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We propose a new field-programmable gate array (FPGA) routing approach, which, when combined with a low-cost architecture change, results in a 40% reduction in router runtime, at the cost of a 6% area overhead and with no increase in critical path delay. Our approach begins with PathFinder-style routing, which we run on a coarsened representation of the routing architecture. This leads to fast generation of a partial routing solution where the signals are assigned to groups of wire segments rather than individual wire segments. A Boolean satisfiability (SAT)-based stage follows, generating a legal routing solution from the partial solution. We explore approximately 165 000 FPGA switch block architectures, showing that the choice of the architecture has a significant impact on the complexity of the SAT formulation, and by extension, on routing runtime. Our approach points to a new research direction, namely, reducing FPGA computer-aided design runtime by exploring FPGA architectures and algorithms together.