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It is well-known that Lenstra's famous algorithm (presented in the paper Integer programming with a fixed number of variables) can solve an ILP problem in $O^*(f(k))$ time where k is the number of variables occur in the ILP.

My question is that whether the algorithm or any of its improved versions can output all feasible solutions in $O^*(f(k))$ time?

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  • $\begingroup$ What does $O^*$ mean? See also this answer to a related question. $\endgroup$
    – Jeffε
    Mar 22, 2013 at 18:16
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    $\begingroup$ The $O^*$ notation usually ignores polynomial factors in the instance size. $\endgroup$
    – Serge Gaspers
    Mar 22, 2013 at 22:10

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No. The number of feasible solutions cannot be upper bounded by $f(k)n^{O(1)}$.

Consider the integer program $I_n: 1 \le x\le 2^n$ with the integer variable $x$. So, $k=1$ and the program can be described with $O(n)$ bits. But it has $2^n$ solutions, which is exponential in the instance size.

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