Advances on strictly -modular IPs

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