Totally $Δ$-modular IPs with two non-zeros in most rows
Stefan Kober
TL;DR
We address the polynomial-time solvability of integer programs with bounded subdeterminants, focusing on totally $\\Delta$-modular constraint matrices with a limited non-zero structure. The authors develop a strongly polynomial-time algorithm by combining a structural graph decomposition with a dynamic-programming framework and proximity arguments to reduce to tractable subproblems. Their contributions include (i) a strongly polynomial-time algorithm for the main class, (ii) an FPT-algorithm for the reduced problem when $A$ is totally unimodular, (iii) connections to the $k$-dimensional partially ordered knapsack and related models, and (iv) a constructive, self-contained decomposition that avoids graph-minor machinery. This work broadens the class of IPs solvable in polynomial time under bounded subdeterminants and two-nonzero-per-row structure, with implications for structured combinatorial optimization problems.
Abstract
Integer programs (IPs) on constraint matrices with bounded subdeterminants are conjectured to be solvable in polynomial time. We give a strongly polynomial time algorithm to solve IPs where the constraint matrix has bounded subdeterminants and at most two non-zeros per row after removing a constant number of rows and columns. This result extends the work by Fiorini, Joret, Weltge \& Yuditsky (J. ACM 72(1), 1-50 (2025)) by allowing for additional, unifying constraints and variables.