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.