On July 14, 2021, Stefan Weltge from the Technical University of Munich gave an online talk at the Virtual Discrete Math Colloquium on an efficient algorithm to solve the Integer Linear Program defined by an integer matrix whose subdeterminants are bounded by a constant and that contains at most two nonzero entries in each row. The title of his talk was “Integer programs with bounded subdeterminants and two nonzeros per row“.
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.
This is joint work with Samuel Fiorini, Gwenaël Joret, and Yelena Yuditsky.