A parameterized linear formulation of the integer hull

Friedrich Eisenbrand EPFL, Switzerland, friedrich.eisenbrand@epfl.ch. Work carried out while the author was visiting at the University of Washington. Thomas Rothvoss University of Washington, USA, rothvoss@uw.edu. Supported by NSF grant 2318620 AF: SMALL: The Geometry of Integer Programming and Lattices.

(September 8, 2025)

Abstract

Let $A\in\mathbb{Z}^{m\times n}$ be an integer matrix with components bounded by $\Delta$ in absolute value. Cook et al. (1986) have shown that there exists a universal matrix $B\in\mathbb{Z}^{m^{\prime}\times n}$ with the following property: For each $b\in\mathbb{Z}^{m}$ , there exists $t\in\mathbb{Z}^{m^{\prime}}$ such that the integer hull of the polyhedron $P=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ is described by $P_{I}=\{x\in\mathbb{R}^{n}\colon Bx\leq t\}$ . Our main result is that $t$ is an affine function of $b$ as long as $b$ is from a fixed equivalence class of the lattice $D\cdot\mathbb{Z}^{m}$ . Here $D\in\mathbb{N}$ is a number that depends on $n$ and $\Delta$ only. Furthermore, $D$ as well as the matrix $B$ can be computed in time depending on $\Delta$ and $n$ only. An application of this result is the solution of an open problem posed by Cslovjecsek et al. (SODA 2024) concerning the complexity of 2-stage-stochastic integer programming problems. The main tool of our proof is the classical theory of Gomory-Chvátal cutting planes and the elementary closure of rational polyhedra.

1 Introduction

An integer program is an optimization problem of the form

\max\{c^{T}x\colon Ax\leq b,\,x\in\mathbb{Z}^{n}\},

(1)

where $c\in\mathbb{Z}^{n}$ , $A\in\mathbb{Z}^{m\times n}$ and $b\in\mathbb{Z}^{m}$ . Many optimization problems can be modeled and solved as an integer program, see, e.g. [18, 22, 6, 21]. Unlike linear programming, which can be solved in polynomial time [16], integer programming is known to be NP-complete [3].

An important special case arises if the polyhedron $P=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ is integral, i.e., if $P$ is equal to its integer hull $P_{I}=\operatorname{conv}(\mathbb{Z}^{n}\cap P)$ . Here $\operatorname{conv}(X)$ denotes the convex hull of a set $X\subseteq\mathbb{R}^{n}$ . In this case, the integer program (1) can be solved in polynomial time with linear programming and integer linear algebra [21]. For example, if the matrix $A$ is totally unimodular, i.e., if each sub-determinant of $A$ is $0$ or $\pm 1$ , then $P$ is integral for each right-hand-side vector $b\in\mathbb{Z}^{m}$ . Here, the notion sub-determinant of $A$ refers to the determinant of some square sub-matrix of $A$ . Efficient algorithms for integer programs defined by matrices $A$ with a fixed sub-determinant bound is an active area of research [2, 11, 1].

In the following we consider polyhedra defined by the matrix $A\in\mathbb{Z}^{m\times n}$ with varying right-hand-side $b\in\mathbb{Z}^{m}$ and denote the thereby parameterized polyhedron by $P(b)=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ . Cook, Gerards, Schrijver and Tardos [8] have shown that there exists a universal integral matrix $M\in\mathbb{Z}^{m^{\prime}\times n}$ that depends only on the matrix $A$ with the following property:

For each right-hand-side $b\in\mathbb{Z}^{m}$ there exists a right-hand-side $t\in\mathbb{Z}^{m^{\prime}}$ with

$P(b)_{I}=\{x\in\mathbb{R}^{n}\colon Mx\leq t\}.$ (2)

Furthermore, Cook et al. show that $\|M\|_{\infty}\leq n^{2n}\delta^{n}$ . Here $\delta$ is an upper bound on the sub-determinants of $A$ and $\|M\|_{\infty}$ denotes the largest absolute value of a component of $M$ .

Contribution

We prove that the right-hand-side $t\in\mathbb{Z}^{m^{\prime}}$ in (2) depends linearly on the right hand side $b\in\mathbb{Z}^{m}$ for all $b$ belonging to the same equivalence class of a certain lattice that depends on $\Delta$ and $n$ only. We make the natural assumption that $A\in\mathbb{Z}^{m\times n}$ does not have repeating rows. Recall that the number $m$ of rows is bounded by $m\leq(2\Delta+1)^{n}$ in this case. Our main contribution is the following.

Theorem 1.

Let $A\in\mathbb{Z}^{m\times n}$ with non-repeating rows and $\|A\|_{\infty}\leq\Delta$ . There exists a $D\in\mathbb{N}$ depending on $n$ and $\Delta$ such that, for each $r\in\{0,\dots,D-1\}^{m}$ , there exist $B_{r}\in\mathbb{Z}^{m^{\prime}\times n}$ , $C_{r}\in\mathbb{Z}^{m^{\prime}\times m}$ and $f_{r}\in\mathbb{Z}^{m^{\prime}}$ such that the following holds:

For each $b\in\mathbb{Z}^{m}$ with $b-r\in{D\cdot\mathbb{Z}^{m}}$ , one has

$P(b)_{I}=\left\{x\in\mathbb{R}^{n}\colon B_{r}x\leq f_{r}+C_{r}b\right\}.$ (3)

Furthermore, the number $D\in\mathbb{N}$ , the matrices $B_{r}$ and $C_{r}$ , as well as the vector $f_{r}$ can be computed in time depending on $\Delta$ and $n$ only.

Remark 1.

The condition that $A\in\mathbb{Z}^{m\times n}$ has non-repeating rows can also be dropped. In this case, $D$ , $B_{r}$ and $C_{r}$ , as well as the vector $f_{r}$ , as well as the running time depend on $\Delta$ , $n$ , and $m$ .

Applications

We describe three almost immediate applications of Theorem 1.

i) A 2-stage stochastic integer programming problem is an integer program (1) where the constraints can be described as maximizing $c^{T}x+\sum_{i=1}^{n}d_{i}^{T}y_{i}$ subject to

U_{i}x+V_{i}y_{i}=b_{i},\,i=1,\dots,n\,\text{ and }\,x,y_{1},\dots,y_{n}\in\mathbb{Z}_{\geq 0}^{k}.

If $\Delta$ denotes the largest absolute value of an entry of the integral constraint matrices $U_{i},V_{i}\in\mathbb{Z}^{k\times k}$ , then the problem can be solved in time $f(\Delta,k)$ -times a polynomial in the input encoding [14, 17, 9]. See also [15] for a doubly exponential lower bound on $f$ . Recently Clovjecsek et al. [10] have proved that the integer feasibility problem of 2-stage stochastic integer programming can be solved in time $f(\Delta,k)$ -times a polynomial in the input encoding. This time, $\Delta$ is a bound on the largest absolute value of the matrices $V_{i}$ only. The authors pose the open problem whether such a complexity bound follows for the full problem (optimization version) as well. We answer this question in the affirmative in Section 4.2.

ii) Clovjecsek et al. [10] provide an important structural result on the convexity of integer cones. In a nutshell, their result shows that the elements of an integer cone generated by a matrix $W\in\mathbb{Z}^{m\times n}$ are partitioned by a finite number of shifts of a lattice $\Lambda=D\cdot\mathbb{Z}^{m}$ . More precisely, for each $r\in\{0,\dots,D-1\}^{m}$ there exists a polyhedron $Q_{r}\subseteq\mathbb{R}^{m}$ such that

(r+\Lambda)\cap\operatorname{intcone}(W)=(r+\Lambda)\cap Q_{r}.

Their original proof is fairly involved. It is implied by our main theorem very quickly as we lay out in Section 4.1.

iii) Finally, we turn our attention to 4-block integer programming problems for which there only exist polynomial-time algorithms, where the degree of the polynomial depends on the parameters $\Delta$ and $k$ [13]. We prove that in time $f(\Delta,k)$ times a polynomial of fixed degree one can at least find an almost feasible solution that violates the $O(1)$ many linking constraints by an additive constant depending on $k$ and $\Delta$ . The details are in Section 4.3.

Proof idea

The proof of Theorem 1 is based on the classical theory of Gomory-Chvátal cutting planes and the elementary closure of polyhedra. We will review this theory in Section 2 at a level of detail that is necessary for us here. The reader shall also be referred to the seminal textbooks [21, 6].

However, the main idea can be easily described by resorting to the following main principles of the theory of cutting planes: The elementary closure of a rational polyhedron $P$ is again a rational polyhedron $P^{\prime}$ that satisfies $P_{I}\subseteq P^{\prime}\subseteq P$ . The computation of $P^{\prime}$ is fixed-parameter-tractable in $\Delta$ and $n$ . Furthermore, if $P^{(i)}$ denotes the result of $i$ successive applications of the closure operator, then the Chvátal rank of $P$ is the smallest $i\in\mathbb{N}$ with $P^{(i)}=P_{I}$ . It is always finite and, for us very important, it is bounded by a function on $n$ and $\Delta$ [8]. Together, this indicates that, in order to prove our main result, it is enough to show some analogous statement for $P^{(i)}$ instead for $P_{I}$ directly. This is done in Theorem 5 and constitutes the heart of the proof.

2 Gomory-Chvátal cutting planes

Let $P=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ be a polyhedron where $A\in\mathbb{R}^{m\times n}$ , $b\in\mathbb{R}^{m}$ . For any $c\in\mathbb{Z}^{n}$ and $\delta\geq\max\{c^{T}x\colon Ax\leq b\}$ , the inequality $c^{T}x\leq\lfloor\delta\rfloor$ is called a Gomory-Chvátal cutting plane [12, 5], or briefly cutting plane of $P$ . It is valid for all integer points $x\in P\cap\mathbb{Z}^{n}$ and therefore also for the integer hull $P_{I}$ of $P$ .

Figure 1: The valid inequality

-x_{1}+x_{2}\leq\delta

yields the cutting plane

-x_{1}+x_{2}\leq\lfloor\delta\rfloor

For the ease of notation we will also write $c^{T}x\leq\delta$ for the set of points $\{x\in\mathbb{R}^{n}:c^{T}x\leq\delta\}$ . The intersection of all cutting planes of $P$ is denoted by

P^{\prime}=\bigcap_{\begin{subarray}{c}(c^{T}x\leq\delta)\supseteq P\\ c\in\mathbb{Z}^{n}\end{subarray}}\big{(}c^{T}x\leq\lfloor\delta\rfloor\big{)}.

(4)

A cutting plane $c^{T}x\leq\lfloor\delta\rfloor$ follows from a simple inference rule that is based on linear programming duality. Duality implies that there exists a $\lambda\in\mathbb{R}^{m}_{\geq 0}$ such that

i)

$\lambda^{T}A=c^{T}$ and
ii)

$\lambda^{T}b\leq\delta$ hold.

Therefore, the elementary close can be described as follows

P^{\prime}=\bigcap_{\begin{subarray}{c}\lambda\in\mathbb{R}^{m}_{\geq 0}\\ \lambda^{T}A\in\mathbb{Z}^{n}\end{subarray}}\big{(}(\lambda^{T}A)x\leq\lfloor\lambda^{T}b\rfloor\big{)}.

(5)

If the polyhedron $P$ is rational, then the matrix $A$ in the representation $Ax\leq b$ , as well as $b$ can be chosen to be integral, i.e., $A\in\mathbb{Z}^{m\times n}$ , $b\in\mathbb{Z}^{m}$ . If this is the case, then $P^{\prime}\subseteq P$ and $P^{\prime}$ is a rational polyhedron as well [20]. We repeat the argument below and provide a useful bound on the $\ell_{\infty}$ -norm of the left-hand-side vector $c$ of a non-redundant cutting plane $c^{T}x\leq\lfloor\delta$ $\rfloor$ .

Theorem 2 (Schrijver [20]).

Let $P=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ be a polyhedron with $A\in\mathbb{Z}^{m\times n}$ , $\|A\|_{\infty}\leq\Delta$ and $b\in\mathbb{Z}^{m}$ . Then

P^{\prime}=\bigcap_{\begin{subarray}{c}\lambda\in[0,1)^{m}\\ \lambda^{T}A\in\mathbb{Z}^{n}\end{subarray}}\big{(}(\lambda^{T}A)x\leq\lfloor\lambda^{T}b\rfloor\big{)}.

In particular, in (4), $P^{\prime}$ is described by all cutting planes $c^{T}x\leq\lfloor\delta\rfloor$ with $c\in\mathbb{Z}^{n}$ and $\|c\|_{\infty}\leq n\cdot\Delta$ .

Proof.

Consider a cutting plane $c^{T}x\leq\lfloor\delta\rfloor$ derived as $(\lambda^{T}A)x\leq\lfloor\lambda^{T}b\rfloor$ with $\lambda\in\mathbb{R}_{\geq 0}^{m}$ , $\lambda^{T}A=c^{T}$ and $\lambda^{T}b\leq\delta$ .

The inequality $(\lambda-\lfloor\lambda\rfloor)^{T}Ax\leq\lfloor(\lambda-\lfloor\lambda\rfloor)^{T}b\rfloor$ is again a cutting plane, since $(\lambda-\lfloor\lambda\rfloor)^{T}A\in\mathbb{Z}^{n}$ . We have

\lambda^{T}Ax=(\lambda-\lfloor\lambda\rfloor)^{T}Ax+\lfloor\lambda\rfloor^{T}Ax\leq\lfloor(\lambda-\lfloor\lambda\rfloor)^{T}b\rfloor+\lfloor\lambda\rfloor^{T}b=\lfloor\lambda^{T}b\rfloor.

This implies that the inequality $(\lambda^{T}A)x\leq\lfloor\lambda^{T}b\rfloor$ is linearly implied by the system $Ax\leq b$ and the inequality $(\lambda-\lfloor\lambda\rfloor)^{T}Ax\leq\lfloor(\lambda-\lfloor\lambda\rfloor)^{T}b\rfloor$ .

Furthermore, we can assume that $\lambda$ is an optimal vertex solution to the linear program

\min\{\lambda^{T}b\colon\lambda^{T}A=c^{T},\,\lambda\in\mathbb{R}^{m}_{\geq 0}\},

(6)

which implies that $\lambda$ has only $n$ nonzero components. From there, it follows that

\|(\lambda-\lfloor\lambda\rfloor)^{T}A\|_{\infty}\leq n\cdot\Delta.

∎

By applying the closure operator $i$ -times successively, one obtains $P^{(i)}$ , the $i$ -th elementary closure of $P\subseteq\mathbb{R}^{n}$ . The next corollary is immediate.

Corollary 3.

Let $P=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ be a rational polyhedron with $A\in\mathbb{Z}^{m\times n}$ , $\|A\|_{\infty}\leq\Delta$ and $b\in\mathbb{Z}^{m}$ . Let $i\in\mathbb{Z}_{\geq 0}$ . Then $P^{(i)}$ is described by a system of inequalities

P^{(i)}=\{x\in\mathbb{R}^{n}\colon Cx\leq d\},

$C\in\mathbb{Z}^{m^{\prime}\times n}$ , $d\in\mathbb{Z}^{m^{\prime}}$ with $\|C\|_{\infty}\leq n^{i}\Delta$ .

The rank of rational polyhedra

The Chvátal rank [20] of a polyhedron $P$ is the smallest natural number $i\in\mathbb{Z}_{\geq 0}$ such that $P^{(i)}=P_{I}$ . Schrijver [20] has shown that the Chvátal rank of a rational polyhedron is always finite.

Cook et al. [7] showed that the Chvátal rank of integer empty polyhedra is bounded by a function on the dimension. More precisely, they show that, the rank of $P\subseteq\mathbb{R}^{n}$ with $P\cap\mathbb{Z}^{n}=\emptyset$ is bounded by a function $g(n)$ that satisfies the recursion

g(n)\leq\omega(n)(1+g(n-1))+1,

(7)

where $\omega(n)$ is the flatness constant in dimension $n$ . The best known bound on $\omega(n)$ is $O(n\log^{3}(2n))$ , see [19], which gives a bound of $g(n)\leq n^{O(n)}$ .

Most relevant for us is another result of Cook et al. [8] that states that the rank can be bounded by $\Delta$ and $n$ . It follows from a proximity theorem, the bound on $g(n)$ (7) and the bound on the $\ell_{\infty}$ -norm of the facet normal-vectors of $P_{I}$ , see also [21, Theorem 23.4]. More precisely, they show that the rank of $P(b)$ is bounded by

\max\left\{g(n),n^{2n+2}\delta^{n+1}+1+n^{2n+2}\delta^{n+1}g(n-1)\right\}

(8)

where $g(n)$ is as in (7) and $\delta$ is an upper bound on the largest sub-determinant of $A$ . With the Hadamard bound on $\delta$ one can see that there exists a constant $C\in\mathbb{N}$ such that (8) is bounded from above by $(n\cdot\Delta)^{C\cdot n^{2}}$ . We summarize this in the following theorem.

Theorem 4 (Cook, Gerards, Schrijver and Tardos [8]).

Let $P=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}$ be a rational polyhedron with $A\in\mathbb{Z}^{m\times n}$ , $\|A\|_{\infty}\leq\Delta$ and $b\in\mathbb{Z}^{m}$ . There exists a function $\operatorname{Rank}(\Delta,n)$ such that

P^{(i)}=P_{I}

for each $i\geq\operatorname{Rank}(\Delta,n)$ . Moreover, there exists a constant $C\in\mathbb{N}$ such that

\operatorname{Rank}(\Delta,n)\leq(n\cdot\Delta)^{C\cdot n^{2}}.

(9)

3 Proof of the main theorem

This section contains the proof of Theorem 1. Throughout, we assume that $A\in\mathbb{Z}^{m\times n}$ satisfies $\|A\|_{\infty}\leq\Delta$ . The strategy is to show an analogous statement to Theorem 1 for the $i$ -th elementary closure. Let us start with the first Gomory-Chvátal closure.

3.1 Linearity of the first elementary closure

Consider a cutting plane $c^{T}x\leq\lfloor\delta\rfloor$ , where $c\in\mathbb{Z}^{n}$ . If this cutting plane is non-redundant, then there exists an optimal vertex solution $\lambda\in\mathbb{R}^{m}_{\geq 0}$ of the linear program (6) with $\delta\geq\lambda^{T}b$ . This means that there are at most $n$ linearly independent rows of $A$ , indexed by $B\subseteq\{1,\dots,m\}$ such that $\lambda_{B}$ is the unique solution of the linear equation

\lambda_{B}^{T}A_{B}=c^{T},

with all other components of $\lambda$ equal to zero. By deleting linearly dependent columns of $A_{B}$ and the corresponding components of $c$ , this shows that there is a $|B|\times|B|$ non-singular sub-matrix $\overline{A}$ of $A$ with

\lambda_{B}^{T}\overline{A}\in\mathbb{Z}^{|B|}.

Cramer’s rule implies then that $\lambda=\mu/D$ where $\mu\in\mathbb{Z}_{\geq 0}^{m}$ , where $D\in\mathbb{N}$ is a multiple of all sub-determinants of $A$ .

Furthermore, we have seen in Theorem 2 that we can replace $\lambda\in\mathbb{R}_{\geq 0}^{m}$ by $\lambda-\lfloor\lambda\rfloor$ . Thus we have

\lambda=\mu/D\,\text{ with }\,\mu\in\{0,\ldots,D-1\}^{m}.

A Gomory-Chvátal cut $(\mu/D)^{T}Ax\leq\lfloor(\mu/D)^{T}b\rfloor$ with $\mu\in\{0,\ldots,D-1\}^{m}$ and $\mu^{T}A\equiv 0\pmod{D}$ is called $\mathrm{mod-}D$ cut [4]. In other words, if $D\in\mathbb{N}$ is an integer multiple of the sub-determinants of $A$ , then the set of $\mathrm{mod-}D$ cuts contains all non-redundant cutting planes.

Let us now fix a $\mu\in\{0,\ldots,D-1\}^{m}$ with $\mu^{T}A\equiv 0\pmod{D}$ as well as a remainder $r\in\{0,\ldots,D-1\}^{m}$ and consider $b\in\mathbb{Z}^{m}$ with $b\equiv r\pmod{D}$ . The cutting plane derived by $\lambda=\mu/D$ can be written as

\begin{array}[]{rcl}\underbrace{\left({\mu}/{D}\right)^{T}A}_{\text{row of }B_{r}}x&\leq&\lfloor(\mu/D)^{T}b\rfloor\\ &=&\underbrace{\lfloor(\mu^{T}r)/D\rfloor-(\mu^{T}r)/D}_{\text{component of }f_{r}}+\underbrace{(\mu^{T}/D)}_{\text{row of }C_{r}}b,\end{array}

(10)

where the equation above follows from the fact that $\mu^{T}(b-r)/D$ is an integer. We conclude that the right-hand-side of the cut induced by $\lambda=\mu/D$ is indeed is linear in $b$ .

For fixed $r\in\{0,\ldots,D-1\}^{m}$ we therefore enumerate all $\mu\in\{0,\ldots,D-1\}^{m}$ with $\mu^{T}A\equiv 0\pmod{D}$ , fill the corresponding rows/components of $B_{r},C_{r}$ and $f_{r}$ as in (10) and append the original inequalities $Ax\leq b$ .

Remark 2.

The above constructed $C_{r}$ and $f_{r}$ are rational and not-necessarily integral. In order to be conform with the notation in Theorem 1 we thus scale with the least common multiple of the denominators to obtain integral data.

3.2 Linearity of the $i$ -th elementary closure

We now proceed with the general statement and its proof. Recall that the Chvátal rank of $P(b)$ is bounded by $\operatorname{Rank}(\Delta,n)$ and that each $P(b)^{(i)}$ can be described by a constraint matrix with infinity norm bounded by $n^{\max\{i,\operatorname{Rank}(\Delta,n)\}}\Delta$ . We define $D\in\mathbb{N}$ to be the least common multiple of nonzero sub-determinants of $n\times n$ -integer matrices with entries bounded by $n^{\operatorname{Rank}(\Delta,n)}\cdot\Delta$ in absolute value. In other words, let

D=\operatorname{lcm}\left\{\det(C)\colon C\in\mathbb{Z}^{k\times k},\,k\leq n,\,\|C\|_{\infty}\leq n^{\operatorname{Rank}(\Delta,n)}\cdot\Delta,\,\det(C)\neq 0\right\}.

(11)

It follows that $P^{(i+1)}$ is defined by $\mathrm{mod-}D$ -cuts derived from the description of $P^{(i)}$ as in Corollary 3. In other words, $D$ is the largest modulus that is ever necessary in the derivation of a valid cutting plane. Theorem 1 is now an immediately corollary of the following statement concerning $P^{(i)}$ .

Theorem 5.

Let $A\in\mathbb{Z}^{m\times n}$ with non-repeating rows and $\|A\|_{\infty}\leq\Delta$ and let $D$ be as in (11). For each $i\in\mathbb{Z}_{\geq 0}$ and $r\in\{0,\dots,D^{i}-1\}^{n}$ , there exist $B_{r}\in\mathbb{Z}^{m^{\prime}\times n}$ , $C_{r}\in\mathbb{Z}^{m^{\prime}\times m}$ and $f_{r}\in\mathbb{Z}^{m^{\prime}}$ such that for each $b\in\mathbb{Z}^{m}$ with $b-r\in D^{i}\cdot\mathbb{Z}^{m}$ , one has

P(b)^{(i)}=\left\{x\in\mathbb{R}^{n}\colon B_{r}\,x\leq f_{r}+\frac{C_{r}(b-r)}{D^{i}}\right\}.

(12)

Furthermore, $B_{r}$ satisfies $\|B_{r}\|_{\infty}\leq n^{i}\Delta$ .

Proof.

We prove the assertion by induction on $i$ . For $i=0$ , one has

P(b)^{(0)}=P(b)=\{x\in\mathbb{R}^{n}\colon Ax\leq b\}.

Each $b\in\mathbb{Z}^{m}$ is congruent to $\bm{0}$ modulo ${1=D^{0}}$ . We set, for $r=\bm{0}$ ,

B_{r}=A,\,f_{r}=\bm{0},\text{ and }C_{r}=I.

We assume that the assertion is true for $i\in\mathbb{Z}_{\geq 0}$ and show that it is true for $i+1$ as well. If $i\geq\operatorname{Rank}(\Delta,n)$ , then $P(b)^{(i+1)}=P(b)^{(i)}$ and we are done.

Otherwise let $r\in\{0,\dots,D^{i+1}-1\}^{m}$ and let $b\equiv r\pmod{D^{(i+1)}}$ . By the induction hypothesis,

P(b)^{(i)}=\left\{x\in\mathbb{R}^{n}\colon B_{r}x\leq f_{r}+\frac{C_{r}(b-r)}{D^{i}}\right\}

for each $b$ with $b\equiv r\pmod{D^{i}}$ and in particular for each $b$ with $b\equiv r\pmod{D^{i+1}}$ . The absolute values of the entries of $B_{r}\in\mathbb{Z}^{m^{\prime}\times n}$ are bounded from above by $n^{i}\Delta$ .

We now construct the cutting planes of $P(b)^{(i)}$ . The discussion above implies that a non-redundant cutting plane of $P(b)^{(i)}$ is of the form

(\mu/D)^{T}B_{r}x\leq\left\lfloor(\mu/D)^{T}\left(f_{r}+\frac{C_{r}(b-r)}{D^{i}}\right)\right\rfloor,

(13)

where $\mu\in\{0,\dots,D-1\}^{m^{\prime}}$ is of support at most $n$ and $\mu^{T}B_{r}\equiv 0\pmod{D}$ . We wish to describe integer matrices $B_{r}^{\prime}$ , $C_{r}^{\prime}$ and an integer vector $f_{r}^{\prime}$ such that

P(b)^{(i+1)}=\left\{x\in\mathbb{R}^{n}\colon B_{r}^{\prime}x\leq f_{r}^{\prime}+\frac{C_{r}^{\prime}(b-r)}{D^{i+1}}\right\}

Let $B_{r}^{\prime}$ be the integer matrix whose rows are comprised first, of the rows of $B_{r}$ and then of the left-hand-side vectors of cutting planes (13). Clearly, $\|B_{r}^{\prime}\|_{\infty}\leq n^{i+1}\Delta$ . Since $b\equiv r\pmod{D^{i+1}}$ , the right-hand-side of (13) is

	$\displaystyle\left\lfloor(\mu/D)^{T}\left(f_{r}+\frac{C_{r}(b-r)}{D^{i}}\right)\right\rfloor$	$\displaystyle=$	$\displaystyle\left\lfloor(\mu/D)^{T}f_{r}+\frac{\mu^{T}C_{r}(b-r)}{D^{i+1}}\right\rfloor$
		$\displaystyle=$	$\displaystyle\underbrace{\left\lfloor(\mu/D)^{T}f_{r}\right\rfloor}_{\text{ component of }f^{\prime}_{r}}+\frac{\overbrace{\mu^{T}C_{r}}^{\text{row of }C^{\prime}_{r}}(b-r)}{D^{i+1}},$

where the last equation follows from the fact that each component of $b-r$ is divisible by $D^{i+1}$ and therefore that

\frac{\mu^{T}C_{r}(b-r)}{D^{i+1}}\in\mathbb{Z}.

∎

Remark 3.

i)

The proof of Theorem 5 contains a finite and constructive procedure to compute $D$ , the matrices $B_{r},C_{r}$ and the vector $f_{r}$ . The input to this procedure is the matrix $A\in\mathbb{Z}^{m\times n}$ . Since $A$ does not have repeated rows, the running time of this procedure is a function in $\Delta$ and $n$ only.
ii)

The condition in Theorem 5 that $A\in\mathbb{Z}^{m\times n}$ has non-repeating rows can be dropped. In this case, we have a dependence on $n$ , $m$ and $\Delta$ .

iii)

Using the Hadamard bound and $\operatorname{Rank}(\Delta,n)\leq(n\cdot\Delta)^{C\cdot n^{2}}$ from Theorem 4, the number $D$ can be bounded from above by

	$\displaystyle D$	$\displaystyle\leq$	$\displaystyle\operatorname{lcm}\left\{1,\dots,n^{{(n\cdot\Delta)}^{C^{\prime}\cdot n^{2}}}\right\}$
		$\displaystyle\leq$	$\displaystyle 2^{n^{{n\cdot\Delta}^{C^{\prime\prime}\cdot n^{2}}}},$

where $C^{\prime},C^{\prime\prime}\in\mathbb{N}_{+}$ are suitable constants. This bound is triple-exponential in $n$ .

4 Applications

4.1 The convexity of integer cones

For a matrix $W\in\mathbb{Z}^{m\times n}$ , we denote the integer cone as $\operatorname{intcone}(W)=\{Wx:x\in\mathbb{Z}_{\geq 0}^{n}\}$ . An integer cone is a discrete set and in general it will have “holes”, i.e., integer points in the rational cone of $W$ that are not in the integer cone. But Cslovjecsek, Koutecký, Lassota, Pilipczuk and Polak [10] proved that integer cones are indeed convex in a discrete sense when intersected with shifts of certain sparse lattices.

Theorem 6 (Convexity of Integer Cones [10]).

Let $W\in\mathbb{Z}^{m\times n}$ with $\|W\|_{\infty}\leq\Delta$ . Then there is a $D\in\mathbb{N}$ dependent only on $m$ and $\Delta$ so that for every $r\in\{0,\ldots,D-1\}^{m}$ there exists a polyhedron $Q_{r}\subseteq\mathbb{R}^{m}$ so that

\Lambda_{r}\cap\operatorname{intcone}(W)=\Lambda_{r}\cap Q_{r}

where $\Lambda_{r}=r+D\cdot\mathbb{Z}^{m}$ .

The original proof of this result takes a substantial amount of work. We demonstrate that it is quickly implied by our main result.

Proof of Theorem 6.

We can assume that $W\in\mathbb{Z}^{m\times n}$ does not have repeated columns. Therefore, $n$ is bounded by $(2\cdot\Delta+1)^{m}$ . Consider $P(b)=\{x\in\mathbb{R}^{n}:Wx=b,x\geq\bm{0}\}$ . Observe that the condition $Wx=b$ and $x\geq\bm{0}$ can be formulated in the usual inequality standard-form as the conjunction of $Wx\leq b$ , $-Wx\leq-b$ and $-Ix\leq\bm{0}$ .

Let $D$ be as in Theorem 1 depending on $\Delta$ and $n$ (and thus $m$ ) only. Now, fix any $r\in\{0,\ldots,D-1\}^{m}$ and let $B_{r},C_{r},f_{r}$ be the vectors so that

P(b)_{I}=\{x\in\mathbb{R}^{n}:B_{r}x\leq f_{r}+C_{r}b\}

for all $b\in\mathbb{Z}^{m}$ with $b\equiv r\pmod{D}$ . Let $Q_{r}=\{b\in\mathbb{R}^{m}\mid\exists x\in\mathbb{R}^{n}:Bx\leq f+Cb\}$ . Then $Q_{r}$ is the linear projection of a polyhedron and hence it is again a polyhedron.

Now, consider a right hand side $b\in\mathbb{Z}^{m}$ with $b\equiv r\pmod{D}$ . Then

$\displaystyle b\in\operatorname{intcone}(W)$	$\displaystyle\Leftrightarrow$	$\displaystyle P(b)\cap\mathbb{Z}^{n}\neq\emptyset$
	$\displaystyle\Leftrightarrow$	$\displaystyle P(b)_{I}\neq\emptyset$
	$\displaystyle\Leftrightarrow$	$\displaystyle b\in Q_{r}.$

∎

4.2 Optimizing over 2-stage stochastic IPs

A 2-stage stochastic integer program is of the form

$\displaystyle\max c^{T}x+\sum_{i=1}^{n}d_{i}^{T}y_{i}$		$\displaystyle(\textsc{2SSIP})$
$\displaystyle U_{i}x+V_{i}y_{i}$	$\displaystyle=$	$\displaystyle b_{i}\quad\forall i=1,\ldots,n$
$\displaystyle x,y_{1},\ldots,y_{n}$	$\displaystyle\in$	$\displaystyle\mathbb{Z}_{\geq 0}^{k},$

with integer matrices $U_{i},V_{i}\in\mathbb{Z}^{k\times k}$ . It will be useful to think of $k$ as a constant and $n$ as large. Then after determining the $k$ variables in $x$ , the rest of the IP decomposes into $n$ disjoint parts, again with $k$ variables each. It was proven in Cslovjecsek et al [10] that the decision version (without an objective function) can be solved in time $g(k,\max_{i}\|V_{i}\|_{\infty})$ times a fixed polynomial in the encoding length. The reader may note that this running time has only a polylogarithmic dependence on $\|U_{i}\|_{\infty}$ . The argument of [10] is based on Theorem 6 which does not allow “access” to the coefficient vectors $y_{i}$ and so the authors left it as an open problem whether the same FPT-type running time is possible for the optimization variant as stated above in $({\sc{2SSIP}})$ . We solve this open problem in the affirmative:

Theorem 7.

The problem $(\textsc{2SSIP})$ can be solved in time $g(k,\Delta)$ times a fixed polynomial in the encoding length where $\Delta=\max\{\|V_{i}\|_{\infty}:i=1,\ldots,n\}$ .

Proof.

We use the following algorithm:

(1)

Let $P_{i}(b^{\prime})=\{y_{i}\in\mathbb{R}^{k}\mid V_{i}y_{i}=b^{\prime},y_{i}\geq\bm{0}\}$ for $i=1,\ldots,n$ .
(2)

Let $D$ be the value as in Theorem 1 that works for parameters $k$ and $\Delta$ .
(3)

Guess $r\in\{0,\ldots,D-1\}^{k}$ so that $x^{**}\equiv r\pmod{D}$ where $(x^{**},y_{1}^{**},\ldots,y_{n}^{**})$ is an optimum integral solution to $(\textsc{2SSIP})$ .
(4)

For all $i=1,\ldots,n$ , let $r_{i}\in\{0,\ldots,D-1\}^{k}$ with $r_{i}=b_{i}-U_{i}r\pmod{D}$ .
(5)

For each $i\in[n]$ , apply Theorem 1 to write $P_{i}(b^{\prime})_{I}=\{y_{i}\in\mathbb{R}^{k}\mid B_{i}y_{i}\leq f_{i}+C_{i}b^{\prime}\}$ for all $b^{\prime}$ with $b^{\prime}\equiv r_{i}\pmod{D}$ .

(6)

Compute an optimum extreme point solution $(x^{*},y_{1}^{*},\ldots,y_{n}^{*})$ to the mixed integer linear program

$\displaystyle\max c^{T}x+\sum_{i=1}^{n}d_{i}^{T}y_{i}$
$\displaystyle B_{i}y_{i}$	$\displaystyle\leq$	$\displaystyle f_{i}+C_{i}(b_{i}-U_{i}x)$
$\displaystyle x$	$\displaystyle\equiv$	$\displaystyle r\pmod{D}$
$\displaystyle x$	$\displaystyle\in$	$\displaystyle\mathbb{Z}_{\geq 0}^{k}$

(7)

Return $(x^{*},y_{1}^{*},\ldots,y_{n}^{*})$

We know that $b_{i}-U_{i}x^{**}\equiv r_{i}$ and so $(x^{**},y_{1}^{**},\ldots,y_{n}^{**})$ is feasible for the MIP in (6). The MIP in $(6)$ has only $k$ many integral variables (and $nk$ many fractional ones) and hence it can be solved within the claimed running time. On the other hand, if $(x^{*},y_{1}^{*},\ldots,y_{n}^{*})$ is an optimum extreme point to (6), then $x^{*}\in\mathbb{Z}_{\geq 0}^{k}$ and each $y^{**}_{i}$ is an extreme point to $P_{i}(b_{i}-U_{i}x^{*})_{I}$ . Each such extreme point must be integral. That shows the claim. ∎

4.3 An almost solution to the 4-block problem

The 4-block problem is the IP of the form

$\displaystyle\max c^{T}x+\sum_{i=1}^{n}d_{i}^{T}y_{i}$		$\displaystyle({\textsc{4BlockIP}})$
$\displaystyle Wx+X_{1}y_{1}+\ldots+X_{n}y_{n}$	$\displaystyle=$	$\displaystyle a\quad(*)$
$\displaystyle U_{i}x+V_{i}y_{i}$	$\displaystyle=$	$\displaystyle b_{i}\quad\forall i=1,\ldots,n$
$\displaystyle x,y_{1},\ldots,y_{n}$	$\displaystyle\in$	$\displaystyle\mathbb{Z}_{\geq 0}^{k}$

where $U_{i},V_{i},W,X_{i}\in\mathbb{Z}^{k\times k}$ and $a,b_{i}\in\mathbb{Z}^{k}$ . This is a strict generalization of $({\textsc{2SSIP}})$ . Note that after deleting $k$ many variables and $k$ many constraints, the problem decomposes into disjoint IPs with $k$ variables each. Indeed, $(\textsc{4BlockIP})$ can be solved in time $n^{f(k,\Delta)}$ times a fixed polynomial in the encoding length where $\Delta=\max\{\|U_{i}\|_{\infty},\|V_{i}\|_{\infty},\|W\|_{\infty},\|X_{i}\|_{\infty}:i\in[n]\}$ , see the work of Hemmecke, Köppe and Weismantel [13]. It is a popular open problem in the theoretical IP community whether there is an FPT-type algorithm as well.

Conjecture 1 (4-block conjecture).

$(\textsc{4BlockIP})$ can be solved in time $f(k,\Delta)$ times a fixed polynomial in the encoding length where $\Delta$ is the largest absolute value appearing in $U_{i},V_{i},W,X_{i}$ .

We prove that one can at least find an almost feasible solution that violates the $O(1)$ many joint constraints by an additive $O(1)$ (assuming $k$ and the maximum entries are bounded by constants):

Theorem 8.

Suppose that $(\textsc{4BlockIP})$ is feasible with value $OPT$ and let $\Delta=\max\{\|V_{i}\|_{\infty}:i\in[n]\}$ . Then in time $f(k,\Delta)$ times a polynomial in the input length one can find a vector $(x^{*},y_{1}^{*},\ldots,y_{n}^{*})$ with $c^{T}x^{*}+\sum_{i=1}^{n}d_{i}^{T}y_{i}^{*}\geq OPT$ that satisfies all constraints except $(*)$ . Moreover, the error for constraint $(*)$ is bounded by

\big{\|}(Wx^{*}+X_{1}y_{1}^{*}+\ldots+X_{n}y_{n}^{*})-a\big{\|}_{\infty}\leq g(k,\max_{i=1,\ldots,n}\|X_{i}\|_{\infty})

Proof.

We use a similar argument to Theorem 7. Again set $P_{i}(b^{\prime})=\{y_{i}\in\mathbb{R}^{k}\mid V_{i}y_{i}=b^{\prime},y_{i}\geq\bm{0}\}$ . Let $(x^{**},y_{1}^{**},\ldots,y_{n}^{**})$ be a solution to $({\textsc{4BlockIP}})$ attaining the value of $OPT$ . Again we can guess a vector $r$ so that $x^{**}\equiv r\pmod{D}$ where $D$ is a parameter so that by Theorem 1 one has

P_{i}(b^{\prime})_{I}=\{y_{i}\in\mathbb{R}^{k}\mid B_{i}y_{i}\leq f_{i}+C_{i}b^{\prime}\}

for all $b^{\prime}$ with $b^{\prime}\equiv r_{i}\pmod{D}$ where $r_{i}=b_{i}-U_{i}r\pmod{D}$ . Consider the mixed integer linear program

$\displaystyle\max c^{T}x+\sum_{i=1}^{n}d_{i}^{T}y_{i}$		$\displaystyle({\textsc{4BlockMIP}})$
$\displaystyle Wx+X_{1}y_{1}+\ldots+X_{n}y_{n}$	$\displaystyle=$	$\displaystyle a\quad(**)$
$\displaystyle y_{i}$	$\displaystyle\in$	$\displaystyle P_{i}(b_{i}-U_{i}x)_{I}\quad\forall i=1,\ldots,n$
$\displaystyle x$	$\displaystyle\equiv$	$\displaystyle r\pmod{D}$
$\displaystyle x$	$\displaystyle\in$	$\displaystyle\mathbb{Z}_{\geq 0}^{k}$

We compute an optimum extreme point solution $(x^{*},\tilde{y}_{1},\ldots,\tilde{y}_{n})$ to $({\textsc{4BlockMIP}})$ . We note that $x^{*}$ will be integral, but as all vertices of $P_{i}(\cdot)_{I}$ are integral and $(**)$ are only $k$ non-trivial constraints, the set $J=\{i\in[n]\mid\tilde{y}_{i}\notin\mathbb{Z}^{k}\}$ has size $|J|\leq k$ . Then by standard proximity bounds, for each $i\in J$ , there is a $y_{i}^{*}\in P_{i}(b_{i}-U_{i}x^{*})_{I}\cap\mathbb{Z}^{k}$ with $\|y_{i}^{*}-\tilde{y}_{i}\|_{\infty}\leq(\Delta k)^{O(k)}$ and $d_{i}^{T}y_{i}^{*}\geq d_{i}^{T}\tilde{y}_{i}$ . We set $y_{i}^{*}=\tilde{y}_{i}$ for all $i\notin J$ and $(x^{*},y_{1}^{*},\ldots,y_{n}^{*})$ satisfies the claim. ∎

References

[1] M. Aprile, S. Fiorini, G. Joret, S. Kober, M. T. Seweryn, S. Weltge, and Y. Yuditsky. Integer programs with nearly totally unimodular matrices: The cographic case. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, SODA 25, 2025. to appear.
[2] S. Artmann, R. Weismantel, and R. Zenklusen. A strongly polynomial algorithm for bimodular integer linear programming. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1206–1219, 2017.
[3] I. Borosh and L. B. Treybig. Bounds on positive integral solutions of linear diophantine equations. Proceedings of the American Mathematical Society, 55(2):299–304, 1976.
[4] A. Caprara, M. Fischetti, and A. N. Letchford. On the separation of maximally violated mod- $k$ cuts. Mathematical Programming, 87(1):37–56, 2000.
[5] V. Chvátal. Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics, 4:305–337, 1973.
[6] M. Conforti, G. Cornuéjols, and G. Zambelli. Integer Programming. Graduate Texts in Mathematics. Springer International Publishing, 2014.
[7] W. Cook, C. R. Coullard, and G. Turán. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25–38, 1987.
[8] W. Cook, A. M. H. Gerards, A. Schrijver, and É. Tardos. Sensitivity theorems in integer linear programming. Mathematical Programming, 34(3):251–264, 1986.
[9] J. Cslovjecsek, F. Eisenbrand, C. Hunkenschröder, L. Rohwedder, and R. Weismantel. Block-structured integer and linear programming in strongly polynomial and near linear time. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1666–1681. SIAM, 2021.
[10] J. Cslovjecsek, M. Kouteckỳ, A. Lassota, M. Pilipczuk, and A. Polak. Parameterized algorithms for block-structured integer programs with large entries. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 740–751. SIAM, 2024.
[11] S. Fiorini, G. Joret, S. Weltge, and Y. Yuditsky. Integer programs with bounded subdeterminants and two nonzeros per row. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 13–24. IEEE, 2022.
[12] R. E. Gomory. Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society, 64:275–278, 1958.
[13] R. Hemmecke, M. Köppe, and R. Weismantel. A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs. In IPCO, volume 6080 of Lecture Notes in Computer Science, pages 219–229. Springer, 2010.
[14] R. Hemmecke and R. Schultz. Decomposition of test sets in stochastic integer programming. Mathematical Programming, 94:323–341, 2003.
[15] K. Jansen, K.-M. Klein, and A. Lassota. The double exponential runtime is tight for 2-stage stochastic ILPs. Mathematical Programming, 197(2):1145–1172, 2023.
[16] L. Khachiyan. A polynomial algorithm in linear programming. Doklady Akademii Nauk SSSR, 244:1093–1097, 1979.
[17] K.-M. Klein. About the complexity of two-stage stochastic IPs. Mathematical Programming, 192(1–2):319–337, Mar. 2022.
[18] G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. John Wiley, 1988.
[19] V. Reis and T. Rothvoss. The subspace flatness conjecture and faster integer programming. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 974–988. IEEE, 2023.
[20] A. Schrijver. On cutting planes. Annals of Discrete Mathematics, 9:291–296, 1980.
[21] A. Schrijver. Theory of Linear and Integer Programming. John Wiley, 1986.
[22] L. A. Wolsey. Integer Programming. John Wiley & Sons, New York, 1998.

A parameterized linear formulation of the integer hull

Abstract

1 Introduction

Contribution

Theorem 1.

Remark 1.

Applications

Proof idea

2 Gomory-Chvátal cutting planes

Theorem 2 (Schrijver [20]).

Proof.

Corollary 3.

The rank of rational polyhedra

Theorem 4 (Cook, Gerards, Schrijver and Tardos [8]).

3 Proof of the main theorem

3.1 Linearity of the first elementary closure

Remark 2.

3.2 Linearity of the ii-th elementary closure

Theorem 5.

Proof.

Remark 3.

4 Applications

4.1 The convexity of integer cones

Theorem 6 (Convexity of Integer Cones [10]).

Proof of Theorem 6.

4.2 Optimizing over 2-stage stochastic IPs

Theorem 7.

Proof.

4.3 An almost solution to the 4-block problem

Conjecture 1 (4-block conjecture).

Theorem 8.

Proof.

References

3.2 Linearity of the $i$ -th elementary closure