Most IPs with bounded determinants can be solved in polynomial time
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In 1983 Lenstra showed that an integer program (IP) is fixed parameter tractable in the number of integer variables or the number of constraints. Since then, an open question has been to identify other parameters for which IP is fixed parameter tractable. One candidate parameter is the largest full-dimensional minor Δ of the constraint matrix. If Δ≤ 2, then Artmann et al. (2017) showed that an IP can be solved in polynomial time. However, it is not known if an efficient algorithm exists for Δ > 2. We consider the family of IPs whose minors are bounded by an arbitrary Δ and provide a fixed parameter tractable algorithm in Δ that solves most IPs in this family. Here, 'most' refers to fixing the constraint matrix and objective function and varying the right hand side.
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