Integer programs with nearly totally unimodular matrices: the cographic case

TL;DR

This work advances the study of integer programs with coefficient matrices that are nearly totally unimodular by solving the Δ-modular IP in the cographic (transpose-of-network) case after deleting a constant number of rows/columns. The authors fuse a strengthened proximity/augmentation framework for TU matrices with a deep graph-minor structure theorem, translating circuits into graph-docsets and leveraging a tree-decomposition to enable dynamic programming. A key contribution is a decomposition theorem for 3-connected rooted graphs without a rooted K_{2,t}-minor, enabling a polynomial-time DP that combines optimized local solutions into a global optimum for the MCIPP and thus the original IP in the cographic setting. This approach not only yields a strongly polynomial-time algorithm for this building block but also illuminates structural routes toward the general totally Δ-modular IP conjecture, highlighting the central role of graph minors and embeddings in controlling combinatorial complexity.

Abstract

It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix $M$ whose subdeterminants are all bounded by a constant $Δ$ in absolute value, can be solved in polynomial time. We answer this question in the affirmative if we further require that, by removing a constant number of rows and columns from $M$, one obtains a submatrix $A$ that is the transpose of a network matrix. Our approach focuses on the case where $A$ arises from $M$ after removing $k$ rows only, where $k$ is a constant. We achieve our result in two main steps, the first related to the theory of IPs and the second related to graph minor theory. First, we derive a strong proximity result for the case where $A$ is a general totally unimodular matrix: Given an optimal solution of the linear programming relaxation, an optimal solution to the IP can be obtained by finding a constant number of augmentations by circuits of $[A\; I]$. Second, for the case where $A$ is transpose of a network matrix, we reformulate the problem as a maximum constrained integer potential problem on a graph $G$. We observe that if $G$ is $2$-connected, then it has no rooted $K_{2,t}$-minor for $t = Ω(k Δ)$. We leverage this to obtain a tree-decomposition of $G$ into highly structured graphs for which we can solve the problem locally. This allows us to solve the global problem via dynamic programming.

Figures (7)

• Figure 1: Subgraph containing a rooted $K_{2,3}$-minor. Roots are indicated with the red squares. Contracting all the edges in each of the five branch sets produces a properly rooted $K_{2,3}$.
• Figure 2: Illustrating the decomposition of \ref{['thm:P1-P6_informal']}. The decomposition tree $T$ is shown on the left. The graphs corresponding to the weak torsos of the two bags $B_u$ and $B_{u'}$ are shown on the right. The top one satisfies \ref{['item:iii_thm:P1-P6_informal']}.\ref{['item:iii_c_thm:P1-P6_informal']}. The vertices above the embedded graph indicate a set of at most $\ell$ vertices, removal of which leaves us with a graph embedded on a surface of bounded genus. The roots are indicated with red squares and are covered by $3$ faces which are drawn in grey. The vertices within the dotted regions in the top and the bottom graphs are the vertices in the intersection of those two graphs. The bottom graph satisfies \ref{['item:iii_thm:P1-P6_informal']}.\ref{['item:iii_b_thm:P1-P6_informal']}.
• Figure 3: An elementary wall of height $6$. Walls of height $h$ are defined as subdivisions of elementary walls of the same height $h$.
• Figure 4: An example of 2-sum of incidence matrices, represented as directed graphs. In order to fit Definition \ref{['def:2sum']}, $A_1$ should be the incidence matrix of $G_1$ with four extra zero rows for the vertices in $V(G_2) \setminus V(G_1)$, and $A_2$ should be defined similarly, so that $A_1$ and $A_2$ have as many rows as the number of vertices of $G$ and share a column $v$ corresponding to arc $(i,j)$. Then it is easy to check that $A_1\oplus_2 A_2$ is the incidence matrix of $G$.
• Figure 5: An example of a 2-sum decomposition of an incidence matrix into four matrices, represented as directed graphs as in Figure \ref{['fig:2-sum']}, with corresponding decomposition tree $T$. Arcs with the same color represent the common vector in the span of two incidence matrices. Given a node $t$ of $T$, $\mathrm{colsp}(A(t))$ is generated by the vectors corresponding to a spanning tree of the corresponding graph, and contains all possible arcs on the respective vertex set.

Theorems & Definitions (115)

• Theorem 1
• Theorem 2: simplified version of \ref{['P1-P6']}
• Lemma 3
• Theorem 4
• Proposition 5
• Lemma 6
• Theorem 7: Theorem 1.2 of boehme_2002
• Theorem 8
• Theorem 9
• Remark 10