Advances on strictly $$\Delta $$-modular IPs

Abstract

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