A note on the parametric integer programming in the average case: sparsity, proximity, and FPT-algorithms
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Parametric integer programming in the average case:sparsity, proximity, and FPT-algorithms
D. V. Gribanov · D. S. Malyshev · P. M. Pardalos the date of receipt and acceptance should be inserted later
Abstract
We consider the Integer Linear Programming (ILP) problemmax { c ⊤ x : Ax ≤ b, x ∈ Z n } , parameterized by a right-hand side vector b ∈ Z m , where A ∈ Z m × n is a matrix of rank n . Let v be an optimal vertexof the Linear Programming (LP) relaxation max { c ⊤ x : Ax ≤ b } and B be acorresponding optimal base. We show that, for almost all b ∈ Z m , an optimalpoint of the square ILP problem max { c ⊤ x : A B x ≤ b B , x ∈ Z n } satisfies theconstraints Ax ≤ b of the original problem, where the system A B x ≤ b B consists of the rows A i x ≤ b i , for i ∈ B . A structure of the ILP problemmax { c ⊤ x : A B x ≤ b B , x ∈ Z n } was perfectly studied in works of R. Gomory.More precisely, from works of Gomory it directly follows that the square ILPproblem can be solved by an algorithm of the arithmetic complexity O ( n · δ · log δ ), where δ = | det A B | . Consequently, it can be shown that, for almostall b ∈ Z m , the original problem max { c ⊤ x : Ax ≤ b, x ∈ Z n } can be solvedby an algorithm of arithmetic complexity O ( n · ∆ · log ∆ ), where ∆ is themaximum absolute value of n × n minors of A . By the same technique, we givenew inequalities on the integrality gap and sparsity of a solution and slackvariables. The article was prepared within the framework of the Basic Research Program at the Na-tional Research University Higher School of Economics (HSE).D. V. GribanovNational Research University Higher School of Economics, 136 Rodionova Ulitsa, NizhnyNovgorod, 603093, Russian FederationE-mail: [email protected]. S. MalyshevNational Research University Higher School of Economics, 136 Rodionova Ulitsa, NizhnyNovgorod, 603093, Russian FederationE-mail: [email protected]. M. PardalosUniversity of Florida, 401 Weil Hall, P.O. Box 116595, Gainesville, FL 326116595, USAE-mail: pardalos@ufl.edu D. V. Gribanov et al.
Another ingredient is a known lemma that states the equality of the max-imum absolute values of rank minors of matrices with orthogonal columns.This lemma gives us an opportunity to transform ILP problems of the typemax { c ⊤ x : Ax = b, x ∈ Z n + } to problems of the previous type, and presentanalogue results for them. Keywords
Integer Linear Programming · Gomory polyhedron · BoundedMinors · FPT-algorithm · Integrality Gap · Sparsity Level A ∈ Z m × n be an integer matrix. We denote by A ij the ij -th element ofthe matrix, by A i ∗ its i -th row, and by A ∗ j its j -th column. The set of integervalues, starting from i and ending in j , is denoted by i : j = { i, i + 1 , . . . , j } .Additionally, for subsets I ⊆ { , . . . , m } and J ⊆ { , . . . , n } , the symbols A I J and A [ I, J ] denote the submatrix of A , which is generated by all the rows withindices in I and all the columns with indices in J . When I or J are replacedby ∗ , it means that all the rows or columns are selected, respectively. When itis clear from the context, we simply write A I instead of A I ∗ and A J insteadof A ∗ J .The maximum absolute value of elements of a matrix A is denoted by k A k max = max i,j | A i j | . The number of non-zero elements of a vector x isdenoted by k x k = |{ i : x i = 0 }| . The l p -norm of a vector x is denoted by k x k p , for p ∈ {∞ , , , . . . } . The vector of diagonal elements of a n × n matrix A is denoted by diag( A ) = ( A , . . . , A n n ) ⊤ . Its adjugate matrix is denotedby A ∗ = det( A ) A − . Definition 1
For a matrix A ∈ Z m × n , by ∆ k ( A ) = max {| det A I J | : I ⊆ m, J ⊆ 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 ), we denote the greatest common divisor of deter-minants of all the k × k submatrices of A . Additionally, let ∆ ( A ) = ∆ rank A ( A )and ∆ gcd ( A ) = ∆ gcd ( A, rank( A )). Definition 2
For a matrix A and a vector b , by P ≤ ( A, b ) we denote thepolyhedron { x ∈ R n : Ax ≤ b } and by P = ( A, b ) we denote the polyhedron { x ∈ R n + : Ax = b } .The set of all vertices of a polyhedron P is denoted by vert( P ). Definition 3
Let A be an integer matrix and b, c be integer vectors.By LP ≤ ( A, b, c ), we denote the problem max { c ⊤ x : x ∈ P ≤ ( A, b ) } .By ILP ≤ ( A, b, c ), we denote the problem max { c ⊤ x : x ∈ P ≤ ( A, b ) ∩ Z n } .By LP = ( A, b, c ), we denote the problem max { c ⊤ x : x ∈ P = ( A, b ) } .By ILP = ( A, b, c ), we denote the problem max { c ⊤ x : x ∈ P = ( A, b ) ∩ Z n } . itle Suppressed Due to Excessive Length 3 Definition 4
For a matrix B ∈ R m × n , cone( B ) = { Bt : t ∈ R n + } is the conespanned by columns of B , conv . hull( B ) = { Bt : t ∈ R n + , P ni =1 t i = 1 } is the convex hull, spanned by columns of B , Λ( B ) = { x = Bt : t ∈ Z n } is the lattice,spanned by columns of B . Definition 5
An algorithm, parameterized by a parameter k , is called fixed-parameter tractable (or, simply, a FPT- algorithm ) if its computational com-plexity can be estimated by a function from the class f ( k ) n O (1) , where n is theinput size and f ( k ) is a computable function that depends on k only. A com-putational problem, parameterized by a parameter k , is called fixed-parametertractable (or, simply, a FPT- problem ) if it can be solved by a FPT-algorithm.For more information about the parameterized complexity theory, see [9,10].1.2 Description of results and related worksLet us fix a vector c ∈ Z n , a matrix A ∈ Z m × n of the rank n and consider theILP ≤ ( A, b, c ) problem, parameterized by b ∈ Z m . Let ∆ = ∆ ( A ). Presentingthe next definition, we follow works [7,30,31,32]. Definition 6
Let Λ be an arbitrary m -dimensional sublattice of Z m and Ω Λ,t = { b ∈ Λ : k b k ∞ ≤ t } . Then, for A ⊆ Z n , we definePr Λ,t ( A ) = | A ∩ Ω Λ,t || Ω Λ,t | and Pr Λ ( A ) = lim inf t →∞ Pr Λ,t ( A ) . For Λ = Z m , we simply denote Pr( A ) = Pr Λ ( A ).Conditional probability of A with respect to G is denoted by the formulaPr Λ ( A | G ) = Pr Λ ( A ∩ G )Pr Λ ( G ) . It was shown in [32] that the ILP ≤ ( A, b, c ) problem is equivalent to theLP ≤ ( A, b, c ) problem with some additional number of integer constraints, thisnumber is called the integrality number . Due to [32], for almost all b ∈ Z m theintegrality number is bounded by O ( √ ∆ ). Hence, for almost all b ∈ Z m , theILP ≤ ( A, b, c ) problem can be solved by a polynomial-time algorithm, when ∆ is fixed. Our first main result strengthens the last fact. Additionally, we givesome bounds on the integrality gap and the sparsity of a solution and the slackvariables.We assume that ∀ x ∈ P ≤ ( A, ) : c ⊤ x ≤ ≤ ( A, b, c )problem is bounded, for all b ∈ Z m . In the opposite case, the LP ≤ ( A, b, c )problem and, consequently, the ILP ≤ ( A, b, c ) problem are unbounded. Thelast condition can be checked by any polynomial-time LP algorithm, see, forexample, [23,25,26,29].Next, we denote an optimal solution of the ILP ≤ ( A, b, c ) problem by z ,an optimal vertex solution of the LP ≤ ( A, b, c ) problem by v , δ = | det A B | ,where B is an optimal base, related to v , and N = 1 : m \ B . Additionally, let D. V. Gribanov et al. F = { b ∈ Z m : P ≤ ( A, b ) = ∅} , hence F means that the LP ≤ ( A, b, c ) problemis feasible.
Theorem 1
Let G = { b ∈ F : b N − A N v ≥ ( δ − } . Then Pr(
G | F ) = 1 ,and for any b ∈ G the following properties of the problem ILP ≤ ( A, b, c ) hold:1. k A ( v − z ) k ≤ log δ + ( m − n ) , k A ( v − z ) k ≤ ( δ −
1) + ( m − n )( ∆ − , k A ( v − z ) k ∞ ≤ ∆ − ,2. k b − Az k ≤ log δ + ( m − n ) , k b − Az k ≤ ( δ −
1) + ( m − n )( ∆ −
1) + k b N − A N v k , k b − Az k ∞ ≤ ∆ − k b N − A N v k ∞ ,3. the point z lies on a face of P , whose dimension is bounded by log δ ,4. the point z can be found by an algorithm with the arithmetic complexity O ( n · ∆ · log ∆ ) .Additionally, if we make some standard normalization of the system Ax ≤ b (the normalization can be done in polynomial time, see Subsection 2.2), thenthe following statements hold for z :1. k z k ≤ δ , k z k < δ log δ , k z k ∞ < δ ,2. k v − z k ≤ δ , k v − z k < δ log δ , k v − z k ∞ < δ / .Remark 1 To make the text more clear, we hide terms of the type poly( s )in O -notation, when we estimate the computational complexity in our work.Here, s denotes the input size.For example, in the previous theorem, the formula O ( n · ∆ · log ∆ ) means O ( n · ∆ · log ∆ + poly( s )). We note, that the computational complexity of thisadditional computations is not greater, than the computational complexity ofthe Hermite Normal Form (HNF, for short) or the Smith Normal Form (SNF,for short) computing. For details, see Section 2.1 and Remark 3.Now, we are going to consider the problems of the type ILP = ( A, b, c ), as-suming that rank A = m . This problem with respect to the parameter b inthe average case was perfectly studied in the works [30,31]. In our next mainresult, we show that if an optimal vertex solution of the LP = ( A, b, c ) problemhas sufficiently big basis components, then the corresponding ILP = ( A, b, c )problem has good properties that are analogues to properties from the pre-vious theorem. We show that the probability of this situation is 1. It givesstrengthening in the average case on the l ∞ -integrality gap and the compu-tational complexity. Additionally, we give a new proof for known inequalities,whose advantage is its simplicity.Again, we assume that ∀ x ∈ P = ( A, ) : c ⊤ x ≤
0, meaning that the LP = ( A, b, c )problem is bounded, for any b ∈ Z m . We denote F = { b ∈ Z m : P = ( A, b ) = ∅} ,meaning that the LP = ( A, b, c ) problem is feasible. Let v B = A − B b be thebasis part of an optimal vertex solution v of the LP = ( A, b, c ) problem and δ = | det A B | .Without loss of generality (see Remark 5) we can assume, that ∆ gcd ( A ) =1. It simplifies the resulting formulas. itle Suppressed Due to Excessive Length 5 Theorem 2
Let A = { b ∈ F : v B ≥ ( δ − } . Then Pr(
A | F ) = 1 , and forany b ∈ A the following properties of the problem ILP = ( A, b, c ) hold:1. k v − z k ≤ log δ + m , k v − z k ≤ ( δ −
1) + m ( ∆ − , k v − z k ∞ ≤ ∆ − ,2. k z k ≤ log δ + m ,3. the point z can be found by an algorithm with the arithmetic complexity O (( n − m ) · ∆ · log ∆ ) . ≤ ( A, b, c ) problem can be solvedby a polynomial-time algorithm. It is well-known that all optimal solutionsof the corresponding LP problem are integer, when ∆ ( A ) = 1. Hence, theILP ≤ ( A, b, c ) problem can be solved by any polynomial-time LP algorithm(like the ones in [23,25,26,29]).The next natural step is to consider the bimodular case, i.e. ∆ ( A ) ≤
2. Thefirst paper that discovers fundamental properties of the bimodular ILP prob-lem is [40]. Recently, using results of [40], a strong polynomial-time solvabilityof the bimodular ILP problem was proved in [4].Unfortunately, not much is known about the computational complexity ofthe ILP ≤ ( A, b, c ) problem, for ∆ ( A ) ≥
3. In [36], a conjecture is establishedthat, for each fixed natural number ∆ = ∆ ( A ), the ILP ≤ ( A, b, c ) problem canbe solved by a polynomial-time algorithm. There are variants of this conjec-ture, where the augmented matrices (cid:18) c ⊤ A (cid:19) and ( A b ) are considered [1,36]. Astep towards deriving its complexity was done by Artmann et al. in [3]. Namely,it has been shown that if the constraints matrix has additionally no singularrank submatrices, then the ILP problem with bounded ∆ can be solved inpolynomial time. The last fact was strengthened to an FPT-algorithm in [19].Some interesting results about polynomial-time solvability of the boolean ILPproblem were obtained in [1,5,17,18].F. Eisenbrand and S. Vempala [12] presented a randomized simplex-typelinear programming algorithm, whose expected running time is strongly poly-nomial if all minors of the constraints matrix are bounded by a fixed constant.As it was mentioned in [4], due to E. Tardos’ results [39], linear programs withthe constraints matrices, whose all minors are bounded by a fixed constant,can be solved in strongly polynomial time. N. Bonifas et al. [6] showed thatany polyhedron, defined by a totally ∆ -modular matrix (i.e., a matrix, whoseany rank minor is ± ∆ ), has a diameter, bounded by a polynomial on ∆ andthe number of variables.For the case, when A is square, a FPT-algorithm can be obtained fromthe classical work of R. Gomory [14]. Due to [19], a FPT-algorithm exists forthe case, when A is almost square, e.g. A has a small number of additionalrows. It was shown in [32] that, for fixed A , c , and varying b , the ILP ≤ ( A, b, c )problem can be solved by a FPT-algorithm with a high probability.
D. V. Gribanov et al.
Due to [13], the number of distinct rows of the system Ax ≤ b can beestimated by ∆ log ∆ · n + 1, for ∆ ≥ P ≤ ( A, b ) hasa relatively small width, i.e. the width is bounded by a function that is linearon the dimension and exponential on ∆ ( A ). Interestingly, due to [21], the widthof any empty lattice simplex, defined by a system Ax ≤ b , can be estimatedby ∆ ( A ). In [16], it has been shown that the width of such simplicies canbe computed by a polynomial-time algorithm. The last result was improvedto a FPT-algorithm in [19]. In [20], a similar FPT-algorithm was given forsimplicies, defined by the convex hull of columns of ∆ -modular matrices. Wenote that, due to [35], this problem is NP-hard in the general case.Important results about the proximity and sparsity of the LP, ILP, andmixed problems in the general case can be found in [2,8,27,33]. Interest-ingly, due to [11], the maximum difference between the optimal values of theLP ≤ ( A, b, c ) and ILP ≤ ( A, b, c ) problems over all right-hand sides b ∈ Z m , forwhich LP ≤ ( A, b, c ) is feasible, can be find by a polynomial-time algorithm ifthe dimension is fixed. A ∈ Z m × n be an integer matrix of the rank n . It is a known fact (see,for example, [34,37,43]) that there exist unimodular matricies P ∈ Z m × m and Q ∈ Z n × n , such that A = P (cid:18) S d × n (cid:19) Q , where d = m − n and S ∈ Z n × n is a diagonal non-degenerate matrix. Moreover, Q ki =1 S i i = ∆ gcd ( k, A ), and,consequently, S i i | S ( i +1) ( i +1) , for i ∈ n − (cid:18) S d × n (cid:19) is calledthe Smith Normal Form (or, shortly, the SNF) of the matrix A .A near-optimal polynomial-time algorithm for constructing the SNF of A is given in [37].2.2 Normalization of a ∆ -modular system of linear inequalitiesLet us consider a system Ax ≤ b , where A be a m × n matrix of rank n thathas already been reduced to the Hermite Normal Form (the HNF) [34,38,43].Let us assume that the matrix A B = A n is non-singular, and let A N be the d × n matrix, generated by the remaining rows of A . In other words, A = (cid:0) A B A N (cid:1) and m = n + d . Let us denote b B and b N in a similar way.Using additional permutations of rows and columns, we can transform A ,such that the matrix A B will have the following form: itle Suppressed Due to Excessive Length 7 A B = . . . . . .
00 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A s +1 1 A s +2 2 . . . A s +1 s A s +1 s +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A n A n . . . . . . . . . . . . . . . . . . . . . . . . A n n , (1)where s is the number of 1’s on the diagonal. Hence, A i i ≥
2, for i ∈ ( s + 1) : n .Let, additionally, k = n − s be the number of diagonal elements that are notequal to 1, ∆ = ∆ ( A ), and δ = | det( A B ) | .The following properties are known for the HNF:1) ≤ A i j < A i i , for any i ∈ n and j ∈ i − ∆ ≥ δ = Q ni = s +1 A i i , and, hence, k ≤ log ∆ ,3) since A i i ≥
2, for i ∈ ( s + 1) : n , we have n X i = s +1 A i i ≤ δ k − + 2( k − ≤ δ. Remark 2
Using integer translations, we can assume that ≤ b B < diag( A B ),so the first s components of b B are equal to 0. Let H = A B [( s + 1) : n, s ] bethe matrix, which is located in the rows of A B right after the s × s identitymatrix. Without loss of generality, we can assume that columns of H arelexicographically sorted. Indeed, any permutations of the first s variables ofthe system Ax ≤ b can be compensated by a permutation of the first s rows. Lemma 1
The following inequality holds: k A N k max ≤ ∆δ ( δ k − + k − ≤ ∆. Hence, k A k max ≤ ∆ . Lemma 2
The adjugate matrix A ∗B has the form δ . . . . . . δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . ∗ ∗ . . . ∗ δ/A s +1 s +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ ∗ . . . . . . . . . . . . . . . . . . . . . δ/A n n . More precisely, ( A ∗B ) i i = δ/A i i , k A ∗B k max ≤ δ / , and the first s rows of A ∗B have the form ( δI s × s ) . D. V. Gribanov et al.
Proof
The structure of A ∗B directly follows from the triangular structure of A B and from the definition of A ∗B .Let the matrix H be obtained from A B by deleting any row and any column.The value of | det H | corresponds to some element of A ∗B . It is easy to see that H is a lower triangular matrix with at most one additional diagonal. We canexpand the determinant of H by the first row, using the Laplace rule. Then, | det H | ≤ k − d d . . . d k , where k is the number of diagonal elements in A B that are not equal to 1, and ( d , d , . . . , d k ) is a sequence diagonal elements.Since k ≤ log δ , we have | det H | ≤ δ / Definition 7
Let A ∈ Z m × n , b ∈ Z m , and rank A = n . Let us consider thesystem Ax ≤ b and a vector v ∈ Z n . The system Ax ≤ b is called v -normalized ,if the matrix A has the form (1), ≤ b B < diag A B , and A B v = b B .The system Ax ≤ b is called c -normalized , if it is v -normalized and c ⊤ v =max { c ⊤ x : Ax ≤ b } . Remark 3 (The computational complexity of the normalization and construc-tion of the HNF )
It can be easily seen that c -normalization of a system Ax ≤ b can be done by a polynomial-time algorithm. Indeed, two most com-plex parts of the normalization are searching of an optimal solution of theproblem max { c ⊤ x : Ax ≤ b } and computing the HNF for A .Due to [26], the computational complexity of the LP problem is polynomial.More efficient algorithms can be found in [23,25,29]. A near-optimal algorithmfor the HNF construction of the matrix A is given in [38].2.3 Some results in the Ehrhart theoryIn this Subsection, we follow [30]. For functions g, h : R > → R > , we write g ∼ h if lim t →∞ g ( t ) h ( t ) = 1 and g . h if lim sup t →∞ g ( t ) h ( t ) ≤ . For a n -dimensional set P ⊆ R n , we denote the n -dimensional Lebesgue mea-sure by vol n ( P ).The next lemma is given in [30], and it is a variation of classical known re-sults in the Ehrhart theory, see, for instance, [28, Theorem 7] and [22, Theorem1.2]. Lemma 3
Let P ⊆ R n be a m -dimensional rational polytope and Λ ⊆ Z n be a n -dimensional affine lattice. There exists a constant η P,Λ > , such that | t P ∩ Λ | . η P,Λ · t m . If m = n , then η P,Λ = vol m ( P ) / det Λ and | t P ∩ Λ | ∼ η P,Λ · t m . itle Suppressed Due to Excessive Length 9 Theorem 3
Let A ∈ Z n × m , B ∈ Z n × ( n − m ) , rank A = m , rank B = n − m ,and A ⊤ B = . Then, for any B ⊆ n , | B | = m , the following equality holds: ∆ gcd ( B ) | det A B ∗ | = ∆ gcd ( A ) | det B N ∗ | , where N = 1 : n \ B . Proof
Consider the n × n matrix C = ( A B ), then C ⊤ C = (cid:18) A ⊤ B ⊤ (cid:19) ( A B ) = (cid:18) A ⊤ A B ⊤ B (cid:19) . Hence, | det C | = p | det( A ⊤ A ) | | det( B ⊤ B ) | . Using the Laplace rule along first m columns of C , we havedet( C ) = X B⊆ n | B | = m ( − σ (1: m )+ σ ( B ) det( A B ∗ ) det( B N ∗ ) , where σ ( B ) is the sum of elements in B . Consider vectors a, b ∈ Z ( nm ) in-dexed by subsets B ⊆ n , | B | = m , such that a B = det A B ∗ and b B =( − σ (1: m )+ σ ( B ) det B N ∗ . Clearly, k a k = p | det( A ⊤ A ) | and k b k = p | det( B ⊤ B ) | .Consider the Euclidean space R ( nm ) with the standard scalar product ( · , · ).Since the equality ( a, b ) = k a k k b k holds, the vectors a, b are proportional: αa = βb for some co-prime α, β ∈ Z . Clearly, α gcd( a ) = β gcd( b ), so, afterthe multiplication of the equality on gcd( a ) β = gcd( b ) α , we achieve the goal of thetheorem. Remark 4
Result of the theorem was strengthened in [41]. Namely, it wasshown that the matrices
A, B have the same diagonal of their Smith NormalForms modulo of gcd-like multipliers.The HNF can be used to solve systems of the type Ax = b , see, for example,[34]. In the following lemma, we show that minors of the considered matricesare related. Lemma 4
Let A ∈ Z m × n , b ∈ Z m , and rank A = m . Let us consider theset M = { x ∈ Z n : Ax = b } of integer solutions of a linear equalities system.Then, there exists a matrix B ∈ Z n × ( n − m ) and a vector r ∈ Z n , such that M = Λ( B ) + r and ∆ ( B ) = ∆ ( A ) /∆ gcd ( A ) . The matrix B and the vector r can be computed by a polynomial-time algorithm.Proof The matrix A can be reduced to the HNF. Let A = ( H ) Q − , where H ∈ Z m × m , ( H ) be the HNF of A , and Q ∈ Z n × n be a unimodular matrix.The original system is equivalent to the system ( H ) y = b , where y = Q − x .Hence, y m = H − b and components of y ( m +1): n can take any integer values.Since x = Qy , we take B = Q ( m +1): n and r = Q m H − b .We have AB = . The matrix Q forms a basis of the lattice Z n , so ∆ gcd ( B ) = 1. Hence, by Theorem 3, we have ∆ ( B ) = ∆ ( A ) /∆ gcd ( A ).Due to Remark 3, the construction of B and r can be done by a polynomial-time algorithm. Corollary 1
Let A ∈ Z m × n , ˆ A ∈ Z n × ( n − m ) , ∆ gcd ( A ) = 1 , rank A = m , rank ˆ A = n − m , b ∈ Z m , ˆ b ∈ Z n , c ∈ Z n , ˆ c ∈ Z n − m .The following propositions hold:1. The ILP = ( A, b, c ) problem can be polynomially transformed to the equiva-lent ILP ≤ ( ˆ A, ˆ b, ˆ c ) problem, such that A ˆ A = and ∆ ( ˆ A ) = ∆ ( A ) .2. Feasible points x and ˆ x of the first and second problems are connected bythe formula x = ˆ b − ˆ A ˆ x .3. Let v be an optimal vertex of the relaxed LP problem LP = ( A, b, c ) , B bea corresponding optimal base, and N = 1 : n \ B . Then, there exists anoptimal vertex solution ˆ v of the relaxed LP problem LP ≤ ( ˆ A, ˆ b, ˆ c ) with acorresponding optimal base b B and the complement b N = 1 : n \ b B , such that v = ˆ b − ˆ A ˆ v, B = b N , N = b B , v B = ˆ b b N − ˆ A b N ˆ v, | det A B | = | det A b B | . Proof
By Lemma 4, there exist a vector r and a matrix B , such that M = { x : x = Bt + r, t ∈ Z n − m } . After the substitution x = Bt + r to the firstproblem formulation, we get an equivalent problem max { c ⊤ B ( t + r ) : − Bt ≤ r, t ∈ Z n − m } . We set ˆ A = − B , ˆ b = r , ˆ c ⊤ = c ⊤ B = − c ⊤ ˆ A .Due to the relation of the systems by the formula x = Bt + r , we have x = ˆ b − ˆ A ˆ x (we set ˆ x = t ). Let us proof the proposition 3. From the elementarytheory of LP, we have v B = A − B b and v N = . Let us consider solutions of thesystem ˆ b − ˆ A ˆ v = v with respect to the variables ˆ v . Since v N = , it followsthat ˆ A N ˆ v = ˆ b N , and, additionally, ˆ b B − ˆ A B ˆ v = v B ≥
0. The matrix ˆ A N ∈ Z ( n − m ) × ( n − m ) is square, and, due to Theorem 3, | det ˆ A N | = | det A B | > A N ˆ v = ˆ b N completely defines the vertex ˆ v . To finish theproof, we need to set b B = N . Remark 5
In the formulation of the previous lemma, we make an assumptionthat ∆ gcd ( A ) = 1. It helps to simplify formulas and can be done without lossof generality, because the original system Ax = b, x ≥ can be polynomiallytransformed to the equivalent system ˆ Ax = ˆ b, x ≥ with ∆ gcd ( ˆ A ) = 1.Definitely, let A = P ( S ) Q , where ( S ) ∈ Z m × m be the SNF of A ,and P ∈ Z m × m , Q ∈ Z n × n be unimodular matricies. For details on the SNF,see Section 2.1. Now, we multiply rows of the original system Ax = b, x ≥ by the matrix ( P S ) − . After this step, the original system transforms to theequivalent system ( I n × n ) Qx = b, x ≥ . Clearly, the matrix ( I n × n ) is theSNF of the matrix ( I n × n ) Q , so its ∆ gcd is equal 1. Everywhere in this Section we assume that A ∈ Z m × n , c ∈ Z n , b ∈ Z m ,rank A = n , and ∆ = ∆ ( A ). Additionally, let v be some optimal vertex of theLP ≤ ( A, b, c ) problem and B be a corresponding optimal base, e.g. v = A − B b B .Denote N = 1 : m \ B and δ = | det A B | . itle Suppressed Due to Excessive Length 11 Definition 8
The ILP ≤ ( A, b, c ) problem is local if there exists an optimalinteger solution z , for which the following inequality for the slack variables y = b B − A B z holds: ( y + 1)( y + 1) . . . ( y n + 1) ≤ δ. (2) Remark 6
It is easy to see that the ILP ≤ ( A, b, c ) problem, defined by a squareinteger full-rank matrix A , is local. Indeed, let P − SQ − = A , where S isthe Smith Normal Form of A . Then, taking y = b − Ax ≥ and applyingthe SNF, the original problem transforms to the problem min { w ⊤ y : y ∈ M } ,where M = { y ∈ Z n + : P y ≡ P b (mod S ) } and w ⊤ = c ⊤ A − ≥ . All thecolumns of P define a group by the addition modulo S , the group order is atmost δ .The inequality (2) is a classical inequality [14,24], investigated by R. Go-mory for vertices of the polyhedron conv . hull( M ). Due to [14,24], the problemmin { w ⊤ y : y ∈ M } can be solved by a dynamic programming algorithm withthe computational complexity O ( n δ ) of group operations. Since columns of P modulo S have at most log δ nonzero coordinates, the arithmetic complexityof the algorithm is O ( n · δ · log δ ). Remark 7
We note that the locality property of the ILP ≤ ( A, b, c ) problemis invariant under the v -normalization procedure (see Subsection 2.2). Or, inanother words, the ILP ≤ ( A, b, c ) problem is local if and only if the ILP ≤ ( ˆ A, ˆ b, ˆ c )problem is local, where the ILP ≤ ( ˆ A, ˆ b, ˆ c ) problem is a v -normalized variant ofthe ILP ≤ ( A, b, c ) problem.
Lemma 5
Let integer vectors y ∈ Z n + and z ∈ Z n satisfy the inequality (2) and the condition A B z + y = b B . Then, k A N ( z − v ) k ∞ ≤ ∆ − . If, additionally,the system Ax ≤ b is v -normalized, then k A N z k ∞ ≤ ∆ − .Proof | ( A N ) i ∗ ( z − v ) | = | ( A N ) i ∗ A − B y | ≤ ∆δ ⊤ y < ∆. If the system Ax ≤ b is v -normalized, then | ( A N ) i ∗ v | = | ( A N ) i ∗ A − B b B | ≤ ∆δ ⊤ b B < ∆, | ( A N ) i ∗ z | = | ( A N ) i ∗ ( z − v ) + ( A N ) i ∗ v | ≤ ∆ − . Corollary 2 If b N − A N v ≥ ( ∆ − , then the ILP(
A, b, c ) problem is local,for any c ∈ cone( A ⊤B ) . Moreover, the ILP ≤ ( A, b, c ) problem is equivalent to the ILP ≤ ( A B , b B , c ) problem. Consequently, it can be solved by an algorithm withthe arithmetic complexity O ( n · δ · log δ ) .Proof We are going to show that any vertex z of conv . hull (cid:16) P ≤ ( A B , b B ) ∩ Z n (cid:17) satisfies the system A N x ≤ b N . It implies that we can solve the ILP ≤ ( A B , b B , c )problem instead of the ILP ≤ ( A, b, c ) problem and that the ILP ≤ ( A, b, c ) prob-lem is local.
If a vector y ∈ Z n + is given by the equality A B z + y = b B , then, by Remark6, y satisfies the inequality (2), and, due to Lemma 5, we have | A N ( v − z ) | ≤ ( ∆ − . Due to the corollary assumptions, we have b N − A N z = A N ( v − z ) + b N − A N v ≥ . Theorem 4
Let z be an optimal solution of the local ILP ≤ ( A, b, c ) problem.Then, the following statements hold:1. k A ( v − z ) k ≤ log δ + ( m − n ) , k A ( v − z ) k ≤ ( δ −
1) + ( m − n )( ∆ − , k A ( v − z ) k ∞ ≤ ∆ − ,2. k b − Az k ≤ log δ + ( m − n ) , k b − Az k ≤ ( δ −
1) + ( m − n )( ∆ −
1) + k b N − A N v k , k b − Az k ∞ ≤ ∆ − k b N − A N v k ∞ ,3. the point z lies on a face of P , whose dimension is bounded by log δ ,If, additionally, the system Ax ≤ b is v -normalized, then1. k z k ≤ δ , k z k < δ log δ , k z k ∞ < δ ,2. k v − z k ≤ δ , k v − z k < δ log δ , k v − z k ∞ < δ / .Proof Let y = b B − A B z = A B ( v − z ). By the inequality (2), we have k y k ≤ log δ and k y k ≤ δ − A ( v − z ) = (cid:18) yA N ( v − z ) (cid:19) , the inequalities of Statement 1 follow fromLemma 5.The inequalities from Statement 2 follow from the equalities b − Az = A ( v − z ) + b − Av and b B − A B v = .Statement 3 trivially follows from the definition of the locality property.Now, assume that the system Ax ≤ b is v -normalized. Due to Lemma2, columns of the matrix A − B have at most log δ + 1 nonzero components.More precisely, columns of A − B have at most log δ last nonzero components.Consequently, k v − z k = k A − B y k ≤ δ, k A − B b B k ≤ log δ, and k z k = k A − B ( b B − y ) k ≤ δ .Since k b B − y k ≤ δ − k z k ∞ ≤ | δ A ∗B ( b B − y ) | ∞ ≤ ( δ − δ < δ and k v − z k ∞ ≤ | δ A ∗B y | ∞ ≤ ( δ − δ < δ / . Now, the inequalities k z k < δ log δ and k v − z k < δ log δ are trivial. itle Suppressed Due to Excessive Length 13 c ∈ Z n , a matrix A ∈ Z m × n of the rank n and considerthe ILP ≤ ( A, b, c ) problem, parameterized by b ∈ Z m . Since the matrix A isfixed, we can assume that A has been reduced to the HNF and it has the form(1). The goal of this Section is to estimate the probability of the situation,when the ILP ≤ ( A, b, c ) problem is local, and prove that it can be solved byan efficient algorithm for fixed ∆ = ∆ ( A ), i.e. to prove Theorem 1. Our proofdirectly follows proof techniques of the work [32].Let Λ be an arbitrary m -dimensional sublattice of Z m . As in the work [32],we define the set G ⊆ Λ in the following way: b ∈ G if the LP ≤ ( A, b, c ) problemis feasible, and, for any optimal base B of the LP ≤ ( A, b, c ) problem, we have b N − A N v ≥ ( ∆ − , where v = A − B b B is the optimal vertex, correspondingto B . Additionally, we set F = { b ∈ Λ : P ≤ ( A, b ) = ∅} . In another words, G = { b ∈ F : b N − A N v ≥ ( δ − } . Due to Corollary 2, if b ∈ G , then the ILP ≤ ( A, b, c ) problem is local and itcan be solved by an algorithm with the arithmetic complexity O ( n · ∆ · log ∆ ).For each basis B ⊆ m , we set ∆ B = | det A B | . Let ¯ G = Λ \ G , it followsfrom the definition of G that¯ G ∩ F ⊆ [ B⊆ m B – basis [ j ∈N ∆ B ( ∆ − [ r =0 { b ∈ F : ∆ B b j = A j A ∗B b B + r } . (3) Lemma 6
The following equality holds: Pr Λ ( G | F ) = 1 . Proof
We are going to prove thatPr Λ (¯ G | F ) = Pr Λ (¯ G ∩ F )Pr Λ ( F ) = lim inf t →∞ | Ω Λ,t ∩ ¯ G ∩ F || Ω Λ,t ∩ F | = 0 . (4)Here, we assume thatPr Λ ( F ) = lim inf t →∞ | Ω Λ,t ∩ F || Ω Λ,t | > . (5)Correctness of this assumption will be shown later.The formula (3) implies that the right-hand part of (4) is at most X B⊆ m B – basis X j ∈N ∆ B ( ∆ − X r =0 |{ b ∈ Ω Λ,t ∩ F : ∆ B b j = A j A ∗B b B + r }|| Ω Λ,t ∩ F | . (6) Let us fix a base B , an index j ∈ N , a value r and consider the fraction inthe right-hand side of (6). Clearly, | Ω Λ,t ∩ F | ≥ | Λ ∩ P t | , where P t is a rationalpolytope, defined by the formula P t = { b ∈ R m : k b k ∞ ≤ t, A N A − B b B ≤ b N } , meaning that the base B is feasible. Clearly, P t = t P and | Λ ∩ P t | 6 = ∅ ,for any sufficiently large t . Without loss of generality, we can assume thatdim P t = dim Λ = m , since, in the opposite case, we can consider the lattice Λ ′ = Λ ∩ L , induced by the intersection of Λ with the affine (linear) hull of P t .Additionally, let P ′ = P ∩ { b ∈ R m : ∆ B b j = A j A ∗B b B + r/t } . Due to Lemma 3, the fraction in the right-hand part of (6) is at most | Λ ∩ P t ∩ { b ∈ R m : ∆ B b j = A j A ∗B b B + r }|| Λ ∩ P t | = | Λ ∩ t P ′ || Λ ∩ t P | . C t , where C = η P ′ ,Λ /η P ,Λ .Additionally, we have | Ω Λ,t ∩ F || Ω Λ,t | = | Λ ∩ t P || Λ ∩ t B ∞ | & η P ′ ,Λ η B ∞ ,Λ > , where B ∞ denotes the unit ball with respect to the l ∞ -norm. The last factproves the correctness of the assumption (5).Finally, we havelim t →∞ | Ω Λ,t ∩ ¯ G ∩ F || Ω Λ,t ∩ F | . (cid:18) mn (cid:19) · ( m − n ) · ∆ · C · t = O (1 /t ) , and consequently Pr Λ (¯ G | F ) = 0 . Theorem 1 holds by taking G as it is stated in the current Subsection. Theremaining required properties follow from Lemma 4.4.2 Proof of Theorem 2Now, let us consider the ILP = ( A, b, c ) problem with a little change that rank A = m . Due to Remark 5, we can assume that ∆ gcd ( A ) = 1.Let A = ( H ) Q , where ( H ) be the HNF of A and Q ∈ Z n × n be a uni-modular matrix. First of all, using Corollary 1, we transform the ILP = ( A, b, c )problem to the ILP ≤ ( ˆ A, ˆ b, ˆ c ) problem. Here,ˆ A = − Q ( m +1): n ∈ Z n × ( n − m ) , ˆ c ⊤ = − c ⊤ ˆ A ∈ Z n − m are fixed, and ˆ b = Q m H − b ∈ Z n varies together with b . itle Suppressed Due to Excessive Length 15 Let Λ = Λ( Q m ) be the lattice, induced by columns of Q m , the set F = = { b ∈ Z m : P = ( A, b ) = ∅} denote the feasibility of the LP = ( A, b, c ) problem, andthe set F ≤ = { ˆ b ∈ Λ : P ≤ ( ˆ A, ˆ b ) = ∅} denote the feasibility of the LP ≤ ( ˆ A, ˆ b, ˆ c )problem.Let ˆ v and v = ˆ b − ˆ A ˆ v be the corresponding optimal vertex solutions of therelaxed LP problems LP ≤ ( ˆ A, ˆ b, ˆ c ) and LP = ( A, b, c ), respectively. Let b B and B be the corresponding optimal bases, and N = 1 : n \ B , b N = 1 : n \ b B . From theelementary theory of LP, we have v B = A B b , v N = , ˆ v = ˆ A − b B ˆ b b B .Due to Corollary 1, we have B = b N , N = b B , v B = ˆ b b N − ˆ A b N ˆ v, δ = | det A B | = | det ˆ A b B | . Now, we define the set
A ⊆ Z m in the following way: b ∈ A , if Q m H − b =ˆ b ∈ G , where the set G = { ˆ b ∈ F ≤ : ˆ b b N − ˆ A b N ˆ v ≥ ( δ − } also has been definedin Section 3. From the definition of G , we have A = { b ∈ F = : v B ≥ ( δ − } .Due to Corollary 2, if b ∈ A , then the ILP ≤ ( ˆ A, ˆ b, ˆ c ) problem is local and itcan be solved by an algorithm with the arithmetic computational complexity O (( n − m ) · ∆ · log ∆ ). Due to Corollary 1, optimal solutions of the ILP = ( A, b, c )and the ILP ≤ ( ˆ A, ˆ b, ˆ c ) problems are connected by the formula z = ˆ b − ˆ A ˆ z . Hence,if b ∈ A , then the ILP = ( A, b, c ) problem can be solved by an algorithm withthe same computational complexity. The remaining properties of Theorem 2follow from Lemma 4 and the formula v − z = ˆ A (ˆ v − ˆ z ).To finish the proof, we need to show that Pr( A | F = ) = 1. Let M = { b ∈ Z m : H − b ∈ Z m } , then the map H − x : M → Z n is a bijective map. We alsohave that b ∈ F = if and only if, then ˆ b ∈ F ≤ . Hence,Pr( A | F = ) = Pr Λ ( G | F ≤ ) = 1 . The authors would like to thank A. Chirkov, S. Veselov, N. Zolotykh andJ. Paat for useful discussions during preparation of this article.
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