  We present simple compact proofs of the strong and weak duality theorems of
tropical linear programming. It follows that there is no duality gap for a pair
of tropical primal-dual problems. This result together with known properties of
subeigenvectors enables us to directly solve a special tropical linear program
with two-sided constraints. We also study the duality gap in tropical integer
linear programming. A direct solution is available for the primal problem. An
algorithm of quadratic complexity is presented for the dual problem. A direct
solution is available provided that all coefficients of the objective function
are integer. This solution yields a good estimate of the optimal objective
function value in the general case.
