Polynomial time reductions using binary search
Is OPT = m? is a coNP decision problem.  A "no" answer has a certificate verifiable in polynomial time, the certificate being a valid bin packing that uses fewer than $m$ bins.

The same is true for the question Is OPT > m?.  There is no polynomial-time verifiable certificate for the "yes" answer to either of these questions unless NP = coNP.

Is OPT &lt; m? is an NP decision problem.  A "yes" answer has a certificate verifiable in polynomial time, the certificate being a valid bin packing that uses fewer than $m$ bins.