Exact exponential-time algorithms for 0-1 programming
If $m$ is superlinear, such an algorithm would disprove the Strong Exponential Time Hypothesis, since formulas in conjunctive normal form are a special case of 0-1 programming and the Sparsification Lemma allows us to reduce $k$-SAT to CNF-SAT on linearly many clauses.

However, there is an algorithm due to Impagliazzo, Paturi, and myself that can solve such a system of inequalities if the number of wires, i.e. the number of nonzero coefficients in $A$ is linear. In particular, if the number of wires is $cn$, the algorithm runs in time $2^{(1-s)n}$, where $s=\frac1{c^{O(c^2)}}$.