A polynomial time algorithm for solving the closest vector problem in zonotopal lattices

In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. Zonotopal lattices are characterized by the fact that their Voronoi cell is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal lattices include lattices of Voronoi's first kind and tensor products of root lattices of type A. The combinatorial structure of zonotopal lattices can be described by regular matroids/totally unimodular matrices. We observe that a linear algebra version of the minimum mean cycling canceling method can be applied for efficiently solving the closest vector problem in zonotopal lattices.

Authors

• 3 publications
• 2 publications
• 1 publication
• 4 publications
• A polynomial-time algorithm to determine (almost) Hamiltonicity of dense regular graphs

We give a polynomial-time algorithm for detecting very long cycles in de...
07/28/2020 ∙ by Viresh Patel, et al. ∙ 0

• Testing isomorphism of circulant objects in polynomial time

Let K be a class of combinatorial objects invariant with respect to a g...
12/15/2018 ∙ by Mikhail Muzychuk, et al. ∙ 0

• Notes on {a,b,c}-Modular Matrices

Let A ∈ℤ^m × n be an integral matrix and a, b, c ∈ℤ satisfy a ≥ b ≥ c ≥ ...
06/28/2021 ∙ by Christoph Glanzer, et al. ∙ 0

• Approximating Permanent of Random Matrices with Vanishing Mean: Made Better and Simpler

The algorithm and complexity of approximating the permanent of a matrix ...
11/27/2019 ∙ by Zhengfeng Ji, et al. ∙ 0

• Recognizing Cartesian products of matrices and polytopes

The 1-product of matrices S_1 ∈R^m_1 × n_1 and S_2 ∈R^m_2 × n_2 is the m...
02/06/2020 ∙ by Manuel Aprile, et al. ∙ 0

• Minkowski Sum of Polytopes Defined by Their Vertices

Minkowski sums are of theoretical interest and have applications in fiel...
12/08/2014 ∙ by Vincent Delos, et al. ∙ 0

• Solving Satisfiability of Polynomial Formulas By Sample-Cell Projection

A new algorithm for deciding the satisfiability of polynomial formulas o...
03/01/2020 ∙ by Haokun Li, et al. ∙ 0

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1. Introduction

A lattice of rank is a discrete subgroup of which spans a linear subspace of dimension . One can specify a lattice by a lattice basis; these are linearly independent vectors so that is given by all their integral linear combinations:

 L={r∑i=1αibi:α1,…,αr∈Z}.

The central computational problems for lattices are the shortest vector problem (SVP) and the closest vector problem (CVP). They have many applications in mathematics, computer science, and engineering, in particular in complexity theory, cryptography, information theory, mathematical optimization, and the geometry of numbers; see for instance

[24].

Solving the shortest vector problem amounts to finding a shortest nonzero vector in a given lattice. In this paper we are concerned with the closest vector problem (CVP): Given a lattice basis of of and given a target vector find a lattice vector which is closest to , i.e.

 determine u∈L with ∥u−t∥=minv∈L∥v−t∥,

where denotes the standard Euclidean norm of a vector . Without loss of generality, after performing an orthogonal projection, we may assume that the target vector lies in the span of , which we denote by .

One can interpret CVP geometrically via the Voronoi cell of the lattice which is defined as

 V(L)={x∈L:∥x∥≤∥v−x∥ for all v∈L}.

The Voronoi cell of is a polytope which tessellates the space by lattice translates for . Now the CVP asks for a lattice vector so that the target vector lies in .

In the past the closest vector problem has been studied intensively. Here we only discuss results on algorithms and complexity which are most relevant for us. We refer to [1] for an up-to-date discussion of the computational complexity of CVP.

Van Emde Boas [13] established the -hardness of exactly solving CVP. Dinur, Kindler, Raz, Safra [10] showed that approximating CVP within a factor of , for some positive constant , is -hard as well. Aharonov and Regev [2] showed that approximating CVP within a factor of lies in .

On the algorithmic side, Micciancio and Voulgaris [23] developed a deterministic algorithm for exactly solving CVP which runs in time and needs space. This was improved by Aggarwal, Dadush, and Stephens-Davidowitz [1] who achieved a -time and space randomized algorithm. Hunkenschröder, Reuland, Schymura [17] considered the possibility to improve the space complexity of the algorithm by Micciancio and Voulgaris if one has a compact representation of the lattice’ Voronoi cell.

In this note we are concerned with the polynomial time solvability of the closest vector problem restricted to a special class of lattices.

That CVP can be solved in polynomial time for special classes of lattices has been proved in the case of lattices of Voronoi’s first kind by McKilliam, Grant, and Clarkson [21] and in the case of tensor products of root lattices of type by Ducas and van Woerden [11].

The main result of this paper unifies and extends these two cases. For this we consider lattices whose Voronoi cell is a zonotope. Zonotopes are defined as projections of cubes; all of their faces (of any dimension) are centrally symmetric. All lattices up to dimension three have a zonotope as Voronoi cell, but starting from dimension four on, there are lattices which do not have this property, for example the root lattice whose Voronoi cell is the -cell whose three-dimensional facets are regular octahedra whose two-dimensional faces are regular triangles and thus are not centrally symmetric.

We show that one can exactly solve CVP for zonotopal lattices in polynomial time using the algorithm of Karzanov and McCormick [19]. Their algorithm can be seen as a linear algebra version of the minimum mean cycle canceling algorithm of Goldberg and Tarjan [16] for finding a minimum-cost circulation in a network. The set up is a follows: A totally unimodular matrix , i.e. every minor of is either equal to , , or , is given. We consider the lattice of all integer points lying in the kernel of . Furthermore, a separable convex objective function is given. Separability means that for every we have a convex function so that

 w(x)=m∑i=1wi(xi)for(x1,…,xm)∈Rm.

Then, under some technical conditions on the separable convex objective function , one can compute in polynomial time a lattice vector so that is as small as possible.

Since the work of Coxeter [9], Shephard [29] and McMullen [22] it is known that the combinatorial structure of zonotopes which tile space by translations is determined by a regular matroid and thus it is related to totally unimodular matrices.

Zonotopal lattices are defined in Section 2 and the relation to regular matroids is reviewed. We develop the theory in such a way that the separability of the objective function which solves the CVP in this setting becomes apparent. We show that lattices of Voronoi’s first kind and that tensor product lattices are zonotopal lattices.

In Section 3 we discuss the algorithm of Karzanov and McCormick. We cast the CVP for zonotopal lattices into a separable convex optimization problem and verify that the technical conditions on the separable convex objective function are fulfilled to ensure the polynomial time solvability.

2. Zonotopal lattices

In this section we collect basic definitions and facts about zonotopal lattices. Zonotopal lattices were first defined by Gerritzen [15] when he gave a metric to Tutte’s regular chain groups (see for example Tutte [33]). The theory of zonotopal lattices was further developed by Loesch [20] and Vallentin [34], [35], [36].

Space tiling zonotopes have been thoroughly investigated in the literature: Main examples of zonotopal lattices are the lattice of integral flows and the lattice of integral cuts on a finite graph which were considered by Bacher, de la Harpe, Nagnibeda [3]

. Lattices whose Voronoi cell are zonotopes can be dually interpreted by Delone subdivisions and hyperplane arrangements; this has been done by Erdahl and Ryshkov

[14] who developed the theory of lattice dicings for this. Zonotopes which tile space by translations were studied by Coxeter [9], Jaeger [18], Shephard [29], and McMullen [22], see also [4].

2.1. Combinatorics: Regular chain groups, regular matroids, totally unimodular matrices

We start by briefly recalling fundamental definitions and results of Tutte’s theory of regular chain groups. Chain groups are defined over general integral domains (commutative rings with a unit element and no divisors of zero). In this paper we only need or . So we sometimes simplify Tutte’s original notation. Regular chain groups are closely related to regular matroids and totally unimodular matrices. We refer, for example, to Camion [5], Oxley [25], Schrijver [27], Tutte [32], [33], Welsh [37] for proofs and more details.

Let be a subspace of . The support of a vector is given by

 suppx={i∈[m]:xi≠0}with[m]={1,…,m}.

A non-zero vector is called an elementary chain if it has minimal (inclusion-wise) support among all non-zero vectors in . An elementary chain is called primitive chain if for all . A subspace is called regular if every elementary chain is a multiple of a primitive chain.

The set of supports of elementary chains in a regular subspace forms the circuits of a regular matroid, a matroid which is representable over every field. If a matrix is totally unimodular, then the kernel of is a regular subspace. Conversely, every regular subspace can be represented as kernel of a totally unimodular matrix.

The orthogonal complement of a regular subspace which is defined by

 L⊥={y∈Rm:m∑i=1xiyi=0% for all x∈L}

is again regular.

Let be a subset of . We define the deletion by

 L∖S={(xi)i∉S:x=(x1,…,xm)∈L,suppx∩S=∅},

and the contraction by

 L/S={(xi)i∉S:x=(x1,…,xm)∈L}.

Both operations preserve regularity. We say that a subspace is a minor of if it is obtained from by a sequence of deletions and contractions.

Two main examples of regular subspaces come from directed graphs. Let be an acyclic, directed graph with vertex set and arc set . By we denote the vertex-arc incidence matrix of which is a totally unimodular matrix. Define the regular subspace as the kernel of :

 L(D)={x∈RA:M(D)x=0}.

The primitive chains of correspond to the simple circuits/cycles (forward and backward arcs are allowed) of the directed graph . Regular subspaces which can be realized by this construction are called graphic. The primitive chains of the orthogonal complement correspond to the simple cuts/bonds (forward and backward arcs are allowed) of . Such a regular subspace is called cographic. Minors of graphic (resp. cographic) subspaces are graphic (resp. cographic). The dimension of equals where is the number of connected components of the underlying undirected graph and the dimension of is .

Tutte [31] gave a characterization of graphic and cographic subspaces in terms of forbidden minors. For this let be the complete graph on vertices and let the complete bipartite graph where one partition has vertices and the other one has vertices. Tutte showed that a regular subspace is graphic if and only if it contains neither nor as minors. Dually, a regular subspace is cographic if and only if it contains neither nor as minors. The central structure theorem about regular subspaces is Seymour’s decomposition theorem [28]: One may construct every regular subspace as -, -, and -sums of regular subspaces starting from graphic, or cographic subspaces, or the special regular subspace called ; see also Truemper [30].

2.2. Geometry: Strict Voronoi vectors, Voronoi cells

A regular subspace comes together with a regular lattice . One can show, see [33, Chapter 1.2], that in a regular lattice every vector is a conformal sum of primitive chains :

 (1) v=w1+⋯+ws with (wj)i(wk)i≥0 for % all i∈[m] and j,k∈[s].

When is a graphic (cographic) subspace we call the associated lattice graphic (cographic) as well. The graphic lattices are the lattices of integral flows and the cographic lattices are the lattices of integral cuts in the framework of [3].

We equip the space with an inner product which is defined by giving positive weights on the set : For a positive vector define the inner product

 (x,y)g=m∑i=1gixiyi.

The standard basis vectors form in this way an orthogonal basis which does not need to be orthonormal. A regular lattice with inner product is called zonotopal lattice. As we explain below, this terminology refers to the fact that the Voronoi cell of a zonotopal lattice is a zonotope. The Voronoi cell of is

 V(L)={x∈L:(x,x)g≤(v−x,v−x)g for all v∈L},

which is a centrally symmetric polytope. Lattice vectors which determine a facet defining hyperplane

 Hv={x∈L:(x,x)g=(v−x,v−x)g}={x∈L:(x,v)g=12(v,v)g}

of are called strict Voronoi vectors (sometimes also called relevant vectors). We denote the set of all strict Voronoi vectors by .

Voronoi showed (see for example [6, Chapter 21, Theorem 10] or [8]), for arbitrary lattices , that a nonzero vector is a strict Voronoi vector if and only if are the only shortest vectors in .

In the following let be a regular subspace and let be the corresponding regular lattice with positive vector . Essentially, the arguments given below can also be found in [12] in the special case of cographic lattices with constant .

Applying Voronoi’s characterization to yields:

Proposition 2.1.

A lattice vector of is a primitive chain if and only if it is a strict Voronoi vector of .

Proof.

Let be a primitive chain and let be a lattice vector with . We have and , for all , which shows . If , then . If , then there exists a factor so that , hence . In both cases are the only shortest vectors in . Hence, is a strict Voronoi vector.

Conversely, let be a strict Voronoi vector. Write as a conformal sum of primitive chains as in (1). Set . Then

 (u,u)g=(v,v)g−4(v−w1,w1)g≤(v,v)g.

since by (1). Hence, is the unique shortest vector in the coset if and only if . ∎

The following special case of Farkas lemma is proved e.g. in [26, Theorem 22.6].

Lemma 2.2.

Let be a vector, and let . Exactly one of the following two alternatives holds:

1. There exists a vector with for all so that

 y′∈x+m∏i=1[−αi,αi].
2. There exists a vector such that

 for all z∈x+m∏i=1[−αi,αi] we % have (y,z)g>0.

If the second condition holds, then one can choose to be a primitive chain of .

Theorem 2.3.

Let be the orthogonal projection of onto . Then, .

Proof.

For a vector inequality holds for all . Decompose orthogonally with . For all we have

 (y,v)g=(x,v)g−(y′,v)g=(x,v)g≤12(v,v)g.

Thus, .

Let be a vector of the Voronoi cell. If there exists with for all so that , then and . Suppose that such a vector does not exist. Then by Lemma 2.2 there is a primitive chain so that

 (v,−x+[−1/2,1/2]m)g>0.

This implies . Hence, ; a contradiction because is centrally symmetric. ∎

This theorem proves that the Voronoi cell of a zonotopal lattice is indeed a zonotope. The operations deleting or contracting correspond to contracting the corresponding zones or projecting along the corresponding zones of , as mentioned in [4, Proposition 2.2.6]. Also the combinatorial structure of , which is independent of , is completely encoded in the covectors of the oriented matroid defined by , see [4, Proposition 2.2.2].

2.3. Example: Lattices of Voronoi’s first kind

McKilliam, Grant, and Clarkson [21] gave a polynomial time algorithm for solving the closest vector problem for lattices of Voronoi’s first kind. Now we show that these lattices correspond to cographic lattices.

Following Conway and Sloane [7] we say that a lattice is of Voronoi’s first kind if has an obtuse superbasis: These are vectors so that the following three conditions hold:

1. is a basis of ,

2. ,

3. for and .

A classical theorem of Voronoi states that every lattice in dimensions and has an obtuse superbasis, see Conway and Sloane [7, Section 7]. However, starting from dimension on, not every lattice is of Voronoi’s first kind.

In the setting of zonotopal lattices, lattices of Voronoi’s first kind appear as cographic lattices: Let be a lattice of Voronoi’s first kind having an obtuse superbasis . Define the directed graph with vertex set where we draw an arc between vertices and whenever and . We assign to the arc the (positive) weight .

The undirected graph which underlies is called Delone graph of , see [7]. In fact, the choice of the directions of the arcs is arbitrary, as long as the graph does not contain a directed cycle.

Proposition 2.4.

The cographic lattices are exactly the lattices of Voronoi’s first kind.

Proof.

The graph is weakly connected (i.e. the underlying undirected graph is connected) since has rank : For suppose not. Then one can partition the vertex set so that there is no arc between and . Consider the spaces spanned by the vectors in , with . These spaces are orthogonal and we have because of (i) and (ii). Hence, , contradicting that the rank of is .

Consider the vertex-arc incidence matrix of and let be the row vectors of . Their integral span coincides with the cographic lattice . Furthermore, the vectors form an obtuse superbasis of and holds when and are adjacent in . Hence, the lattice which is of Voronoi’s first kind is isometric to the cographic lattice .

Clearly, this construction can be reversed. Starting from a vertex-arc incidence matrix of a weakly connected acyclic directed graph defining a cographic lattice one can get an obtuse superbasis of this lattice. What happens when the graph defining the cographic lattice is not weakly connected? Then one can make it weakly connected by identifying vertices of distinct connected components without changing the cographic lattice. ∎

For instance, the root lattice

 An={x∈Zn+1:n+1∑i=1xi=0}

is a lattice of Voronoi’s first kind. Its Delone graph is the cycle graph of length . The dual lattice is again a lattice of Voronoi’s first kind. Its Delone graph is the complete graph on vertices. The Voronoi cell of is the -dimensional permutahedron.

2.4. Example: Tensor product of root lattices of type A

Ducas and van Woerden [11] gave a polynomial time algorithm for solving the closest vector problem for tensor products of the form . Now we show that these lattices correspond to the graphic lattices for the complete bipartite graph .

Let be a lattice of rank with basis and let be a lattice of rank with basis . Then their tensor product is the lattice having basis with and .

Proposition 2.5.

The tensor product lattice coincides with the graphic lattice of the complete bipartite graph .

Proof.

Recall that the Delone graph of is the cycle graph . A basis of is

 b1=e1−e2,b2=e2−e3,…,bm=em−em+1,

where are the standard basis vectors of . A basis of is , with where are the standard basis vectors of . This defines the following basis of

 (2) bi⊗cj=ei⊗fj−ei⊗fj+1−ei+1⊗fj+ei+1⊗fj+1for i∈[m],j∈[n].

One can inductively (by adding nodes successively and verifying the necessary conditions) orient the arcs of consistently (see the example below) so that the basis (2) lies in the graphic lattice . Since the dimension of the circuit space of the complete bipartite graph is

 dimL(Km+1,n+1)=(m+1)(n+1)−(m+1+n+1)+1=mn,

we see that the basis (2) also forms a basis of the graphic lattice. ∎

Example 2.6.

We give a basis of the graphic lattice corresponding to the complete bipartite graph and a corresponding consistent orientation of .

 b1⊗c1 = e1⊗f1−e1⊗f2−e2⊗f1+e2⊗f2 b2⊗c1 = e2⊗f1−e2⊗f2−e3⊗f1+e3⊗f2

3. Minimum mean cycle canceling algorithm for CVP

Let be a totally unimodular matrix and let be a positive vector. By we denote the kernel of which is a regular subspace. This defines the zonotopal lattice with inner product . Let be any (target) vector. Define the separable convex function by

 wi(vi)=gi(vi−ti)2so thatw(v)=m∑i=1gi(vi−ti)2=(v−t,v−t)g.

Then solving the closest vector problem for given the target vector amounts to finding a minimizer for among all lattice vectors . So we can apply the results of McCormick and Karzanov to solve the closest vector problem for zonotopal lattices.

The minimum mean cycle canceling method gives a polynomial time algorithm for solving the closest vector problem here. To see this we have to verify some technical conditions for which we will do now.

We describe how the minimum mean cycle canceling method works in our setting and discuss which arguments of the paper of McCormick and Karzanov have to be applied to prove that the algorithm runs in polynomial time.

We start by setting up notation. The (discrete) right derivative of is

 c+i(vi)=wi(vi+1)−w(vi)=gi(vi+1−ti)2−gi(vi−ti)2=gi(2(vi−ti)+1).

Similarly the (discrete) left derivative of is

 c−i(vi)=wi(vi)−wi(vi−1)=gi(2(vi−ti)−1).

The cost of the strict Voronoi vector at a lattice vector is

 c(v,u)=∑i∈u+c+i(vi)−∑i∈u−c−i(vi),

where

 u+={i∈suppu:ui=+1}andu−={i∈suppu:ui=−1}.

If the cost is negative, then is closer to than because we have

 (v+u−t,v+u−t)g=(v−t,v−t)g+c(v,u),

which is easily verified.

The mean cost of at is

 ¯¯c(v,u)=c(v,u)|suppu|.

A strict Voronoi vector is called a minimum mean strict Voronoi vector for if its mean cost is a small as possible. The following quantity is used to measure the progress of the algorithm:

 λ(v)=max{0,−minu∈Vor(L)¯¯c(v,u)},

where denotes the set of strict Voronoi vectors. Now [19, Lemma 3.1] says that is a solution of the closest vector problem if and only if . [19, Proof of Lemma 3.2]

shows that the following linear program computes

:

 min m∑i=1(c+i(vi)x+i−c−i(vi)x−i) M(x+−x−)=0 eT(x++x−)=1 x+,x−∈Rm≥0,

where is the all-ones vector. One can furthermore find a minimum mean strict Voronoi vector at by first determining and then finding a vector with minimal support with . This can be done by solving at most auxiliary linear programs where one probes to set coordinates to .

Now the minimum mean cycle canceling algorithm works as follows: We start at the origin . As long as is positive, we improve , moving it closer to the target vector by finding a minimum mean strict Voronoi vector at and updating to . The step size is determined by [19, (16)] which is the minimum integer so that

 Δ∈[λ(v)gi,λ(v)gi+1]for i∈[m].

Choosing the step size like this makes sure that , see [19, Lemma 3.3]. By [19, Lemma 3.4] we see that after iterations the value of decreases by a factor of at most , so that we have a geometric decrease.

We start with and we assume that . Let be a minimum mean strict Voronoi vector at . If and are rational, it is immediate to see that the binary encoding length of is polynomial in the input size. If and are rational, then we can also derive a stopping criterion for the algorithm. Because of rationality, there exists an integer so that is an integer for all and all strict Voronoi vectors . The binary encoding length of is polynomial in the input size. If for then is a closest vector to because from [19, Proof of Lemma 6.1] it follows that

 c(v,u)>−mδ≥−1K

and so is nonnegative. The bound on , the stopping criterion together with the geometric decrease of show that only a polynomial number of iterations are needed to find a closest vector.

Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie agreement No 764759. The fourth named author is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” and by the project “Spectral bounds in extremal discrete geometry” (project number 414898050), both funded by the DFG.

References

• [1] D. Aggarwal, D. Dadush, and N. Stephens-Davidowitz, Solving the Closest Vector Problem in Time: The Discrete Gaussian Strikes Again!, FOCS, 2015.
• [2] D. Aharonov, O. Regev, Lattice problems in , J. ACM 52 (2005), 749–765.
• [3] R. Bacher, P. de la Harpe, T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125 (1997), 167–198.
• [4] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids, Cambridge University Press, 1993.
• [5] P. Camion, Unimodular modules, Discrete Mathematics 306 (2006), 2355–2382.
• [6] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer, 1988.
• [7] J.H. Conway and N.J.A. Sloane, Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices, Proc. Roy. Soc. London Ser. A 436 (1992), 55–68.
• [8] J.H. Conway, The sensual (quadratic) form (With the assistance of Francis Y. C. Fung), Mathematical Association of America, 1997.
• [9] H.S.M. Coxeter, The classification of zonohedra by means of projective diagrams, J. Math. Pure Appl. 41 (1962), 137–156.
• [10] I. Dinur, G. Kindler, R. Raz, S. Safra, Approximating CVP to within almost-polynomial factors is NP-hard, Combinatorica 23 (2003), 205–243.
• [11] L. Ducas, W.P.J. van Woerden, The closest vector problem in tensored root lattices of type A and in their duals, Des. Codes Cryptogr. 86 (2018), 137–150.
• [12] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Complexity and algorithms for computing Voronoi cells of lattices Math. Comp. 78 (2009), 1713–1731.
• [13] P. van Emde Boas, Another NP-complete problem and the complexity of computing short vectors in a lattice. Tech. rep., University of Amsterdam, Department of Mathematics, Netherlands. Technical Report 8104.
• [14] R.M. Erdahl, S.S. Ryshkov, On lattice dicing European J. Combin. 15 (1994), 459–481.
• [15] L. Gerritzen, Die Jacobi-Abbildung über dem Raum der Mumfordkurven, Math. Ann. 261 (1982), 81–100.
• [16] A. Goldberg, R.E. Tarjan, Finding minimum-cost circulations by canceling negative cycles, J. Assoc. Comput. Mach. 36 (1989), 873–886.
• [17] C.  Hunkenschröder, G. Reuland, M. Schymura, On compact representations of Voronoi cells of lattices, pp. 261–274 in: Lecture Notes in Comput. Sci., 11480, Springer, 2019.
• [18] F. Jaeger, On space-tiling zonotopes and regular chain-groups, Ars Combin. 16 (1983), B, 257–270.
• [19] A.V. Karzanov and S.T. McCormick, Polynomial methods for separable convex optimization in unimodular linear spaces with applications, SIAM J. Comput. 26 (1997), 1245–1275.
• [20] H.-F. Loesch, Zur Reduktionstheorie von Delone-Voronoi für matroidische quadratische Formen, Dissertation, Ruhr-Universität Bochum, 1990.
• [21] R.G. McKilliam, A. Grant, and I.V. Clarkson, Finding a closest point in lattices of Voronoi’s first kind, SIAM J. Discrete Math. 28 (2014), 1405–1422.
• [22] P. McMullen, Space tiling zonotopes, Mathematika 22 (1975), 202–211.
• [23] D.  Micciancio, P. Voulgaris, A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations, SIAM J. Comput. 42 (2013), 1364–1391.
• [24] P.Q. Nguyen, B. Vallée (eds.). The LLL Algorithm — Survey and Applications, Springer, 2010.
• [25] J. Oxley, Matroid theory (second edition), Oxford University Press, 2011.
• [26] R.T. Rockafellar, Convex analysis, Princeton University Press, 1970.
• [27] A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1986a.
• [28] P.D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305–359.
• [29] G.C. Shephard, Space-filling zonotopes, Mathematika 21 (1974), 261–269.
• [30] K. Truemper, Matroid decomposition, Academic Press, 1992.
• [31] W.T. Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958), 144–174.
• [32] W.T. Tutte, Lectures on matroids, J. Res. Natl. Bur. Stand. B 69B (1965) 1–47.
• [33] W.T. Tutte, Introduction to the theory of matroids, American Elsevier Publishing Company, 1971.
• [34] F. Vallentin. Über die Paralleloeder-Vermutung von Voronoï. Diploma thesis, University of Dortmund, 2000.
• [35] F. Vallentin, Sphere coverings, lattices, and tilings, Dissertation, Technische Universität München, 2003.
• [36] F. Vallentin, A note on space tiling zonotopes, arXiv:math/0402053 [math.MG], 2004, 7 pages.
• [37] D.J.A. Welsh, Matroid Theory, Academic Press, 1976.