Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in polynomial time ($(mn)^c$ arithmetic operations on $(mn)^c$ sized words suffices).

Supposing we have the promise that the number of feasible solutions is polynomial then *under what conditions* can the problem be solved in polynomial time using Kannan's and Barnivok's algorithm (other than total unimodularity)?

It may be possible that naturally defining IP with poly # solutions is in P unless somehow defined by Unique-SAT.

Is it possible that if perfect basis reduction for given problem is in P then Kannan's algorithm is in P?

Is there necessary and sufficient conditions on categories of inputs on which the algorithms run in poly time?

It is unlikely that Kannan's algorithm or Barnivok's algorithm runs uniformly in $n^{O(n)}$ time for all inputs.

There is no evidence whatsoever that Kannan's or Barnivok's algorithm ever runs in poly time for any non-trivial class of inputs which is why the query. It seems unbelievable that there cannot be input families for which these terminate in poly time.

By **non-trivial** class of inputs I mean an exponential set of inputs.