Integer programs with bounded subdeterminants and two nonzeros per row

TL;DR

This work proves that integer programs with totally Δ-modular coefficient matrices and at most two nonzeros per row or per column admit strongly polynomial-time solutions, unifying polytime solvability for a broad IP class. The authors reduce such IPs to weighted stable set problems on graphs with bounded odd cycle packing number by exploiting proximity results and structured coefficient reductions. Central to the approach is a deep graph-minor–inspired structure theorem: graphs with bounded ocp admit near-embeddings into surfaces with a controlled arrangement of bipartite vortices and a surface core, enabling a dynamic-programming solution over topological decompositions. The combination of gadget-based preprocessing, edge-induced weights, slack-vector duality, and a carefully crafted DP over windows and sketches yields a principled, strongly polynomial-time algorithm with a clear pathway to practical reductions via b-matching for the column-two-nonzeros case. This advances exact polynomial-time solvability for stable-set-like IPs and highlights powerful connections between integer programming, graph structure theory, and topological embeddings.

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.

Figures (15)

• Figure 1: An Escher wall of height $5$. (The coloring highlights the bipartition of the wall.)
• Figure 2: Theorem \ref{['thm:structure_bounded_OCP']} illustrated. The surface part $G_0$ is shown in black. Small vortices are depicted in green, and large vortices in blue.
• Figure 3: An elementary wall of height $6$.
• Figure 4: The graph $H_2$ from the proof of Theorem \ref{['thm:bipartite_enlarged']}. The disk $\Delta(V)$ of vortex $V$ is the shaded blue area. One parity-breaking path is highlighted in red.
• Figure 5: Illustration of the proof of Theorem \ref{['thm:bipartite_enlarged']}. The figure shows the initial segments $\ell_i Q_i m_i$, $i \in [s]$ of a transaction $\mathcal{Q} := \{Q_1,\ldots,Q_s\}$ and the cutset $Y$ (in red).

Theorems & Definitions (72)

• Theorem 1
• Theorem 2
• Theorem 3: Cook et al. CGST86
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6
• Lemma 7: ocpgenus
• Lemma 8