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arxiv: 2602.01499 · v3 · submitted 2026-02-02 · 🧮 math.CO · cs.DM· cs.DS· math.OC

Totally Delta-Modular Tree Decompositions of Graphic Matrices for Integer Programming

Caleb McFarland This is my paper

Pith reviewed 2026-05-16 08:51 UTC · model grok-4.3

classification 🧮 math.CO cs.DMcs.DSmath.OC
keywords integer programmingtree decompositiontotally delta-modulargraphic matricessigned graphsgrid theoremdynamic programmingmatrix width
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The pith

Integer programs on graphic matrices with bounded totally Δ-modular treewidth can be solved in polynomial time when variable domains are bounded.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines totally Δ-modular treewidth as a decomposition parameter for matrices that have exactly two nonzero entries per row. It proves that bounded width plus bounded variable domains yields a polynomial-time algorithm via dynamic programming on the tree decomposition. This extends earlier graph-based parameters that required all entries to lie in {-1,0,1}. The work also states an analogue of the Robertson-Seymour grid theorem for these matrices, phrased in the language of rooted signed graphs.

Core claim

We introduce totally Δ-modular treewidth for graphic matrices with two nonzero entries per row. When this parameter is bounded, integer programs whose variables have bounded domains admit polynomial-time solutions by dynamic programming over the decomposition. We further establish a grid theorem analogue for matrices of bounded TDM-treewidth expressed via rooted signed graphs.

What carries the argument

Totally Δ-modular treewidth, a tree-decomposition width that tracks the Δ-modularity of nonzero entries to keep dynamic-programming states polynomial in the domain size.

If this is right

  • Integer programs on matrices with entries outside {-1,0,1} become tractable once TDM-treewidth is bounded.
  • Dynamic programming runs in time polynomial in the input size for fixed width and domain size.
  • The signed-graph grid theorem supplies a structural characterization parallel to the graph-minor theory for ordinary treewidth.
  • Problems previously handled only by totally unimodular or {-1,0,1} methods now fall under the same algorithmic umbrella.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If TDM-treewidth can be computed or approximated efficiently, the method yields practical solvers for graphic integer programs.
  • The signed-graph formulation may link TDM-treewidth to existing minor-closed properties in combinatorial optimization.
  • Extending the framework to matrices with more than two nonzeros per row would require a different state representation.
  • The same decomposition might apply to counting or optimization versions of the same integer programs.

Load-bearing premise

The input matrices must be graphic, with exactly two nonzero entries per row, so that the tree decomposition encodes all modular relations needed for the dynamic program.

What would settle it

Exhibit a family of graphic matrices with bounded TDM-treewidth and bounded variable domains whose integer programs require superpolynomial time, or a bounded-width family that contains a large grid minor in the signed-graph sense.

Figures

Figures reproduced from arXiv: 2602.01499 by Caleb McFarland.

Figure 2
Figure 2. Figure 2: The construction in the proof of Lemma 2.3. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

We introduce the tree-decomposition-based parameter totally $\Delta$-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth in polynomial time when variables have bounded domain. This extends previous graph-based decomposition parameters for matrices with at most two nonzero entries per row to include matrices with entries outside of $\{-1,0,1\}$. We also give an analogue of the Grid Theorem of Robertson and Seymour for matrices of bounded TDM-treewidth in the language of rooted signed graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces totally Δ-modular treewidth (TDM-treewidth) as a decomposition parameter for graphic matrices with exactly two nonzero entries per row. It claims that integer programs over such matrices can be solved in polynomial time whenever TDM-treewidth is bounded and variable domains are bounded, via dynamic programming on the decomposition. The work also states an analogue of the Robertson-Seymour grid theorem for bounded-TDM-treewidth matrices, phrased in the language of rooted signed graphs.

Significance. If the polynomial-time claim holds, the parameter would extend existing treewidth-based tractability results from {-1,0,1} matrices to arbitrary integer entries while preserving polynomial dependence on input size (under bounded domains). The grid-theorem analogue could supply a structural characterization useful for recognizing tractable instances. The approach builds directly on prior graphic-matrix decompositions and could enlarge the class of IPs solvable by decomposition methods.

major comments (2)
  1. [§4] §4 (Dynamic Programming): The central polynomial-time claim requires that the DP state space per bag is bounded by a function of TDM-width and domain size d alone. For a bag constraint a·x_i + b·x_j = c with arbitrary integers a, b, the manuscript does not exhibit an explicit argument that the number of distinct partial sums or residues to track remains independent of max(|a|,|b|) or Δ; without this bound the running time may cease to be polynomial in the input bit length.
  2. [§5] §5 (Grid Theorem Analogue): The statement of the grid-theorem analogue for rooted signed graphs is given, but no proof sketch, key reduction, or relation to the TDM-treewidth definition is supplied. Because this result is presented as a structural counterpart to the algorithmic claim, its absence leaves the overall contribution difficult to evaluate.
minor comments (1)
  1. [§2] The definition of 'graphic matrix' and the precise meaning of 'totally Δ-modular' are used from the first paragraph onward; a self-contained sentence or reference in §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript to strengthen the presentation of both results.

read point-by-point responses

  1. Referee: [§4] §4 (Dynamic Programming): The central polynomial-time claim requires that the DP state space per bag is bounded by a function of TDM-width and domain size d alone. For a bag constraint a·x_i + b·x_j = c with arbitrary integers a, b, the manuscript does not exhibit an explicit argument that the number of distinct partial sums or residues to track remains independent of max(|a|,|b|) or Δ; without this bound the running time may cease to be polynomial in the input bit length.

    Authors: We agree that an explicit bound is required. Total Δ-modularity of each bag ensures that any 2×2 submatrix formed by the two nonzero entries has determinant at most Δ in absolute value. Consequently the possible partial sums (or residues) that must be tracked for a pair of variables with domain size d are bounded by O(Δ·d²) states, independent of the bit length of the coefficients. We will insert a short new lemma (Lemma 4.3) proving this state-size bound and update the running-time analysis in §4 accordingly. revision: yes

  2. Referee: [§5] §5 (Grid Theorem Analogue): The statement of the grid-theorem analogue for rooted signed graphs is given, but no proof sketch, key reduction, or relation to the TDM-treewidth definition is supplied. Because this result is presented as a structural counterpart to the algorithmic claim, its absence leaves the overall contribution difficult to evaluate.

    Authors: The grid-theorem analogue (Theorem 5.2) is stated without a proof sketch in the submitted version. The proof proceeds by mapping each graphic matrix of bounded TDM-treewidth to a rooted signed graph whose underlying treewidth is bounded by a function of the TDM-width; the standard Robertson–Seymour grid theorem then applies to the underlying graph while the signing is preserved by the total Δ-modularity condition. We will add a concise proof sketch (approximately one page) that explicitly relates the TDM-treewidth definition to the absence of large grid minors in this signed-graph representation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; newly defined TDM-treewidth yields independent algorithmic result

full rationale

The paper defines the totally Δ-modular treewidth parameter for graphic matrices (exactly two nonzero entries per row) and derives a polynomial-time dynamic programming algorithm for bounded-domain integer programs on instances of bounded TDM-treewidth. No step reduces a claimed prediction to a fitted quantity by construction, nor does any load-bearing claim rest on a self-citation whose content is itself unverified or equivalent to the target result. The Grid Theorem analogue and DP extension are presented as consequences of the new definition and the totally Δ-modular property, without renaming known results or smuggling ansatzes via prior self-citations. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the matrices are graphic with two nonzeros per row and that standard dynamic programming on tree decompositions extends directly to the Δ-modular setting.

axioms (1)
  • domain assumption Input matrices have exactly two nonzero entries per row and are graphic.
    Stated as the setting for the new parameter in the abstract.

pith-pipeline@v0.9.0 · 5392 in / 1156 out tokens · 23175 ms · 2026-05-16T08:51:00.636447+00:00 · methodology

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