arxiv:2110.01732v1 [cs.cc] 4 oct 2021

Noname manuscript No.(will be inserted by the editor)

Faster algorithm for counting of the integer points

number in ∆-modular polyhedra

D. V. Gribanov, D. S. Malyshev

the date of receipt and acceptance should be inserted later

Abstract Let a polytope P be defined by one of the following ways:

(i) P = {x ∈ Rn : Ax ≤ b}, where A ∈ Z(n+m)×n, b ∈ Q(n+m), rank(A) = nand d := dim(P) = n;

(ii) P = {x ∈ Rn+ : Ax = b}, where A ∈ Z

m×n, b ∈ Zm, rank(A) = m and

d := dim(P) = n−m;

and let all the rank minors of A be bounded by ∆ in the absolute values. Weshow that | P ∩Zn | can be computed with an algorithm, having the arithmeticcomplexity bound

O(ν(d,m,∆) · d3 ·∆4 · log(∆)

where ν(d,m,∆) is the maximal possible number of vertices in a d-dimensionalpolytope P, defined by one of the systems above.

Using the obtained result, we have the following arithmetical complexitybounds to compute | P ∩Zn |:

– The bound O( dm+1)m·d3 ·∆4 ·log(∆) that is polynomial on d and∆, for any

fixed m. Takingm = 1, it gives an O(d4 ·∆4 ·log(∆)

)-algorithm to compute

the number of integer points in a simplex or the number of solutions of theunbounded Subset-Sum problem, where ∆ means the maximal weight ofan item.

The article was prepared under financial support of Russian Science Foundation grant No21-11-00194.

D. V. GribanovNational Research University Higher School of Economics, 25/12 Bolshaja PecherskajaUlitsa, Nizhny Novgorod, 603155, Russian FederationE-mail: dimitry.gribanov@gmail.com

D. S. MalyshevNational Research University Higher School of Economics, 25/12 Bolshaja PecherskajaUlitsa, Nizhny Novgorod, 603155, Russian FederationE-mail: dsmalyshev@rambler.ru

2 D. V. Gribanov, D. S. Malyshev

– The bound O(md +1

)d2 ·d4 ·∆4 · log(∆) that is polynomial on m and ∆, for

any fixed d. The last bound can be used to obtain a faster algorithm forthe ILP feasibility problem, when the parameters m and ∆ are relativelysmall. For example, taking m = O(d) and ∆ = 2O(d), the above boundbecomes 2O(d), which is faster, than the state of the art algorithm, due to[10,11] with the complexity bound O(d)d · poly(size(A, b));

– The bound O(d)4+d2 ·∆4+d · log(∆) that is polynomial on ∆, for any fixed

d. Taking ∆ = O(1), the last bound becomes O(d)4+d2 , which again gives

a faster algorithm for the ILP feasibility problem, than the state of the artalgorithm, due to [10,11].

The unbounded and parametric versions of the above problem are alsoconsidered.

Keywords Integer Linear Programming · Short Rational GeneratingFunction · Bounded Sub-Determinants · Multidimensional KnapsackProblem · Subset-Sum Problem · Counting Problem

1 Introduction

1.1 Basic definitions and notations

Let A ∈ Zm×n. We denote by Aij its ij-th element, by Ai∗ its i-th row, andby A∗j its j-th column. For subsets I ⊆ {1, . . . ,m} and J ⊆ {1, . . . , n}, thesymbol AIJ denote the sub-matrix of A, which is generated by all the rowswith indices in I and all the columns with indices in J . If I or J is replacedby ∗, then all the rows or columns are selected, respectively. Sometimes, wesimply write AI instead of AI∗ and AJ instead of A∗J , if this does not lead toconfusion.

The maximum absolute value of entries of a matrixA is denoted by ‖A‖max =maxi,j

∣∣Ai j

∣∣. The lp-norm of a vector x is denoted by ‖x‖p. The number of non-

zero components of a vector x is denoted by ‖x‖0 =∣∣{i : xi 6= 0}

∣∣.For x ∈ R, we denote by ⌊x⌋, {x}, and ⌈x⌉ the floor, fractional part, and

ceiling of x, respectively.For c, x ∈ R

n, by 〈c, x〉 we denote the standard scalar product of twovectors. In other words, 〈c, x〉 = c⊤x.

Definition 1 For a matrix A ∈ Zm×n, by

∆k(A) = max{∣∣det(AIJ )

∣∣ : I ⊆ {1, . . . ,m}, J ⊆ {1, . . . , n}, |I| = |J | = k},

we denote the maximum absolute value of determinants of all the k × k sub-matrices of A. By ∆gcd(A, k) and ∆lcm(A, k), we denote the greatest commondivisor and the least common multiplier of nonzero determinants of all thek × k sub-matrices of A, respectively. Additionally, let ∆(A) = ∆rankA(A),∆gcd(A) = ∆gcd(A, rank(A)), and ∆lcm(A) = ∆lcm(A, rank(A)).

If ∆(A) ≤ ∆, for some ∆ > 0, then A is called ∆-modular.

Title Suppressed Due to Excessive Length 3

Definition 2 For a matrix B ∈ Rm×n, cone(B) = {Bt : t ∈ Rn≥0} is the cone,

spanned by columns of B.

Definition 3 For A ∈ Zk×n and b ∈ Z

k, we denote

P(A, b) = {x ∈ Rn : Ax ≤ b},

xz := xz11 · . . . · xzn

n and f(P ;x) =∑

z∈P ∩Zn

Definition 4 Let A ∈ Z(n+m)×n, rank(A) = n, b ∈ Q

n+m and ∆ = ∆(A).Consider a polytope P = P(A, b) and a cone C = cone(A⊤).

By ν(n,m,∆) and µ(n,m,∆) we denote the maximal possible number ofvertices in P, and the maximal possible number of n-dimensional simplices inan arbitrary triangulation of C respectively.

1.2 Valuations and indicator functions of polyhedra

In this Subsection, we mainly follow to the monographs [4,5] in the most ofdefinitions and notations.

Let V be a d-dimensional real vector space and L ⊂ V be a lattice.

Definition 5 Let A ⊆ V be a set. The indicator [A] of A is the function[A] : V → R defined by

[A](x) =

{1, if x ∈ A

0, if x /∈ A .

The algebra of polyhedra P(V) is the vector space defined as the span of theindicator functions of all the polyhedra P ⊂ V .

Definition 6 A linear transformation T : P(V) → W , where W is a vectorspace, is called a valuation. We consider only L-valuations or lattice valuationsthat satisfy

T ([P +u]) = T ([P ]), for all rational polytopes P and u ∈ L,

see [28, pp. 933–988], [29].

Remark 1 [3]. Let us denote g(P) = T ([P ]) for a lattice valuation T . Thegeneral result of P. McMullen [27] states that if P ⊂ V is a rational polytope,d = dim(P), and t ∈ N is a number, such that t · P is a lattice polytope, thenthere exist functions gi(P , ·) : N → C, such that

g(α · P) =

gi(P , α) · αi, for all α ∈ N , and

gi(P , α+ t) = gi(P, α), for all α ∈ N .

If we compute g(q · P), for q ∈ {α, α+ t, α+2t, . . . , α+ dt}, then we canobtain gi(P, α) by interpolation.

We are mainly interested in two valuations, the first is the counting val-uation E([P ]) = | P ∩Z

d |. Applying the result of P. McMullen to E([P ]), weconclude that

|α · P ∩Zd | =

ei(P , α) · αi, for all α ∈ N , and

ei(P , α+ t) = ei(P, α), for all α ∈ N .

The function on the right hand side is called the Ehrhart quasi-polynomial ofP named after E. Ehrhart, who discovered the existence of such polynomials[15], see also [33, Section 4.6].

The second valuation F([P ]), which will be significantly used in our paper,is defined by the following theorem, proved by J. Lawrence [23], and, indepen-dently, by A. Khovanskii and A. Pukhlikov [30]. We borrowed the formulationfrom [4, Section 13]:

Theorem 1 ([23,30]) Let R(Cd) be the space of rational functions on Cd

spanned by the functions of the type

(1 − xu1) . . . (1− xud),

where v ∈ Zd and ui ∈ Zd \{0}, for any i ∈ {1, . . . , d}. Then there exists alinear transformation (a valuation)

F : P(Qd) → R(Cd),

such that the following properties hold:

1. Let P ⊂ Rd be a non-empty rational polyhedron without lines, and let

C be its recession cone. Let C be generated by rays w1, . . . , wn, for somewi ∈ Z

d \{0}, and let us define

WC ={x ∈ C

d : |xwi | < 1, for any i ∈ {1, . . . , n}}.

Then, WC is a non-empty open set and, for all x ∈ WC, the series

f(P ;x) =∑

z∈P ∩Zd

converges absolutely and uniformly on compact subsets of WC to the func-tion f(P ;x) = F([P ]) ∈ R(Cd).

2. If P contains a line, then f(P;x) = 0.

If P is a rational polyhedron, then f(P ;x) is called its short rational gen-erating function.

1.3 The problems under consideration and results of the paper

1.3.1 The lattice points counting problem

The main problem that we consider is following:

Problem 1 Let P be a rational polytope defined by one of the following ways:

1. The polytope P is defined by a system in the canonical form: P = {x ∈

Rn : Ax ≤ b}, where A ∈ Z

(n+m)×n, b ∈ Zn+m, and rank(A) = n;

2. The polytope P is defined by a system in the standard form: P = {x ∈R

n≥0 : Ax = b}, where A ∈ Z

m×n, b ∈ Zm, rank(A) = m, and ∆gcd(A) = 1.

The problem at state is to compute the value of | P ∩Zn |.

The first polynomial-time in a fixed dimension algorithm, which finds theshort generating function (its definition will be presented later) for P andsolves Problem 1, was proposed by A. Barvinok in [2]. Further modificationsand details were given in [4,5,6,14]. An alternative approach was presented in[20,22].

The paper [19] gives a polynomial-time algorithm for Problem 1 parame-terised by m and ∆ = ∆(A). More precisely, the paper [19] gives an algorithmwith the arithmetic complexity bound

O(TSNF (d) · d

m · dlog2(∆)),

where TSNF (d) is the complexity of computing the Smith Normal Form ford × d integer matrices and d is the dimension of the corresponding polytope(d ≤ n for the canonical form and d ≤ n − m for the standard form). Weimprove the mentioned result from [19] in the following way:

Theorem 2 Problem 1 can be solved with an algorithm, having the arithmeticcomplexity bound

O(ν(d,m,∆) · d3 ·∆4 · log(∆)

where ∆ = ∆(A), d = dim(P) (d ≤ n, for the canonical form, and d ≤ n−m,for the standard form).

In contrast to [19], in Theorem 2 we do not implicitly compute the corre-sponding short rational generating function for the formal power series f(P ;x).Instead of that, we only find the result of the substitution xi = eci·τ to f(P ;x),where the vector c ∈ Zn is chosen as the sum of some rows of the matrix A andτ is a variable. Such a substitution gives a shorter formula that can be com-puted using dynamic programming techniques, and, due to [4, Chapter 14],the value | P ∩Zn | is exactly the constant therm in a Teilor’s expansion of theresulting function with respect to τ .

Using Theorem 2 and results of the papers [24,26] that can help to boundthe value of ν(d,m,∆), we present new complexity bounds for Problem 1.Additionally, we show how to handle the case of unbounded polyhedra.

Corollary 1 The arithmetic complexity of an algorithm by Theorem 2 can bebounded with the following relations:

1. The bound O(

)m· d3 ·∆4 · log(∆) that is polynomial on d and ∆, for

any fixed m;

2. The bound O(md + 1

) d2 · d4 ·∆4 · log(∆) that is on m and ∆, for any fixed

d;3. The bound O(d)4+

d2 ·∆4+d · log(∆) that is polynomial on ∆, for any fixed

To handle the case, when P is an unbounded polyhedron, we need to pay anadditional factor of O( d

m + 1) · d4 in the first bound and O(d4) in the secondbound. The third bound stays unchanged.

Proofs of Theorem 2 and Corollary 1 will be given in Section 2 and Subsection2.4, respectively.

The first bound can be used to count the number of integer points in asimplex or the number of solutions of the unbounded Subset-Sum problem

{w⊤x = w0

x ∈ Zn≥0 .

For the both problems, it gives the arithmetic complexity bound O(n4 ·∆4 ·

log(∆)), where ∆ = ‖w‖max for the Subset-Sum problem.

The second and third bounds can be used to obtain a faster algorithmfor the ILP feasibility problem, when the parameters m and ∆ are relativelysmall. For example, taking m = O(d) and ∆ = 2O(d) in the second bound,it becomes 2O(d), which is faster, than the state of the art algorithm, dueto [10,11] (see also [7,17,37], for a bit more general setting) that has thecomplexity bound O(d)d · poly(size(A, b)). Substituting ∆ = O(1) to the third

bound, it gives O(d)4+d2 , which again is better, than the general case bound

O(d)d · poly(size(A, b)).Using the Hadamard’s inequality, we can write trivial complexity estimates

in terms of ∆1 := ∆1(A) = ‖A‖max for Problem 1 in the standard form.

Corollary 2 Let a polytope P be defined in the following way:

P = {x ∈ Rn≥0 : Ax = b},

where A ∈ Zm×n, b ∈ Z

m, rank(A) = m, and ∆gcd(A) = 1.Then, the value of | P ∩Zn | can be computed with an algorithm, having the

following arithmetic complexity bounds:

– O(d+m)m · d3 ·mm+1 ·∆4m1 · log(m∆1),

– O(md + 1

)d2 · d4 ·m2m+1 ·∆4m

1 · log(m∆1),

where ∆1 = ∆1(A) and d = dim(P) ≤ n−m. To handle the case, when P isan unbounded polyhedron, we need to pay an additional factor of O(d+m

m ) · d4

in the first bound and O(d4) in the second bound.

Remark 2 We are interested in development of algorithms that will be poly-nomial, when we bound some of the parameters d, m, and ∆. Due to [19,Corollary 3], the problem in the standard form can be reduced to the problemin the canonical form maintaining values of m and ∆, see also [18, Lemmas 4and 5] and [38] for a more general reduction. Hence, in the proofs we will onlyconsider polytopes defined by systems in the canonical form.

Remark 3 To simplify analysis, we assume that ∆gcd(A) = 1 for ILP prob-lems in the standard form. It can be done without loss of generality, becausethe original system Ax = b, x ≥ 0 can be polynomially transformed to theequivalent system Ax = b, x ≥ 0 with ∆gcd(A) = 1.

Indeed, let A = P(S 0

)Q, where

)∈ Z

m×m be the SNF of A, and

P ∈ Zm×m, Q ∈ Z

n×n be unimodular matrices. We multiply rows of theoriginal system Ax = b, x ≥ 0 by the matrix (PS)−1. After this step, theoriginal system is transformed to the equivalent system

(Im×m 0

)Qx = b,

x ≥ 0. Clearly, the matrix(Im×m 0

)is the SNF of

(Im×m 0

)Q, so its ∆gcd(·)

is equal 1.

1.3.2 Parametric versions of the counting problem

The most natural parametric generalization of Problem 1 is the following state-ment:

Problem 2 Let P be a polytope from the definition of Problem 1. We areinterested in computing of the coefficients ei(P , α) of the Ehrhart’s quasi-polynomial (see Subsection 1.2):

|α · P ∩Zd | =d∑

ei(P , α)αi.

After this, the coefficients are known and stored in some data structure upto their periods, we can compute |α · P ∩Zd | with a linear-time algorithm.

Let P be a polyhedron of Problem 1. There exists t ∈ Z>0, such thatt·P ⊆ Z

n and t ≤ ∆lcm(A). Corollary 1 gives a straightforward way to calculateall the coefficients of the Ehrhart’s quasipolynomial for P just by interpolation(see Subsection 1.2). Since the period of the coefficients is bounded by∆lcm(A),we need to compute at most n ·∆lcm(A) values of |α · P ∩Z

n | to complete theinterpolation. In other words, the following Corollary holds:

Corollary 3 Problem 2 can be solved with an algorithm having the followingcomplexity bounds

1. The polynomial on d bound O(

dm + 1

)m· d4 · g(∆);

2. The polynomial on m bound O(md + 1

) d2 · d5 · g(∆);

3. The bound O(d)5+d2 ·∆4+d · g(∆);

where d = dim(P), g(∆) = ∆lcm(A) ·∆4 · log(∆), and ∆ = ∆(A).

The previous Corollary gives an FPT-algorithm to compute coefficients ofthe Ehrhart quasipolynomials of ∆-modular simplices. A problem with a closeformulation was solved in [3], where it was shown that the last k coefficientsof the Ehrhart quasipolynomial of a rational simplex can be found with apolynomial-time algorithm, for any fixed k.

A more general variant of the parametric counting problem is consideredin the papers [8,19,21,25,35,36], Let us formulate some definitions followingto [36].

Definition 7 Let P ⊂ Qp ×Q

n be a rational polyhedron, such that, for ally ∈ Q

p, the set Py = {x ∈ Rn :

)∈ P} is bounded, and cP : Z

p → Z isdefined by

cP(y) =∣∣Py ∩Z

n∣∣ =

∣∣∣{x ∈ Zn :

)∈ P}

∣∣∣ .

We call P a parametric polytope, because, if

P = {(y

)∈ Q

p ×Qn : B y+Ax ≤ b},

for some matrices B ∈ Z(n+m)×p, A ∈ Z(n+m)×n, rank(A) = n, and a vectorb ∈ Q

n+m, then

Py = {x ∈ Qn : Ax ≤ b−B y}.

Problem 3 Let P ⊂ Qp ×Qn be a rational parametric polytope defined byone of the following ways:

1) P = {(y

)∈ Qp ×Qn : B y+Ax ≤ b}, whereA ∈ Z(n+m)×n,B ∈ Z(n+m)×p,

b ∈ Qn+m, and rank(A) = n;2) P = {

)∈ Q

p ×Qn+ : B y+Ax = b}, where A ∈ Z

m×n, B ∈ Zm×p,

b ∈ Zm, rank(A) = m, and ∆gcd(A) = 1.

Our aim is to find an efficient representation of the counting functioncP(y) = | Py ∩Zn |. It can be some formula or an efficient data structurethat after a preprocessing one can efficiently substitute arguments to cP(y).

The main result of the paper [35] states that, in a fixed dimension, thepiece-wise step-polynomial representation of the counting function cP(y) canbe computed with a polynomial-time algorithm. After such representation isconstructed, the values of cP(y) can be computed by an even faster algorithm.

One of the main results of the paper [19] states that such a piece-wise step-polynomial representation can be computed in varying dimension if we fix theparameters m and ∆ = ∆(A). The resulted algorithm has the arithmeticcomplexity n(m−1)(p+1)+const · nlog2(∆) and the length of the resulted step-polynomial can be bounded by O(nm · nlog2(∆)).

As the second main result of the current paper, we give a faster algorithmthat computes a more efficient representation of cP(y) than in the paper [19].We call this representation a periodical piece-wise step-polynomial.

Definition 8 A periodical step-polynomial g : Qn → Q is a function of theform

g(x) =

(⌊T j(x)⌋

) dj∏

⌊Fkj(x)⌋, (1)

where F jk : Qn → Q and T j : Qn → Qs (for some s) are affine functions, andπj : Z

s → Z are periodical functions (e.g. πj(x) = πj(x+t), for some t ∈ Zs).

We say that the degree of g(x) is maxj

{dj} and length is l.

A periodical piece-wise step-polynomial c : Zn → Q is a collection of ratio-nal polyhedra Qi together with the corresponding functions gi : Qi ∩Z

n → Q,such that

1) Qn =

⋃i rel.int(Qi) and rel.int(Qi) ∩ rel.int(Qj) = ∅, for different i, j;

2) c(x) = gi(x), for x ∈ rel.int(Qi) ∩ Zn;

3) each gi is a periodical step-polynomial.

We say that the degree of c(x) is maxi{deg(gi)} and the length is maxi{length(gi)}.

Now, we are able to present our second main result.

Theorem 3 Let P ⊂ Qp ×Qn be a rational parametric polytope defined byone of the following ways:

1) P = {(y

)∈ Qp ×Qn : B y+Ax ≤ b}, where A ∈ Z(n+m)×n, B ∈ Z(n+m)×p,

b ∈ Qn+m, and rankA = n;2) P = {

)∈ Q

p ×Qn+ : B y+Ax = b}, where A ∈ Z

m×n, B ∈ Zm×p,

b ∈ Zm, and rankA = m.

Let ν := ν(n,m,∆), µ := µ(n,m,∆) and η be the number of (p − 1)-dimensional faces in P. Then the periodical piece-wise step-polynomial repre-sentation of the counting function cP(y) can be computed with an algorithmhaving the arithmetic complexity bound

(nmp)O(1) · η2p · ν · µ · log(ν) ·∆4 · log(∆).

The length and degree of the resulted function is bounded by O(µ ·n3 ·∆2) andn− 1, respectively.

Having such representations, the queries to the values of cP(y) can be per-formed with O(µ · n3 ·∆2) arithmetic operations.

The proof of Theorem 3 will be given in Section 3.

Remark 4 It is important to note that we do not need completely constructedperiodical piece-wise step-polynomial if we want to perform queries to cP(y).After a preprocessing with the better arithmetic complexity

(nmp)O(1) · ηp · ν · µ ·∆4 · log(∆),

we can perform the queries cP(y) on the fly without knowledge of a completefunction preserving the same complexity bound. The complexity bounds thatare given in [19,35] follow to this rule.

Using Lemma 1 and the previous remark, we give the following complexitybounds.

Corollary 4 The following arithmetical complexity bounds of Theorem 3 hold:

Preprocessing Query

Poly. on n and ∆: (nmp)O(1) · O(n+mm

)(m−1)(p+2)·∆4 · log(∆) O

)m· n3 ·∆2

Poly. on m and ∆: (nmp)O(1) ·O(m+nn

)(p+1)(n+1)·∆4 · log(∆) O

)n/2· n3 ·∆2

Poly. on ∆: (∆nmp)O(1) · n(p+1)(n+1) ·∆2(p+1)(n+1) n3+n2 ·∆2+n

Here, when we say ”Poly.”, we mean that the formula is polynomial on givenparameters if other parameters are fixed numbers.

Good surveys on the related ∆-modular ILP problems and parameterisedILP complexity are given in [16,18,19,34].

1.4 Auxiliary facts from the polyhedral analysis

In this Subsection, we mainly follow to [4,5].

Theorem 4 (Theorem 2.3 of [5]) Let V and W be finite-dimensional realvector spaces, and let T : V → W be an affine transformation. Then

1) for every polyhedron P ⊂ V, the image T (P) ⊂ W is a polyhedron;2) there is a unique linear transformation (valuation) T : P(V) → P(W),

such that

T ([P ]) =[T (P)

], for every polyhedron P ⊂ V .

Let us fix a scalar product 〈·, ·〉 in V , just making V Euclidean space.

Definition 9 Let P ⊂ V be a non-empty set. The polar P◦ of P is definedby

P◦ ={x ∈ V : 〈x, y〉 ≤ 1 ∀y ∈ P

Definition 10 Let P ⊂ V be a non-empty polyhedron, and let v ∈ P be apoint. We define the tangent cone of P at v by

tcone(P , v) ={v + y : v + εy ∈ P, for some ε > 0

We define the cone of feasible directions at v by

fcone(P , v) ={y : v + εy ∈ P, for some ε > 0

Thus, tcone(P , v) = v + fcone(P, v).

Remark 5 If an n-dimensional polyhedron P is defined by a system Ax ≤ band P contains no lines, then, for any v ∈ vert(P), it holds

tcone(P , v) = {x ∈ V : AJ (v)∗x ≤ bJ (v)},

fcone(P , v) = {x ∈ V : AJ (v)∗x ≤ 0},

fcone(P , v)◦ = cone(A⊤J (v)∗),

where J (v) = {j : Aj∗v = bj}.

It is widely known that a slight perturbation in the right-hand sides of asystem Ax ≤ b can transform the polyhedron P(A, b) to a simple one. Here,we need an algorithmic version of this fact, presented in the following technicaltheorem.

Theorem 5 Let A ∈ Zk×n, rank(A) = n ≤ k, b ∈ Q

k, γ = max{‖A‖max, ‖b‖∞},and P = P(A, b) be the n-dimensional polyhedron.

Then, for 1/ε = 1 + 2n · n⌈n/2⌉ · γn and the vector t ∈ Qk, with ti = εi−1,

the polyhedron P ′ = P(A, b+ t) is simple.

Proof Let us suppose by the contrary that there exists a vertex v of P ′ and aset of indices J such that AJ v = (b + t)J , | J | = n + 1 and rank(AJ ) = n.The last is possible iff det(M) = 0, where M =

(AJ (b + t)J

). Note that

M = B +D, where B =(AJ bJ

)and D =

(0(n+1)×n tJ

). We have,

det(M) = det(B) +

n+1∑

det(B[i, tJ ]) =

= det(B) +

n+1∑

(−1)i+j · (tJ )j · det(BJ \{j} I \{i}),

where I = {1, . . . , n+1} and B[i, tJ ] is the matrix induced by the substitutionof the column tJ instead of i-th column of B.

Let us assume that (tJ )j = εdj , for j ∈ I, where dj ∈ Z and 0 ≤ d1 < d2 <· · · < dn+1 ≤ k − 1. Consequently, the condition det(M) = 0 is equivalent tothe following condition:

det(B) +

n+1∑

εdj ·

(−1)i+j · det(BJ \{j} I \{i})

= 0. (2)

Note that the polynomial (2) is not zero, because in the opposite case thesystem AJ x = (b+ t)J will have infinitely many solutions that contradicts tothe equality rank(AJ ) = n.

As a corollary of the Rouche’s theorem, we have that |ε∗| ≥ 11+αmax/β

β+αmax, where ε∗ is any root of (2), αmax is the maximal absolute value of

the coefficients, and β is the absolute value of the non-zero coefficient with aminimal index.

Finally, 1/|ε∗| ≤ 2αmax ≤ 2n ·nn/2 ·γn, which contradicts to the Theorem’scondition on ε.

Lemma 1 The following relations hold for ν := ν(n,m,∆) and µ := µ(n,m,∆):

1. ν, µ = O(nm + 1

2. ν = n ·O(mn + 1

)⌊n2⌋, µ = n ·O

(mn + 1

)⌊n+1

3. ν = O(n)1+⌊n2⌋ ·∆n, µ = O(n)1+⌊ n+1

2⌋ ·∆1+n.

Proof The first bounds for ν and µ are follow from the trivial identities ν, µ ≤(n+mn

(nm + 1

W.l.o.g. we can assume that dim(P) = n. Let us denote by ζ(k, n) themaximal number of vertices in a polytope that is dual to the n-dimensionalcyclic polytope with k vertices. Due to the seminal work of P. McMullen [26],ν ≤ ζ(n+m,n). Analogously, due to the seminal work of R. Stanley [31] (seealso [13, Corollary 2.6.5] and [32]), µ ≤ ζ(n+m+ 1, n+ 1)− (n+ 1).

Due to [40, Corollary 8.28],

ζ(k, n) ≤

⌊n2⌋∑

(k − (n+ 1) + i

≤ n ·

(k − (n+ 1) + ⌊n

⌊n2 ⌋

)= n ·O

)⌊n2⌋.

Consequently, ν = n ·O(n+mn

)n2 and µ = (n+1) ·O

(n+m+1n+1

)⌊n+1

2⌋. So, the

second bounds hold.Finally, due to [24], we can assume that n+m = O(n2 ·∆2). Substituting

the last formula to the second bounds, we finish the proof.

2 Proof of Theorem 2

2.1 A recurrent formula for the generating function of a group polyhedron

Let G be a finite Abelian group and g1, . . . , gn ∈ G. Let, additionally, ri =∣∣〈gi〉

∣∣be the order of gi, for i ∈ {1, . . . , n}, and rmax = max

iri. For g0 ∈ G and

k ∈ {1, . . . , n}, let PG(k, g0) be the polyhedron induced by the convex hull ofsolutions of the following system:

k∑i=1

xigi = g0

x ∈ Zk≥0 .

Let us consider the formal power series

fk(g0;x) =∑

z∈PG(k,g0)∩Zk

For k = 1, we clearly have

f1(g0;x) =xs1

1− xr11

, where s = min{x1 ∈ Z≥0 : x1g1 = g0}.

If such s does not exist, then we put f1(g0;x) = 0.

Note that, for any value of xk ∈ Z≥0, the system (3) can be rewritten as

k−1∑i=1

xigi = g0 − xkgk

x ∈ Zk−1≥0 .

Hence, for k ≥ 1, we have

fk(g0;x) =

=fk−1(g0;x) + xk · fk−1(g0 − gk;x) + · · ·+ xrk−1

k · fk−1(g0 − gk · (rk − 1);x)

1− xrkk

·rk−1∑

xik · fk−1(g0 − i · gk;x). (4)

Consequently,

fk(g0;x) =

r1−1∑i1=0

· · ·rk−1∑ik=0

ǫi1,...,ikxi11 . . . xik

(1 − xr11 )(1 − xr2

2 ) . . . (1− xrkk )

where the nominator is a polynomial with coefficients ǫi1,...,ik ∈ {0, 1} anddegree at most (r1 − 1) . . . (rk − 1). Additionally, the formal power seriesfk(g0;x) converges absolutely to the given rational function if |xri

i | < 1, foreach i ∈ {1, . . . , k}.

2.2 Simple ∆-modular polyhedral cone and its generating function

Let A ∈ Zn×n, b ∈ Z

n, ∆ = | det(A)| > 0, P = P(A, b), and let us considerthe formal power series

f(P;x) =∑

z∈P ∩Zn

Let A = P−1SQ−1 and σ = Snn = ∆/∆gcd(A, n − 1), where S ∈ Zn×n is

the SNF of A and P,Q ∈ Zn×n are unimodular matrices. After the unimodularmap x = Qx′ and introducing slack variables y, the system Ax ≤ b becomes

Sx+ Py = Pb

x ∈ Zn

y ∈ Zn≥0 .

Since P is unimodular, the last system is equivalent to the system

{Py = Pb (mod S)

y ∈ Zn≥0 .

Note that points of P ∩Zn and the system (6) are connected by the bijectivemap x = A−1(b− y).

The system (6) can be interpreted as a group system (3), where G =ZnmodS with an addition modulo S, k = n, g0 = Pb mod S and gi =

P∗i mod S, for i ∈ {1, . . . , n}. Clearly, | G | = | det(S)| = ∆ and rmax ≤ σ.Following to the previous Subsection, for k ∈ {1, . . . , n} and g0 ∈ G, let

Mk(g0) be the solutions set of the system

k∑i=1

yigi = g0

y ∈ Zk≥0,

fk(g0;x) =∑

y∈Mk(g0)

where hi is the i-th column of the matrix A∗ = ∆ ·A−1.Note that

f(P ;x) =∑

z∈P ∩Zn

xz =∑

y∈Mn(Pb mod S)

xA−1(b−y) =

= xA−1b ·∑

y∈Mn(Pb mod S)

x− 1∆A∗y = xA−1b · fn(Pb mod S;x

1∆ ). (7)

Due to (4) and (5), for k = 1, we have

f1(g0;x) =x−sh1

1− x−r1h1, where s = min{y1 ∈ Z≥0 : y1g1 = g0}. (8)

For k ≥ 2, we have

fk(g0;x) =1

1− x−rkhk·rk−1∑

x−ihk · fk−1(g0 − i · gk;x) and (9)

fk(g0;x) =

r1−1∑i1=0

· · ·rk−1∑ik=0

ǫi1,...,ik x−(i1h1+···+ikhk)

(1 − x−r1h1)(1− x−r2h2) . . . (1− x−rkhk), (10)

where the nominator is a Laurent polynomial with coefficients ǫi1,...,ik ∈ {0, 1}.Clearly, the power series fk(g0;x) converges absolutely to the given function if|x−rihi | < 1, for each i ∈ {1, . . . , k}.

Due to the formulae (10) and (7), we have

f(P ;x) =

r1−1∑i1=0

· · ·rn−1∑in=0

ǫi1,...,in x1∆A∗(b−(i1,...,in)

(1− x−

)(1− x−

). . .

(1− x− rn

) . (11)

Note that ri∆hi is an integer vector, for any i ∈ {1, . . . , n}, and 1

∆A∗(b −(i1, . . . , in)

⊤) is an integer vector, for any (i1, . . . , in), such that ǫi1,...,in 6= 0.Indeed, by definition of ri, we have riP∗i ≡ 0 (mod S), so ri

∆hi = (riA−1)∗i =

(QS−1Pri)∗i, which is an integer vector. Vectors (i1, . . . , in)⊤ correspond to

solutions y of the system (6), and 1∆A∗(b − (i1, . . . , in)

⊤) = A−1(b − y) is aninteger vector.

Additionally, note that the vectors − ri∆hi represent extreme rays of the

recession cone of P.Let c ∈ Zn be chosen, such that (c⊤A∗)i 6= 0, for any i. Let us consider

the exponential sum

fk(g0; τ) =∑

y∈Mk(g0)

e−τ ·〈c,∑k

i=1hiyi〉

that is induced by fk(g0;x), substituting xi = eτ ·ci.The formulae (8), (9), and (10) become

f1(g0; τ) =e−〈c,sh1〉·τ

1− e−〈c,r1h1〉·τ, (12)

fk(g0;x) =1

1− e−〈c,rkhk〉·τ·rk−1∑

e−〈c,ihk〉·τ · fk−1(g0 − i · gk; τ), (13)

fk(g0; τ) =

r1−1∑i1=0

· · ·rk−1∑ik=0

ǫi1,...,ike−〈c,i1h1+···+ikhk〉·τ

(1− e−〈c,r1h1〉·τ

)(1− e−〈c,r2h2〉·τ

). . .

(1− e−〈c,rkhk〉·τ

) . (14)

Let χ = maxi∈{1,...,n}

{|〈c, hi〉|

}. Since 〈c, hi〉 ∈ Z 6=0, for each i, the number of

therms e−〈c,·〉·τ is bounded by 1 + 2 · k · rmax · χ ≤ 1 + 2 · k · σ · χ. So, aftercombining similar therms, the nominator’s length becomes O(k · σ · χ).

In other words, there exist coefficients ǫi ∈ Z≥0, such that

fk(g0; τ) =

k·σ·χ∑i=−k·σ·χ

ǫi · e−i·τ

(1− e−〈c,r1·h1〉τ

)(1− e−〈c,r2h2〉·τ

). . .

(1− e−〈c,rkhk〉·τ

) . (15)

Let us discuss the group-operations complexity issues to find the represen-tation (15) of fk(g0; τ), for any k ∈ {1, . . . , n} and g0 ∈ G.

Clearly, to find the desired representation of f1(g0; τ), for all g0 ∈ G, weneed r1 ·∆ group operations.

Fix g0 ∈ G and k ∈ {1, . . . , n}. To find fk(g0; τ), for k ≥ 2, we can use the

formula (13). Each nominator of the therm e−〈c,ihk〉·τ ·fk−1(g0−igk; τ) contains

at most 1 + 2 · (k − 1) · σ · χ non-zero therms of the type ǫ · e−〈c,·〉·τ . Hence,the summation can be done with O(k ·σ2 ·χ) group operations. Consequently,the total group-operations complexity can be expressed by the formula

O(∆ · n2 · σ2 · χ).

Finally, since the diagonal matrix S can have at most log2(∆) thermsthat are not equal to 1, the arithmetic complexity of one group operation isO(log(∆)). Hence, the total arithmetic complexity is

O(∆ · log(∆) · n2 · σ2 · χ

Finally, let us show how to find the exponential form

f(P ; τ) =∑

z∈P ∩Zn

e〈c,z〉·τ

of the power series f(P;x) induced by the map xi = eci·τ .

Due to the formula (7), we have

f(P ; τ) = e〈c,A−1b〉·τ · fn(Pb mod S;

Due to the last formula and the formulae (11) and (15), we have

f(P; τ) =

n·σ·χ∑i=−n·σ·χ

ǫi · e1∆(〈c,A∗b〉−i)·τ

(1− e−〈c,

·h1〉·τ)(1− e−〈c,

·h2〉·τ). . .

(1− e−〈c, rn

∆·hn〉·τ

Again, due to (11), we have 〈c, ri∆hi〉 ∈ Z 6=0, for any i ∈ {1, . . . , n}, and

1∆(〈c, A∗b〉 − i) ∈ Z, for any i, such that ǫi > 0.

We have proven the following:

Theorem 6 Let A ∈ Zn×n, b ∈ Zn, ∆ = | det(A)| > 0, and P = P(A, b). Let,additionally, σ = Snn, where S is the SNF of A, and χ = max

i∈{1,...,n}

{|〈c, hi〉|

where hi is the i-th column of A∗ = ∆ · A−1.

Then, the formal exponential series f(P ; τ) can be represented as

f(P ; τ) =

ǫi · eαi·τ

(1− eβ1·τ

)(1− eβ2·τ

). . .

(1− eβn·τ

where ǫi ∈ Z≥0, βi ∈ Z6=0, and αi ∈ Z.

This representation can be found with an algorithm having the arithmeticcomplexity bound O

(∆ · log(∆) · n2 · σ2 · χ

Moreover, the coefficients {ǫi} depend only on Pb mod S, so they take atmost ∆ possible values. All these values are computed during the algorithm.

2.3 Handling the general case

Following to Remark 2, we will only work with systems in the canonical form.Let A ∈ Z

(n+m)×n, b ∈ Qn+m, rank(A) = n, and ∆ = ∆(A). Let us consider

the polytope P = P(A, b).Let us choose γ = max{‖A‖max, ‖b‖∞}, β = min

i∈{1,...,n+m}{⌈bi⌉−bi : bi /∈ Z},

and ε = min{β/2, (1 + 2n · n⌈n/2⌉ · γ)−1}. Then, by Theorem 5, the polytopeP ′ = P(A, b+t) is simple, where the vector t is chosen, such that ti = εi−1, fori ∈ {1, . . . , n+m}. By the construction, P ∩Z

n = P ′ ∩Zn. From this moment,

we assume that P = P(A, b) is a simple polytope.Due to Remark 5, the Brion’s Theorem gives:

[P ] =∑

v∈vert(P)

[tcone(P , v)

v∈vert(P)

[P(AJ (v), bJ (v))

]modulo polyhedra with lines .

Due to the seminal work [1], all vertices of the simple polyhedra P can beenumerated with O

((m+ n) · n · | vert(P)|

)arithmetic operations.

Denote f(P;x) = F([P ]) ∈ R(V), for any rational polyhedra P, where Fis the evaluation considered in Theorem 1.

Note that f(P(B, u);x) = f(P(B, ⌊u⌋);x), for any B ∈ Qn×n and u ∈ Q

n.So, due to Theorem 1, we can write

f(P;x) =∑

v∈vert(P)

f(P(AJ (v), ⌊bJ (v)⌋

Due to results of the previous Subsection, each therm f(P(AJ (v), ⌊bJ (v)⌋

has the form (11).To find the value of | P ∩Z

n | = limx→1

f(P ;x), we follow to Chapters 13 and

14 of [4].Let us choose c ∈ Z

n, such that any element of the row-vector c⊤(AJ (v))−1

is non-zero, for each v ∈ vert(P). Substituting xi = eci·τ , let us consider theexponential function

f(P ; τ) =∑

v∈vert(P)

f(P(AJ (v), ⌊bJ (v)⌋

); τ).

Due to [4, Chapter 14], the value | P ∩Zn | is a constant therm in the Tailor

series of the function f(P ; τ), so we just need to compute it.

Let us fix some therm f(P(B, u); τ

)of the previous formula. Due to The-

orem 6, it can be represented as

f(P(B, u); τ

ǫi · eαi·τ

(1− eβ1·τ

)(1− eβ2·τ

). . .

(1− eβn·τ

where ǫi ∈ Z≥0, βi ∈ Z 6=0, and αi ∈ Z.

Again, due to [4, Chapter 14], we can see that the constant therm in Tailor

series for f(P(B, u); τ

)is exactly

n·σ·χ∑

i=−n·σ·χ

ǫiβ1 . . . βn

j!· tdn−j(β1, . . . , βn), (16)

where tdj(β1, . . . , βn) are homogeneous polynomials of degree j, called the j-thTodd polynomial on β1, . . . , βn. Due to [12, Theorem 7.2.8, p. 137], the valueof tdj(β1, . . . , βn) can be computed with an algorithm that is polynomial onj, n, and the bit-encoding length of β1, . . . , βn.

Since σ ≤ ∆, due to Theorem 6, the total arithmetic complexity to findthe value of (16) can be bounded by O

(∆3 · log(∆) · n2 · χ

The constant therm in Tailor series for the complete function f(P ; τ) canbe found just by summation. It gives the arithmetic complexity bound

O(ν(n,m,∆) · n2 ·∆3 · log(∆) · χ

Finally, we choose c⊤ as the sum of rows of some non-degenerate n × nsub-matrix of A. Note that elements of the matrix A · A∗

J (v) are included to

the set of all n × n sub-determinants of A, where A∗J (v) = ∆ · A−1

J (v), for

all v ∈ vert(P). Hence, χ ≤ n∆, and the total arithmetic complexity boundbecomes

O(ν(n,m,∆) · n3 ·∆4 · log(∆)

). It finishes the proof of Theorem 2.

2.4 Proof of Corollary 1

The presented complexity bounds follow by the different ways to estimate thevalue ν(m,n,∆) that are given in Lemma 1.

Finally, let us show how to handle the case, when P is an unboundedpolyhedron. We need to distinguish between two possibilities: | P ∩Z

n | = 0and | P ∩Zn | = ∞. Let us choose any vertex v of P and consider a set ofindices J , such that | J | = n, AJ v = bJ and rank(AJ ) = n. For the firstand second bounds, we add a new inequality c⊤x ≤ c0 to the system Ax ≤ b,where c⊤ =

∑ni=1(AJ )i∗ and c0 = c⊤v + ‖c‖1 · n∆ + 1. Let A′x ≤ b′ be the

new system. Due to [9], | P ∩Zn | = 0 iff | P(A′, b′)∩Z

n | = 0. Since P(A′, b′) isa polytope and ∆(A′) ≤ n∆, we just need to add an additional multiplicativefactor of O( d

m + 1) · n4 to the first bound and O(n4) to the second bound.

To deal with third bound, we just need to add additional inequalitiesAJ x ≥ bJ − ‖AJ ‖max · n2∆ · 1 to the system Ax ≤ b. The polyhedron be-comes bounded and the sub-determinants stay unchanged, and we follow tothe original scenario.

3 Proof of the Theorem 3

3.1 The chamber decomposition

Due to [8, Section 3], there exists a collection Q of p-dimensional polyhedraQ, such that

1. Qp =

⋃Q∈Q

2. dim(Q1 ∩Q1) < p, for any Q1,Q2 ∈ Q, Q1 6= Q2;3. for any Q ∈ Q and any y ∈ Q, the polytopes {Py} have a fixed collection

of vertices, given by affine transformations of y.

All the polytopes from Q are also called chambers. Here, we will call themas full-dimensional chambers. Following to [35, Lemma 3], let us consider thehyperplanes in the parameter space Qp formed by the affine hulls of the (p−1)-dimensional faces (facets) of full-dimensional chambers. Let H be the set of allsuch hyperplanes. Due to [39] (see also [36, Lemma 3.5]), the hyperplanes fromH divide the parameter space into at most O(|H |p) cells. These hyperplanescorrespond to the projections of the generic (p−1)-dimensional faces of {

Qp ×Q

n : B y+Ax ≤ b} into the parameters space. Since a part of the cellsforms a subdivision of the chambers, the total number of full-dimensionalchambers can be bounded by O(ηp), where |H | ≤ η.

The paper [8] of P. Clauss and V. Loechner gives an algorithm that com-putes the collection Q, which, for each Q ∈ Q, also computes a finite setT Q of affine functions T ∈ T Q, T : Q

p → Qn, such that all the parametric

vertices of Py are given by {T (y) : T ∈ T Q}.Due to the proof of [35, Lemma 3], the complexity of the Clauss and Loech-

ner’s algorithm is bounded by the number of chambers, which is O(ηp), timesthe number of parametric vertices, which is bounded by ν. So the Clauss andLoechner algorithm’s complexity is ηp · ν · (nmp)O(1).

Due to [8], for any fixedQ ∈ Q and y ∈ int(Q), the vertices T (y) : T ∈ T Q

are unique. But, for y ∈ Q1 ∩Q2, where Q1 6= Q2, some of the vertices cancoincide. Due to [35,36], this problem can be resolved by working with cham-bers of any dimension, induced by various intersections of the full-dimensionalchambers from Q. Let us denote this new collection of chambers by D .

Due to [35,36], we have

1. Qp =

⋃D∈D

rel.int(D);

2. rel.int(D1) ∩ rel.int(D2) = ∅, for any D1,D2 ∈ D , D1 6= D2;3. for any D ∈ D and any y ∈ rel.int(D), the polytope Py have a fixed set

of the unique vertices {T (y) : T ∈ T D}, where T D is a finite set of affinefunctions T : Qp → Qn.

We call the polyhedra from the new collection D as the low-dimensional cham-bers or just chambers. Let us give an algorithm to find the collection D withthe corresponding lists of the unique parametric vertices.

Briefly, we enumerate all the possible affine sub-spaces induced by all thepossible intersections of the hyperplanes from H . For any such an affine sub-space L ⊆ Q

p and any full-dimensional chamber Q ∈ Q, we consider theintersection L∩Q. By the definition, if L∩Q 6= ∅, then the intersection L∩Qwill form some low-dimensional chamber D ∈ D .

There are three main difficulties:

1. Two different full-dimensional chambers Q1 and Q2 can have the sameintersection with L: Q1 ∩L = Q2 ∩L;

2. If L′ ⊂ L is a sub-space of L, then it is possible that Q∩L = Q∩L′, forsome full-dimensional chamber Q ∈ Q. In other words, the dimension ofL∩Q is strictly less than dim(L);

3. For any Q ∈ Q, the set of the affine functions T Q is already given in theprevious stage by P. Clauss and V. Loechner’s algorithm. As it was alreadynoted, if y ∈ Q1 ∩Q2, then some of the vertices given by different func-tions from T Q can coincide. In other words, T 1(y) = T 2(y), for differentT 1, T 2 ∈ T Q and y ∈ Q1 ∩Q2. How we able to find such duplicates?

Let us first deal with the 1-th and 3-th difficulties. Let us fix L and afull-dimensional chamber Q ∈ Q and consider the lower-dimensional cham-ber D = L∩Q with the collection of parametric vertices T Q. Assume thatdim(L) = dim(L∩Q), we will break it later. Resolving the system of equali-ties corresponding to L, we can find a basis of L. In other words, there existsa matrix B ∈ Z

p×dL , where dL = dim(L), such that L = span(B). Supposethat T (y) ∈ T Q is given as T (y) = T y+t, where T ∈ Q

n×p is a rationaltransformation matrix and t ∈ Q

n is a rational translation vector. Let us showhow to find duplicates in the parametric vertices list for y ∈ L∩Q. If T 1 andT 2 form a duplicate, then

T1 y+t1 = T2 y+t2, for any y ∈ D,

that is equivalent to

(T1 − T2)y = t2 − t1, for any y ∈ aff.hull(D).

Due to the assumption dim(L) = dim(L∩Q) = dim(D),

(T1 − T2)Bx = t2 − t1, for any x ∈ RdL .

Since the solutions set of the last system is dL-dimensional, we have rank((T2−

T1)B)= 0. Consequently, T1B = T2B and t1 = t2, so the matrices {TB} and

vectors {t} must serve as a unique representation of affine functions T ∈ T Q

for fixed sub-space L. We just compute the basis B of L, compute the pairs{(TB, t)}, sort the resulted list, and delete all duplicates. This work can bedone in

(nmp)O(1) · |T Q | · log |T Q | = (nmp)O(1) · ν · log(ν)

arithmetic operations. Since |Q | = O(ηp), with enumerating of all the full-dimensional chambers Q inQ, it gives the complexity bound:

(nmp)O(1) · ηp · ν · log(ν).

Similarly, we can resolve the 1-th difficulty, for chambers Q∩L withdim(Q∩L) = dim(L). Any such a chamber is uniquely represented by the listof unique pairs

{(TB, t) : T (y) = T y+t, T ∈ T Q}.

So, we can consider this list as a unique identifier of Q∩L ∈ D and allthe duplicates can be eliminated just by sorting. Since the length of eachlist is bounded by ν(n,m,∆) and |Q | = O(ηp), the total complexity of thisprocedure is again

(nmp)O(1) · ηp · ν · log(ν).

Let us discuss simultaneously, how to resolve the 2-th difficulty and howto break the assumption dim(L) = dim(L∩Q), for Q ∈ Q. To do that, weneed first to sketch the full algorithm that constructs the collection D oflower-dimensional chambers. First of all, we use P. Clauss and V. Loechner’salgorithm to construct the full-dimensional chambers Q together with thecorresponding parametric vertices. Next, we enumerate all affine sub-spacesinduced by intersections of hyperplanes from H . The enumeration follows tothe following partial order: if L′ and L are sub-spaces, such that L′ ⊂ L,then L′ will be processed first. So, we start from 0-dimensional sub-spacescorresponding to intersections of p linearly independent hyperplanes from H

and move forward from (k− 1)-dimensional chambers to k-dimensional cham-bers. Clearly, for a 0-dimensional sub-space L and a full-dimensional chamberQ ∈ Q, the assumption dim(L) = dim(L∩Q) holds. Since the assumptiondim(L) = dim(L∩Q) holds, using the method discussed earlier, we can foundall the 0-dimensional chambers from the collection D with the sets of theirunique parametric vertices.

Let k ≥ 1 and suppose inductively that we want process k-dimensional sub-space L, when the sub-space L′ ⊂ L, with dim(L′) = k−1, is already processed.When we say ”processed”, we mean that all unique low-dimensional chambersL′ ∩Q have already built and k − 1 = dim(L′) = dim(L′ ∩Q). We againenumerate all the full-dimensional chambers Q ∈ Q. If dim(Q∩L) = k − 1,then we dismiss the chamber Q∩L, because it just coincides with the chamberQ∩L′. In the opposite case, when dim(Q∩L) = k = dim(L), we can usethe approach presented earlier to construct the unique set of the parametricvertices for Q∩L. So, we need to learn how to distinguish between the casesdim(Q∩L) = k and dim(Q∩L) = k − 1.

Let B ∈ Zp×k and B′ ∈ Zp×(k−1) be bases of L and L′, chosen, such thatB =

(B′ b

), for b ∈ Z

p. The equality dim(Q∩L) = k holds, iff there existsaffine functions T 1(y) = T1 y+t1 and T 2(y) = T2 y+t2, from the set of affinefunctions T Q, such that

∀y ∈ Q∩L′ : T 1(y) = T 2(y) and

∃y ∈ Q∩L : T 1(y) 6= T 2(y),

which is equivalent to

∀y ∈ aff.hull(Q∩L′) : T 1(y) = T 2(y) and

∃y ∈ aff.hull(Q∩L) : T 1(y) 6= T 2(y).

The last is equivalent to

∀x ∈ Rk−1 :(T1 − T2

)B′x = t2 − t1 and

∃x ∈ Rk :

(T1 − T2

)(B′ b

)x 6= t2 − t1.

By the same reasoning, rank((T1−T2

)B′) = 0,

(T1−T2

)B′ = 0 and t2−t1 = 0.

Hence, the second property can be satisfied only if T1b 6= T2b.

So, we have the following criterion:

dim(Q∩L) = k ⇐⇒ ∃T 1, T 2 ∈ T Q : T 1(b) 6= T 2(b). (17)

Let us summarize the whole algorithm:

1. Construction of full-dimensional chambers. Using the P. Clauss andV. Loechner’s algorithm, we construct the collection Q of full-dimensionalchambers together with the sets of affine functions T Q, for each Q ∈ Q.It takes (nm∆)O(1) · ηp · ν arithmetic operations.

2. Construction of lower-dimensional chambers. To construct the col-lection D , we consider intersections of sub-spaces L, induced by all the pos-sible intersection of hyperplanes H with full-dimensional chambers fromQ.Assume that all unique (k − 1)-dimensional chambers of the type Q∩L′,where L′ is the (k − 1)-dimensional intersection of hyperplanes from H

and Q ∈ Q, have already constructed with their unique sets of parametricvertices. For all k-dimensional sub-spaces L and all Q ∈ Q, we performthe following operations:

3. Dimension check. Let D = Q∩L. Using the criteria (17), we check thatdim(D) = k = dim(L). If it does not holds, then we skip the chamber D;

4. Erase duplicated parametric vertices. Using the algorithm presentedearlier, we erase all the duplicated parametric vertices of D and append Dto the collection D .

5. Erase duplicated chambers. After all Q ∈ Q for fixed L are processed,we remove the duplicated chambers D = Q∩L from D .

For any fixed L, the complexity is bounded by (nmp)O(1) · ηp · ν · log(ν).All the sub-spaces {L} can be enumerated with O(ηp) arithmetic operations.The total arithmetic complexity to construct the full chambers decompositioncan be bounded by:

(nmp)O(1) · η2p · ν · log(ν).

3.2 Dealing with a fixed chamber D ∈ D

Let us fix a low-dimensional chamber D chosen from the collection D , con-structed in the previous stage. As it was noted before, for any y ∈ D, polytopes{Py} have a fixed set of unique vertices {T (y) : T ∈ T } for T = T D. With-out loss of generality we ca assume that dim(Py) = n.

Again, due to the Brion’s Theorem, we have

[Py] =∑

T ∈T

[tcone

(Py, T (y)

T ∈T

[T (y) + fcone

(Py, T (y)

)]modulo polyhedra with lines .

Let us fix a vertex T (y) and consider the cone C = fcone(Py, T (y)

Clearly, C = P(AJ (T (y)),0

), and, consequently, C◦ = cone

J (T (y))

apply the decomposition (triangulation) of C◦ into simple cones Si. Let qTbe the total number of simple cones in this decomposition. Clearly, for anyi ∈ {1, . . . , qT }, we can write that Si = cone(B⊤

i ), where B⊤i is a non-singular

n× n integer sub-matrix of A⊤J (T (y)). In other words, we have

[C◦] =⋃

i∈{1,...,qv}

[cone(B⊤

modulo lower-dimensional rational cones .

Next, we use the duality trick, see [5, Remark 4.3]. Due to [4, Theorem 5.3](see also [4, Theorem 2.7]), there is a unique linear transformationD : P(V) →P(V), for which D([P ]) = [P ◦].

Consequently, due to Remark 5, we have

[C] =⋃

i∈{1,...,qT }

[cone(B⊤

i )◦]=

i∈{1,...,qT }

[P(Bi,0)

]modulo polyhedra with lines,

and, due to Theorem 4, we have

[T (y) + C

i∈{1,...,qT }

[T (y) + P(Bi,0)

i∈{1,...,qT }

[P(Bi, Bi T (y)

The triangulation of all cones cone(A⊤J (T (y))) induces the triangulation of

the cone cone(A⊤). Hence,∑

T ∈TqT ≤ µ.

Consequently,

[Py] =∑

[P(Bi, Bi T i(y)

where | I | ≤ µ, {Bi} are some n×n non-singular sub-matrices of A and T i(y)are some parametric vertices of Py.

Denote f(P;x) = F([P ]), for any rational polyheda P , where F is theevaluation considered in Theorem 1.

Note that

f(P(Bi, Bi T i(y)

(P(Bi, ui(y)

where ui(y) = ⌊Bi T i(y)⌋. Hence, due to Theorem 1, we can write

f(Py;x

f(P(Bi, ui(y)

Let us again fix i and B = Bi, u(y) = ui(y). Due to the proof of Theorem2, each therm f

(P(B, u(y));x

)has the form

f(P(B, u(y));x

∑r1−1i1=0 · · ·

∑rn−1in=0 ǫi1,...,in x

1δB∗(u(y)−(i1,...,in)

(1− x−

r1δh1

)(1− x−

r2δh2

). . .

(1− x− rn

Here δ = | det(B)|, B∗ = δ · B−1, (h1, . . . , hn) are the columns of B∗ and(r1, . . . , rn) are orders of (h1, . . . , hn) modulo S, where S is the SNF of B.

Let us consider now the exponential function f(P(B, u(y)); τ

)induced

from f(P(B, u(y));x

)by substitution xi = eciτ , where c ∈ Zn is chosen, such

that c⊤hi 6= 0, for any i.It follows that

f(Py; τ) =

f(P(Bi, ui(y)

); τ), (18)

and, due to Theorem 6, for fixed i,

f(P(B, u(y)); τ) =

ǫi(y) · eαi(y)·τ

(1− eβ1·τ

)(1− eβ2·τ

). . .

(1− eβn·τ

where ǫi(y) ∈ Z≥0, αi(y) =1δ

(〈c, B∗u(y)〉 − i

)and βi = −〈c, ri

δ hi〉.Here, due to Theorem 6, the coefficients ǫi(y) are functions depending on

Pu(y) mod S. They can be computed in O(δ · log(δ) · n2 · σ2 · χ) arithmeticoperations for all δ possible values of Pu(y) mod S. As we noted before inthe proof of Theorem 2, cP(y) is equal to the constant therm in the Teilor’s

expansion of f(Py; τ). It can be represented as the sum of constant therms of

the Teilor’s expansions of f(P(Bi, ui(y)); τ).

Us in the formula (16), we have that constant therm in the Teilor’s expan-

sion of f(P(B, u(y)); τ) is

n·σ·χ∑

−n·σ·χ

ǫi(Pu(y) mod S)

β1 · · · · · βn

n−1∑

αji (y)

j!· tdn−j(β1, . . . , βn). (19)

Here, the internal sum is an (n − 1)-degree step-polynomial and ǫi(·) are pe-riodical coefficients with the period given by the formula ǫi(y) = ǫi(y+S · 1).

Taking c as in the proof of Theorem 2, we have χ ≤ n ·∆. So, the lengthand degree of the last formula are bounded by n3 · δ2 and n− 1, respectively.

Moreover, such a representation can be found, usingO(δ4 ·log(δ)·n3) arithmeticoperations.

Consequently, the total length and degree of the full periodical step-polynomialrepresenting cP(y) for y ∈ D are bounded by O(µ · n3 · δ2) and n− 1, respec-tively. The total arithmetic complexity is O(µ · n3 · δ4 · log(δ)).

3.3 How to store the data and what is the final preprocessing time?

As it was considered in Subsection 3.1, the collection of full-dimensional cham-bers Q with their parametric vertices can be computed with (nmp)O(1) · ηp ·ν · log(ν) arithmetic operations. The collection D of lower dimensional cham-bers with their collections of unique parametric vertices can be computed with(nmp)O(1) · η2p · ν · log(ν) arithmetic operations.

Each chamber of Q or D is represented as a cell that induced by subset ofhyperplanes from H . Consequently, we have an injection from D ∪Q to theset {−1, 0, 1}|H |. So, we can use a hash-table to store the chambers of D ∪Q,mapping them to keys from {−1, 0, 1}|H |. This hash table can be constructedwith (nmp)O(1) · η2p · ν · log(ν) arithmetic operations.

For fixed chamber D from Q ∪D and y ∈ D, the counting function cP(y)is represented as the periodical step-polynomial given by the formulae (18)and (19). Its length is O(n3 ·∆2), its degree is n− 1, and it can be computedby O(µ · n3 ·∆4 · log(∆)) arithmetic operations.

To store such a representation, for each i, in the formula (18) we need tostore the formula (19) as the matrix Bi, the sequence of coefficients β1, . . . , βn

and the values of at most O(n2 ·∆2) periodical coefficients {ǫi(·)}. The valuesof {ǫi(·)} can be stored in a hash-table of the size | det(S)| ≤ ∆, where thekeys are vectors from Z

n modS.Consequently, the total preprocessing arithmetic complexity is

(nmp)O(1) · η2p · ν · µ · log(ν) ·∆4 · log(∆).

3.4 What is a query time?

Let us estimate complexity of a query to compute cP(y) given y ∈ Qp. First,we need to find a chamber D ∈ Q ∪D , such that y ∈ rel.int(D). To do that, wesubstitute y to the inequalities corresponding to the hyperplanes H , and mapy to {−1, 0, 1}|H |. Using the corresponding hash table, we find the chamberD. All of that take O(p · |H |) arithmetic operations.

After that, for each parametric vertex T (y), given by T ∈ T D, we performthe following operations:

1. Compute u(y) = ⌊B T (y)⌋ with O(n2 · p) operations;2. Access the coefficients ǫi(Pu(y) mod S) using the corresponding hash-

table. It takes O(n2 + log(∆)) operations;

3. For each i ∈ {−n · σ ·χ, . . . , n · σ ·χ}, compute αi(y) =1δ

(〈c, B∗u(y)〉 − i

It takes O(n2 · ∆2) operations. And compute the powers αji (y), for j ∈

{0, . . . , n− 1}, totally it takes O(n3 ·∆2) operations;4. Compute the formula (19) with O(n2 ·∆2) operations.

After that we just need to combine the results of different at most µ thermsin (18). Hence, the total query time is

O(µ · n3 ·∆2).

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