Tight complexity lower bounds for integer linear programming with few constraints

We consider the ILP Feasibility problem: given an integer linear program ${Ax = b, x\geq 0}$, where $A$ is an integer matrix with $k$ rows and $\ell$ columns and $b$ is a vector of $k$ integers, we ask whether there exists $x\in\mathbb{N}^\ell$ that satisfies $Ax = b$. Our goal is to study the complexity of ILP Feasibility when both $k$, the number of constraints (rows of $A$), and $|A|\infty$, the largest absolute value in $A$, are small. Papadimitriou [J. ACM, 1981] was the first to give a fixed-parameter algorithm for ILP Feasibility in this setting, with running time $\left((|A\mid b|\infty) \cdot k\right)^{O(k2)}$. This was very recently improved by Eisenbrand and Weismantel [SODA 2018], who used the Steinitz lemma to design an algorithm with running time $(k|A|\infty)^{O(k)}\cdot |b|\infty^2$, and subsequently by Jansen and Rohwedder [2018] to $O(k|A|\infty)^{k}\cdot \log |b|\infty$. We prove that for ${0,1}$-matrices $A$, the dependency on $k$ is probably optimal: an algorithm with running time $2^{o(k\log k)}\cdot (\ell+|b|\infty)^{o(k)}$ would contradict ETH. This improves previous non-tight lower bounds of Fomin et al. [ESA 2018]. We then consider ILPs with many constraints, but structured in a shallow way. Precisely, we consider the dual treedepth of the matrix $A$, which is the treedepth of the graph over the rows of $A$, with two rows adjacent if in some column they both contain a non-zero entry. It was recently shown by Kouteck\'{y} et al. [ICALP 2018] that ILP Feasibility can be solved in time $|A|\infty^{{2{O(td(A))}}\cdot} (k+\ell+\log |b|_\infty)^{O(1)}$. We present a streamlined proof of this fact and prove optimality: even assuming that all entries of $A$ and $b$ are in ${-1,0,1}$, the existence of an algorithm with running time $2^{{2{o(td(A))}}\cdot} (k+\ell)^{O(1)}$ would contradict ETH.