The cavity method has played and is playing a major role in the solution of optimization problems like the K-satisfiability and the graph coloring in present days.
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Since the ATPG problem is NP-complete (by reduction from the Boolean satisfiability problem) there will be cases where patterns exist, but ATPG gives up since it will take an incredibly long time to find them (assuming P≠NP, of course).
3-SAT is NP-complete (like any other k-SAT problem with k>2) while 2-SAT is known to have solutions in polynomial time.