Advances on strictly Δ -modular IPs.

Abstract

There has been significant work recently on integer programs (IPs) $min\{c^{\top }x:Ax\le b,x\in Z^n\}$ with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $\Delta \in Z_{>0}$ , $\Delta$ -modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in Z^{m\times n}$ has full column rank and all $n\times n$ minors of A are within $\{-\Delta ,\cdots ,\Delta \}$ . Previous progress on this question, in particular for $\Delta =2$ , relies on algorithms that solve an important special case, namely strictly $\Delta$ -modular IPs, which further restrict the $n\times n$ minors of A to be within $\{-\Delta ,0,\Delta \}$ . Even for $\Delta =2$ , such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $\Delta$ -modular IPs. Prior advances were restricted to prime $\Delta$ , 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 $\Delta$ -modular IPs in strongly polynomial time if $\Delta \le 4$ .