IntegerHull: Convex Hulls of Lattice Points

Updated 10 July 2026

IntegerHull is the convex hull of lattice points in a set, serving as the integral counterpart to basic feasible solutions in integer programming.
Its structural properties include quasipolynomial behavior, periodic affine descriptions, and complexity bounds that vary with the dimensional and planar settings.
Algorithmic paradigms—ranging from cutting plane methods to fixed-dimension techniques—enable efficient computation of integer hulls in both polyhedral and planar cases.

Integer hull denotes the convex hull of the lattice points contained in a set. In the standard polyhedral setting, for $P \subseteq \mathbb{R}^n$ , the integer hull is

$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$

while in the planar convex-body setting the same construction is also called the discrete hull,

$C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$

For linear and combinatorial optimization, vertices of $P_I$ are the integral analogue of basic feasible solutions: every optimal solution of an integer program can be chosen to be a vertex of $P_I$ , and every vertex of $P_I$ is optimal for some objective vector $c$ (Eisenbrand et al., 4 Jan 2025, Har-Peled, 1 Jan 2026, Berndt et al., 2020).

1. Definitions and formal variants

The defining operation is always the same: intersect with a lattice, then take a convex hull. In the polyhedral case, the integer hull is the smallest convex polyhedral set containing every lattice point of the original set. In two dimensions, if the input is a convex polygon or polyhedral set, the integer hull is again a polygon whose vertices are integer points. For compact planar convex bodies, the discrete hull is equivalently the smallest convex polygon whose vertices are lattice points and that contains all grid points of the body.

Setting	Notation	Description
Polyhedron $P\subseteq \mathbb{R}^n$	$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$	Convex hull of all integer points in $P$
Planar convex body $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 0	$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 1	Also called the discrete hull
Mixed-integer polyhedron $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 2	$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 3	Mixed-integer analogue of the integer hull

The literature also considers symbolic inputs that are not initially given by linear inequalities. If $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 4 is a regular language over digit vectors and $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 5, then the convex hull $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 6 is polyhedral and effectively computable. In that setting, Number Decision Diagrams (NDDs) provide the finite symbolic representation, and the geometric output is a finite half-space or ray description (0812.1951, Hildebrand et al., 2014).

2. Structural properties and combinatorial complexity

Integer hulls can be substantially more complicated than their inputs. A particularly sharp warning is that even a triangle can have an integer hull with arbitrarily many vertices. This invalidates the common expectation that passing from a rational polyhedron to its lattice-point hull should be only a minor combinatorial refinement (Calegari et al., 2010).

For polyhedra of the form

$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 7

the cited work studies vertices of $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 8 through their support $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 9. The key obstruction to vertexhood is the existence of a nonzero integer vector $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 0 on the support with $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 1 and $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 2: if such a vector exists, then $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 3, so $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 4 is not a vertex of $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 5. From this criterion one obtains quantitative sparsity bounds. In the knapsack case $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 6,

$C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 7

and in the general $C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 8-row case,

$C^0=\operatorname{ConvexHull}(C\cap \mathbb{Z}^2).$ 9

The same work derives bounds on the number of vertices and on enumeration: for general $P_I$ 0-constraint polytopes,

$P_I$ 1

with the same asymptotic bound for enumeration, and for knapsack the number of integer vertices and an enumeration algorithm are bounded by

$P_I$ 2

Lower-bound constructions show that support can grow like $P_I$ 3, so these estimates are nearly optimal in the sense stated in the paper (Berndt et al., 2020).

In the plane, complexity can be much smaller than ambient scale. For a compact convex body $P_I$ 4 of diameter $P_I$ 5, the number of vertices of the discrete hull satisfies

$P_I$ 6

The proof splits edge vectors into “long” and “short” ones, uses the perimeter bound $P_I$ 7, and balances the corresponding contributions with $P_I$ 8. For disks this yields the same asymptotic estimate, and the bound is also noted to extend to smooth bodies with curvature at least $P_I$ 9, where $P_I$ 0 (Har-Peled, 1 Jan 2026).

3. Parametric, quasipolynomial, and symbolic descriptions

A major theme in recent work is that integer hulls can vary in a highly structured way when the input varies. One formulation concerns right-hand-side families

$P_I$ 1

with $P_I$ 2, no repeated rows, and $P_I$ 3. The parameterized theorem states that there exists an integer $P_I$ 4, depending only on $P_I$ 5 and $P_I$ 6, such that for every residue class $P_I$ 7 there are matrices

$P_I$ 8

and a vector $P_I$ 9 with

$P_I$ 0

for every $P_I$ 1 satisfying $P_I$ 2. Thus, on each congruence class modulo $P_I$ 3, the facet offsets of the integer hull depend affinely on $P_I$ 4; the paper describes this as a strong periodicity statement for integer hulls (Eisenbrand et al., 4 Jan 2025).

A different regularity phenomenon appears in one-parameter families of polyhedra whose vertices are rational functions of $P_I$ 5 with size $P_I$ 6. For such a family $P_I$ 7, the integer hull is eventually a quasipolynomial family: after passing to a cycle, the vertices of $P_I$ 8 are given by integer polynomials in $P_I$ 9. The proof uses simplex decompositions, unimodular normalization via Hermite normal form, and bounded lattice approximation near moving vertices. The $c$ 0 assumption is essential in the argument (Calegari et al., 2010).

Symbolic encodings via finite automata yield a third type of structure. If $c$ 1 for a regular language $c$ 2, then $c$ 3 is polyhedral and can be computed in exponential time from a regular expression. The proof reduces arbitrary regular languages to starred decompositions, uses the affine map

$c$ 4

and shows that Kleene-star operations can be eliminated at the level of convex hulls through the identity $c$ 5. This produces a finite ray representation of the convex hull from an automata-theoretic description (0812.1951).

4. Algorithmic paradigms for computation

One algorithmic line uses cutting planes. The DDM Cuts algorithm computes all vertices and all facets of the convex hull of all integer points of a polyhedron given by linear inequalities. It iteratively chooses a non-integer vertex, generates Gomory-type cuts that exclude that vertex but preserve all integer points, adds the cuts, and uses the dynamic double description method to maintain the exact vertex and facet representations of the current polyhedron. The construction of cuts proceeds from a square tight subsystem $c$ 6, the determinant $c$ 7, and a Smith normal form parameterization of the relevant residue classes. The method returns the full combinatorial description of the integer hull rather than only a feasibility certificate (Semenov et al., 2020).

A second line exploits fixed dimension. For a rational polyhedron $c$ 8, the mixed-integer hull

$c$ 9

can be computed in polynomial time from an inequality description when $P\subseteq \mathbb{R}^n$ 0 and $P\subseteq \mathbb{R}^n$ 1 are fixed. The proof scales the continuous coordinates by an integer $P\subseteq \mathbb{R}^n$ 2 so that the scaled mixed-integer hull becomes an ordinary integer hull, computes that hull, and scales back. For vertex-described input $P\subseteq \mathbb{R}^n$ 3 with fixed $P\subseteq \mathbb{R}^n$ 4, one can compute $P\subseteq \mathbb{R}^n$ 5 in polynomial time in $P\subseteq \mathbb{R}^n$ 6; the structural input is that every mixed-integer-hull vertex lies in a face of $P\subseteq \mathbb{R}^n$ 7 of dimension at most $P\subseteq \mathbb{R}^n$ 8 (Hildebrand et al., 2014).

A third line is implementation-oriented and two-dimensional. Maple’s $P\subseteq \mathbb{R}^n$ 9 library contains an $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 0 function. The baseline method translates the supporting hyperplanes of the facets inward until each hyperplane encounters at least one integer point, subdivides the set into smaller regions, and uses brute force—specifically Fourier–Motzkin elimination followed by integer-point enumeration—on the remaining pieces. A later 2D algorithm modifies the geometric translation rule: it translates each supporting hyperplane inward from the relative opposite vertex, producing smaller residual regions in edge cases where integer points already lie on the facet hyperplanes. The paper presents this as a practical improvement to the 2D base case rather than as a formal complexity theorem (Mukherjee, 11 Sep 2025).

5. Planar convex bodies, continued fractions, and hyperbolas

For compact convex bodies in the plane, an output-sensitive algorithm computes the entire discrete hull in time

$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 1

and constant space, assuming constant-time line intersection queries with $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 2. The method does not scan all lattice points in the body. Instead, it walks from one vertex of $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 3 to the next. The local step starts from a known hull vertex $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 4, interprets the unknown next edge direction as a primitive lattice vector, and simulates a continued-fraction process. Convergents serve as successive best lattice approximations to the unknown slope, and the usual line-based ray shooting is replaced by ray shooting against the convex body itself. A crucial lemma shows that the segment associated with each convergent intersects the true edge segment, while the sequence of convergents grows at least as fast as Fibonacci numbers, yielding $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 5 work per hull edge. Under lattice-connectivity assumptions, a starting vertex can be found in $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 6 time, and if only a point $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 7 is known, the full hull can be computed in

$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 8

For disks of radius $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n)$ 9, this specializes to

$P$ 0

time and $P$ 1 space (Har-Peled, 1 Jan 2026).

Hyperbolic regions form a particularly rich planar family. For

$P$ 2

one study proved that

$P$ 3

and more generally obtained analogous bounds for hyperbolas with rational asymptotic slopes in terms of the discriminant $P$ 4. The same work gives a geometric enumeration algorithm that computes the next hull vertex in logarithmic time per vertex, by maintaining a search basis $P$ 5 and updating it in a manner that mirrors the Euclidean algorithm (Alcántara et al., 31 Jan 2025).

A later paper sharpened the standard-hyperbola count for

$P$ 6

showing that the number of vertices is of order $P$ 7. It also proved that every vertex lies in the thin strip

$P$ 8

and that the area of the part of $P$ 9 inside $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 00 is also of order $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 01. The upper bound uses Minkowski’s theorem and divisor-sum estimates; the lower bound uses a count of primitive vectors together with a detailed analysis of lattice lines through the strip (Balog et al., 6 Feb 2026).

Integer hull methods appear in several distinct application domains. For bounded-coefficient systems $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 02, the affine-on-residue-classes description of integer hulls implies that on each lattice class $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 03, an integer cone $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 04 coincides with a rational polyhedron, and it yields an FPT-type algorithm for optimization in 2-stage stochastic integer programming with running time $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 05 times a polynomial in the input size (Eisenbrand et al., 4 Jan 2025).

For automata-based verification, the polyhedrality of $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 06 for NDD-recognized regular sets connects symbolic representations of infinite-state systems to convex over-approximations and widening. The stated significance is that one can pass mechanically from a finite automaton to a finite polyhedral representation, which is then amenable to standard polyhedral algorithms (0812.1951).

In geometric group theory, the quasipolynomial behavior of integer hulls in rational $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 07-families yields the corollary that for a surgery family $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 08, $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 09 is eventually a ratio of quasipolynomials in $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 10, and unit balls in the scl norm quasi-converge in finite-dimensional surgery families (Calegari et al., 2010).

In computational number theory, enumerating vertices of $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 11 is motivated by deterministic factorization of integers, because integer points on $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 12 are exactly factor pairs of $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 13. The corresponding full hull computation runs in

$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 14

arithmetic operations, where $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 15 is the number of vertices, and using the vertex bound yields

$P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 16

for the standard hyperbola setting (Alcántara et al., 31 Jan 2025).

Implementation work makes the concept operational under the explicit name $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 17. In Maple’s $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 18 library, the $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 19 function computes the integer hull of a polyhedral set, and the newer 2D algorithm was proposed precisely to improve difficult edge cases in which integer points already lie on facet-supporting hyperplanes (Mukherjee, 11 Sep 2025).

The nearby mixed-integer notion is both a natural extension and a caution. Although $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 20 reduces to the ordinary integer hull when $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 21, the mixed-integer hull may have exponentially many vertices and facets in $P_I=\operatorname{conv}(P\cap \mathbb{Z}^n),$ 22 even with only one integer variable. This does not contradict fixed-dimension tractability, but it shows that combinatorial explosion remains intrinsic once continuous dimension is allowed to grow (Hildebrand et al., 2014).