Advances on Strictly Δ-Modular IPs
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There has been significant work recently on integer programs (IPs) min{c^⊤ x Ax≤ 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.
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