Diagonal Frobenius Number in Integer Cones

• The diagonal Frobenius number is a saturation threshold for integer cones, ensuring that sufficiently deep fractional solutions imply integer feasibility.
• The main bound shows that F_diag(A) is at most Δ plus an additive O(log k) term, with Δ defined by the maximum absolute k×k subdeterminant and largely independent of n.
• The proof leverages Gomory’s corner relaxation and discrepancy-based rounding, yielding constructive, algorithmic results for integer programming feasibility.

The diagonal Frobenius number is a saturation parameter for integer cones associated with an integer matrix. For a matrix $A \in \mathbb{Z}^{k \times n}$ of rank $k$, with $k \le n$, and $\Delta=\Delta(A)$ denoting the maximum absolute value of all $k \times k$ subdeterminants of $A$, the diagonal Frobenius number $F_{\text{diag}}(A)$ is the smallest $t \in \mathbb{Z}_{\ge 1}$ such that every right-hand side $b \in \operatorname{span}_{\mathbb{Z}}(A)$ admitting a nonnegative representation $b=Ax$ with all coordinates of $k$0 at least $k$1 also admits a nonnegative integer representation $k$2. In "Diagonal Frobenius Number via Gomory's Relaxation and Discrepancy" the governing quantity is shown to be $k$3, up to an additive logarithmic term in $k$4, and the proof is based on Gomory’s corner polyhedron relaxation together with discrepancy-theoretic rounding (Gribanov et al., 6 Sep 2025).

1. Formal definition and ambient setting

Let

$k$5

The nonnegative orthants are $k$6 and $k$7, the all-ones vector is denoted by $k$8, and the integer image of $k$9 is

$k \le n$0

Under the normalization $k \le n$1, where $k \le n$2 is the greatest common divisor of all $k \le n$3 subdeterminants, one has $k \le n$4; the case $k \le n$5 is reduced to the normalized case by a Smith normal form based polynomial-time reduction (Gribanov et al., 6 Sep 2025).

The defining implication for $k \le n$6 is

$k \le n$7

for every $k \le n$8. Equivalently, if $k \le n$9 lies in the continuous cone $\Delta=\Delta(A)$0 and has a representation that is sufficiently deep along the diagonal direction $\Delta=\Delta(A)$1, then $\Delta=\Delta(A)$2 also lies in the integer cone $\Delta=\Delta(A)$3.

For $\Delta=\Delta(A)$4, the theorem specializes to $\Delta=\Delta(A)$5 because $\Delta=\Delta(A)$6. The paper interprets this as consistency with the intuition that sufficiently deep fractional solutions force integrality in the one-row case (Gribanov et al., 6 Sep 2025).

2. Main bounds and their significance

The central theorem states that there exists an absolute constant $\Delta=\Delta(A)$7 such that

$\Delta=\Delta(A)$8

In particular,

$\Delta=\Delta(A)$9

The dominant term is therefore independent of $k \times k$0, and one of the two additive terms is also independent of $k \times k$1 (Gribanov et al., 6 Sep 2025).

This improves several previously cited estimates after standard inequalities such as Hadamard’s inequality are applied. The paper compares its result with the Aharoni–Harnik bound

$k \times k$2

with recent “total regime” bounds

$k \times k$3

and with the bound

$k \times k$4

where $k \times k$5 is the maximum entry magnitude. The new estimate depends only on $k \times k$6 and is independent of $k \times k$7 in the leading term (Gribanov et al., 6 Sep 2025).

The paper establishes tightness, up to constants, for the intermediate Gomory threshold $k \times k$8 that appears in the corner-polyhedron step, by means of a simple example. By contrast, it does not provide general lower bounds matching the additive $k \times k$9 term for $A$0, and explicitly leaves such lower bounds for future work. This distinguishes the sharpness of the underlying relaxation threshold from the sharpness of the final Frobenius bound.

3. Gomory relaxation and discrepancy as the proof mechanism

The proof is carried out mainly through a generalized slack version in canonical form, and the standard-form statement is obtained through parameter-preserving reductions. In the canonical-form view, the corner relaxation around a base $A$1 keeps only the basic constraints

$A$2

If

$A$3

then the canonical system $A$4 has an integer feasible solution, and this solution is findable in polynomial time (Gribanov et al., 6 Sep 2025).

The geometric role of the threshold $A$5 is central. The proof sketch transforms $A$6 to Hermite normal form $A$7 via a unimodular matrix $A$8, solves $A$9, obtains an integer point with slack $F_{\text{diag}}(A)$0 of bounded $F_{\text{diag}}(A)$1-norm at most $F_{\text{diag}}(A)$2, and then uses bounds on the entries of $F_{\text{diag}}(A)$3 to ensure the nonbasic inequalities remain satisfied. The threshold is stated to be tight, with a diagonal example showing that slack exactly $F_{\text{diag}}(A)$4 can fail to force integrality (Gribanov et al., 6 Sep 2025).

To supply the missing slack, the argument invokes discrepancy theory. The paper uses

$F_{\text{diag}}(A)$5

together with the determinant lower bound $F_{\text{diag}}(A)$6 and the estimates

$F_{\text{diag}}(A)$7

For $F_{\text{diag}}(A)$8, this yields

$F_{\text{diag}}(A)$9

The rounding lemma used in the construction is

$t \in \mathbb{Z}_{\ge 1}$0

This provides uniform $t \in \mathbb{Z}_{\ge 1}$1 control when rounding from $t \in \mathbb{Z}_{\ge 1}$2 to $t \in \mathbb{Z}_{\ge 1}$3 (Gribanov et al., 6 Sep 2025).

The high-level synthesis is as follows. One reduces to canonical form, fixes a feasible base $t \in \mathbb{Z}_{\ge 1}$4, constructs basic and nonbasic slacks from a deep fractional point, applies discrepancy rounding to the matrix $t \in \mathbb{Z}_{\ge 1}$5, adjusts the basic right-hand side, and then invokes the Gomory relaxation theorem. When $t \in \mathbb{Z}_{\ge 1}$6, the adjusted nonbasic slack remains at least $t \in \mathbb{Z}_{\ge 1}$7, so the corner-relaxation certificate yields an integer feasible point for the original system.

4. Constructive and algorithmic results

The results are not merely existential. The paper gives algorithms in the Word-RAM model with unit-cost arithmetic on words of size polynomial in $t \in \mathbb{Z}_{\ge 1}$8, where $t \in \mathbb{Z}_{\ge 1}$9 bounds the entries of $b \in \operatorname{span}_{\mathbb{Z}}(A)$0 and $b \in \operatorname{span}_{\mathbb{Z}}(A)$1 (Gribanov et al., 6 Sep 2025).

For polynomial-time construction under weaker thresholds, sufficiently large absolute constants $b \in \operatorname{span}_{\mathbb{Z}}(A)$2 are fixed and

$b \in \operatorname{span}_{\mathbb{Z}}(A)$3

If there exists $b \in \operatorname{span}_{\mathbb{Z}}(A)$4 with $b \in \operatorname{span}_{\mathbb{Z}}(A)$5 and $b \in \operatorname{span}_{\mathbb{Z}}(A)$6, then a polynomial-time algorithm finds $b \in \operatorname{span}_{\mathbb{Z}}(A)$7 with $b \in \operatorname{span}_{\mathbb{Z}}(A)$8. The paper summarizes this informally as two polynomial-time regimes: one with $b \in \operatorname{span}_{\mathbb{Z}}(A)$9 of order $b=Ax$0 in $b=Ax$1, and one with $b=Ax$2 of the form $b=Ax$3 plus a $b=Ax$4-dependent additive term (Gribanov et al., 6 Sep 2025).

A stronger constructive statement is obtained if either $b=Ax$5 preprocessing is allowed or a base $b=Ax$6 with $b=Ax$7 is provided. In that case one can take

$b=Ax$8

and construct $b=Ax$9 with $k$00 in $k$01 time. The constant $k$02 in this algorithmic statement is stated to be $k$03 times larger than the constant in the existential Frobenius bound (Gribanov et al., 6 Sep 2025).

The need for preprocessing is tied to the difficulty of locating a determinant-maximizing base. The paper states that finding a base $k$04 with $k$05 is NP-hard. It also gives two base-construction procedures. Using Nikolov’s algorithm, one can find a base $k$06 with $k$07 in deterministic polynomial time, after which repeated augmentation yields $k$08 for $k$09, with total cost $k$10. A simpler polynomial-time selection yields $k$11 with $k$12 per iteration and $k$13 iterations (Gribanov et al., 6 Sep 2025).

5. Slack generalization and canonical-form formulation

The paper introduces a more general invariant for canonical-form systems. If $k$14 has rank $k$15 and the system has $k$16 constraints, the generalized diagonal Frobenius number for slacks is

$k$17

defined as the minimum $k$18 such that

$k$19

This is presented as a generalization of $k$20 for canonical-form systems such as $k$21, and all proofs are mainly carried out for $k$22 (Gribanov et al., 6 Sep 2025).

The corresponding main bound is

$k$23

The paper also gives polynomial-time and $k$24 algorithmic variants for $k$25 that mirror the standard-form statements, with $k$26 replacing $k$27 in the threshold formulas (Gribanov et al., 6 Sep 2025).

The standard-form and canonical-form results are connected by parameter-preserving reductions. The equivalence asserted in the proof sketch is that a standard-form condition

$k$28

can be translated into a canonical-form slack condition

$k$29

without changing the relevant parameters $k$30 and $k$31. This makes the slack formulation the conceptual center of the argument, with the standard diagonal Frobenius number appearing as a corollary.

A consequence is that the same discrepancy-plus-Gomory mechanism controls both exact equalities in standard form and feasibility under inequalities in canonical form. This suggests that the saturation phenomenon is more naturally expressed in terms of slack depth than in terms of equality form alone, although the paper formulates the standard diagonal version explicitly (Gribanov et al., 6 Sep 2025).

6. Relations to classical Frobenius theory, examples, and limitations

The terminology “diagonal Frobenius number” also appears in a distinct matrix-ring generalization of the classical Frobenius problem. In "Frobenius templates in certain $k$32 matrix rings," upper triangular $k$33 matrices with constant diagonal are represented by pairs $k$34, and a natural invariant controlling the first coordinate is

$k$35

the classical Frobenius number of the diagonal entries. In that setting, the diagonal coordinate reduces exactly to the classical coin problem, while the off-diagonal coordinate introduces additional coupling through the same coefficients $k$36 (Eller et al., 2021).

Over the integers, if $k$37 and $k$38, the matrix-ring paper proves

$k$39

and for two generators it gives the exact formula

$k$40

with $k$41. For $k$42, the exact Frobenius set can be strictly larger than the lower-bound rectangle or even a union of rectangles (Eller et al., 2021). This use of the term is conceptually related but mathematically different from the subdeterminant-controlled invariant studied for integer cones in (Gribanov et al., 6 Sep 2025).

Within the integer-programming formulation, the paper gives the illustrative example $k$43, so $k$44, $k$45, $k$46, and $k$47. Taking

$k$48

one has

$k$49

and the theorem guarantees a nonnegative integer solution to $k$50; for instance $k$51 (Gribanov et al., 6 Sep 2025). The example is used to illustrate the role of $k$52 in forcing integrality once all coordinates of the fractional solution are at least $k$53.

Several limitations are explicit. The additive $k$54 term is not matched by known lower bounds in full generality. The precise constructive threshold $k$55 does not directly translate into polynomial time because finding a base attaining $k$56 is NP-hard. Alternative bounds with $k$57 show a mild dependence on $k$58, but the main theorem’s dominant term remains $k$59. The paper also notes robustness under $k$60 through the Smith normal form reduction (Gribanov et al., 6 Sep 2025).

From the viewpoint of integer programming, the practical implication stated in the paper is that sufficiently deep fractional solutions certify the existence of integer solutions, and the discrepancy-based rounding plus Gomory relaxation yields constructive procedures under $k$61-governed thresholds. In that sense, the diagonal Frobenius number quantifies when the integer cone saturates the continuous cone along the diagonal direction.