  In this paper we study the computation of Markov bases for contingency tables
whose cell entries have an upper bound. In general a Markov basis for unbounded
contingency table under a certain model differs from a Markov basis for bounded
tables. Rapallo, (2007) applied Lawrence lifting to compute a Markov basis for
contingency tables whose cell entries are bounded. However, in the process, one
has to compute the universal Gr\"obner basis of the ideal associated with the
design matrix for a model which is, in general, larger than any reduced
Gr\"obner basis. Thus, this is also infeasible in small- and medium-sized
problems. In this paper we focus on bounded two-way contingency tables under
independence model and show that if these bounds on cells are positive, i.e.,
they are not structural zeros, the set of basic moves of all $2 \times 2$
minors connects all tables with given margins. We end this paper with an open
problem that if we know the given margins are positive, we want to find the
necessary and sufficient condition on the set of structural zeros so that the
set of basic moves of all $2 \times 2$ minors connects all incomplete
contingency tables with given margins.
