Advances on Strictly Δ-Modular IPs

There has been significant work recently on integer programs (IPs) min { c ⊤ x : A x ≤ b , x ∈ Z n } with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant Δ ∈ Z > 0 , Δ -modular IPs are efficiently solvable, which are IPs where the constraint matrix A ∈ Z m × n has full column rank and all n × n minors of A are within { - Δ , ⋯ , Δ } . Previous progress on this question, in particular for Δ = 2 , relies on algorithms that solve an important special case, namely strictly Δ -modular IPs , which further restrict the n × n minors of A to be within { - Δ , 0 , Δ } . Even for Δ = 2 , such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly Δ -modular IPs. Prior advances were restricted to prime Δ , which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly Δ -modular IPs in strongly polynomial time if Δ ≤ 4 .