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arXiv · 2106.05947

Integer programs with bounded subdeterminants and two nonzeros per row

Samuel Fiorini·Gwenaël Joret·Stefan Weltge·Yelena Yuditsky
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Abstract

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than $k$ vertex-disjoint odd cycles, where $k$ is any constant. Previously, polynomial-time algorithms were only known for $k=0$ (bipartite graphs) and for $k=1$. We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to $b$-matching.

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Samuel Fiorini, Gwenaël Joret, Stefan Weltge, Yelena Yuditsky. 2021-06-10. Integer programs with bounded subdeterminants and two nonzeros per row. https://doi.org/10.1145/3695985

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