Integer programs with bounded subdeterminants and two nonzeros per row

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.

Neighborhood persistency of the linear optimization relaxation of integer linear optimization

“…We introduced neighborhood persistency of the linear optimization (LO) relaxation of integer linear optimization (ILO), which is a property stronger than persistency, and show that ILO on unit-two-variable-per-inequality (UTVPI) systems is a maximal subclass of ILO with its LO relaxation having (neighborhood) persistency. Our persistency result generalizes known results on special cases of ILO on UTVPI systems [14,7,4]. Using neighborhood persistency, we obtain fixed-parameter algorithms (where the parameter is the solution size) and another proof of the two-approximability for special cases of ILO on UTVPI systems.…”

“…Note that persistency and neighborhood persistency are the same when the variables are binary. Fiorini et al [4] gave another generalization that persistency holds for the LO relaxation of ILO on UTVPI systems if each inequality is of the form x i + x j ≤ c for some integer c. It should be noted that optimal solutions of the LO relaxation are assumed to be half-integral in these persistency results, while not in our (neighborhood) persistency result in this paper.…”

“…Indeed, persistency was first shown for the LO relaxation of an ILO formulation of the vertex cover problem [14], which is ILO on special UTVPI systems, and used to obtain a fixed-parameter algorithm for the vertex cover problem [11]. The persistency result in [14] is generalized to special cases of ILO on UTVPI systems [7,4].…”

Congruency-Constrained TU Problems Beyond the Bimodular Case

“…A problem that can be cast as a bounded subdeterminant integer program, has gained substantial interest recently [BFMR14; CFHJW20; CFHW21], and was resolved in [FJWY21], is the stable set problem in graphs G with bounded odd cycle packing number ocp(G), i.e., graphs for which the maximum number of disjoint odd cycles is bounded. The incidence matrix of such a graph has maximum subdeterminant 2 ocp(G) (see, e.g., [GKS95]).…”

“…The result in [VC09] does not extend to ∆ > 2, and it remains open whether another reduction to CCTU problems may exist. The relevance of CCTU problems is also highlighted by recent progress of Fiorini, Joret, Weltge, and Yuditsky [FJWY21], who obtained an efficient algorithm for totally ∆-modular ILPs with a constraint matrix having at most two non-zeros in each row. This algorithm needs to compute certain circulations with parity constraints, which can be interpreted as CCTU problems.…”