analyze various properties of these swapping operations in financial networks. We first show that there can be no positive swap for any pair of banks in a static financial system, or when a shock hits each bank in the network proportionally. We then study worst-case shock models, when a shock of given size is distributed in the worst possible way for a specific bank. If the goal of banks is to minimize their losses in such a worst-case setting, then a positive swap can indeed exist. We analyze the effects of such a positive swap on other banks of the system, the computational complexity of finding a swap, and special cases where a swap can be found efficiently. Finally, we also present some results for more complex swapping operations when the banks swap multiple contracts, or when more than two banks participate in the swap. CCS Concepts: • Applied computing → Economics; • Theory of computation → Market equilibria; Network games.

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