"There can’t be any efficient way of answering this question (unless P = NP), but is it beyond our ability to close the gap between n = 11 and n = 14?" No. Th

"There can’t be any efficient way of answering this question (unless P = NP), but is it beyond our ability to close the gap between n = 11 and n = 14?" No. There might be an "efficient" way even if P is not equal NP. Why? Integer linear programming is in NP-complete, still there are at least countably infinitely many integer linear programs that can solve just by inspection. There may be a simple solution the OP's question. More generally, that a problem X is and example of a class of problems in NP-complete doesn't mean that problem X is difficult to solve. In particular, integer linear programming problems are common in production and scheduling in business; for some of them, can get solutions; for some of them, the solutions are valuable for the business. E.g., once I had a 0-1 integer linear programming problem with 40,000 constraints and 600,000 variables. In 905 seconds on a 90 MHz PC (yes, Virginia, at one time that was considered a fast computer!) I found a feasible solution and showed that its objective function value had to be within 0.025% of optimality. Then once when discussing design of IP networks in Plano, TX, some people concluded that I must be fibbing because integer linear programming is in NP-complete. No fibbing. Not all integer linear programming problems are difficult. Besides, likely I didn't find an optimal solution, just a solution close to optimality. But for a related point, in business being close to optimality might save nearly all the money and be plenty good. Moral: That a practical problem is in NP-complete (in the sense of the OP) is no reason to give up on the problem.