On the optimality of pseudo-polynomial algorithms for integer programming

In the classic Integer Programming Feasibility (IPF) problem, objective is to decide whether, for a given $$m \times n$$ matrix A and an m-vector $$b=(b_1,\dots , b_m)$$ there non-negative integer n-vector x such that $$Ax=b$$ . Solving important step in numerous algorithms it obtain understanding of precise complexity this problem as function natural parameters input. The pseudo-polynomial time algorithm Papadimitriou [J. ACM 1981] instances with constant number constraints was only recently improved upon by Eisenbrand Weismantel [SODA 2018] Jansen Rohwedder [ITCS 2019]. designed running $${\mathcal {O}}(m \varDelta )^m$$ $$\log (\varDelta ) \log +\Vert b\Vert _{\infty })+{\mathcal {O}}(mn)$$ Here, $$\varDelta $$ upper bound on absolute values entries A. We continue line work show under Exponential Time Hypothesis (ETH), nearly optimal, proving lower $$n^{o(\frac{m}{\log m})} \cdot \Vert }^{o(m)}$$ also prove assuming ETH, cannot be solved $$f(m)\cdot (n })^{o(\frac{m}{\log m})}$$ any computable f. This motivates us pick up research initiated Cunningham Geelen [IPCO 2007] who studied solving matrices which may unbounded, but branch-width column-matroid corresponding constraint constant. optimal results respect closely related parameter, path-width. Specifically, we matching bounds when path-width