Fast integer programming in fixed dimension

Question

1. What are the contributions in "Fast integer programming in fixed dimension" ?

2. What is the simplest way to reduce a lattice basis?

3. What is the simplest way to perform the LLL-algorithm?

4. What is the greatest common divisor of two integers?

Accepted Answer

In this paper, it was shown that the optimum of an integer program in fixed dimension can be computed with O ( s ) basic arithmetic operations, where s is the binary encoding length of the input.

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Reduction algorithms apply unimodular transformations of a lattice basis from the right, to obtain a basis whose vectors are more and more orthogonal.

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Let A ∈ Zn×n be a lattice basis and let A0 be the number A0 = max{‖a j‖ | j = 1, . . . ,n}. The LLL-algorithm performs O(n4 logA0) arithmetic operations on rational numbers, whose binary encoding length is O(n log A0).

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The greatest common divisor of two integers can be formulated as an integer program in fixed dimension with a fixed number of constraints, see, e.g., [11].

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The parametric shortest vector problem for A, U and k can be solved with O(n5(logU + logA0)) basic arithmetic operations with rational numbers of binary encoding length O(n(log A0 + logU)).

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An integer program which is defined by m constraints can be solved with O(m) basic operations and O(logm) calls to an algorithm which solves an integer program defined by a fixed size subset of the constraints, see also [7].

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The linear complexity of the parametric shortest vector problem carries over to the integer programming problem with a fixed number of constraints, if the authors can ensure the following conditions for each reduction step.(C-1) A problem of size s is reduced to a constant number of problems of size O(s).

Accepted Answer

(i) |µi, j|6 1/2, for every 1 6 i < j 6 n; (ii) ‖b∗j+1 + µ j+1, j b∗j‖2 > 3/4‖b∗j‖2, for j = 1, . . . ,n−1.The LLL-algorithm iteratively normalizes the basis, which means that the basis is unimodularly transformed into a basis which meets condition (i), and swaps two columns if these violate condition (ii).

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A two-layer simplex is a simplex, whose vertices can be partitioned into two sets V and W , such that the first components of the elements in V and W agree, i.e., for all v1,v2 ∈ V one has v1(1) = v2(1) and for all w1,w2 ∈W one has w1(1) = w2(1).

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This shows that the complexity of PSV in fixed dimension n is linear in the input size and operates on rationals whose size is also linear in the input.

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This means that the authors have reduced the integer programming problem over a two-layer simplex in dimension n to a constant number of integer programming problems in dimension n− 1 with a fixed number of constraints.

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The optimal solution of IP over P is the largest optimal solution of IP over polytopesPi = P∩ (x(1) 6 αi)∩ (x(1) > αi+1), i = 1, . . . , ℓ−1. (2)Carathéodory’s theorem, see [15, p. 94], implies that each Pi is covered by the twolayer simplices, which are spanned by the vertices of Pi.

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There is a subscript j which violates condition (ii) (8) Swap columns b j and b j+1 of B (9) Update GSO b∗j , µ ji (10) until B is LLL-reduced (11) return BThe key to the termination argument of the LLL-algorithm is the following potential function φ(B) of a lattice basis B ∈ Zn×n:φ(B) = ‖b∗1‖2n‖b∗2‖2(n−1) · · · ‖b∗1‖2. (15)The potential of an integral lattice basis is always an integer.

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In this way, the authors assure that the optimum to the integer programming problem lies in the truncation Pπ which is still flat along some integer vector c, thereby reducing the integer programming problem over an n-dimensional polyhedron to a constant number of integer programming problems over the (n−1)-dimensional polyhedraPπ ∩ (cT x = δ ), where δ ∈ Z∩ [min{cT x | x ∈ Pπ)},max{cT x | x ∈ Pπ}].

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If the algorithm for PLW asserts that w(Σ) 6 γ K, then Σ itself is already flat along an integral direction c. Similarly in this case, the optimization problem can be reduced to a constant number of optimization problems in lower dimension.

Accepted Answer

1 Clarkson claims a complexity of O(m + log(m)s) because he mistakenly relied on algorithms from the literature [13,9,5] for the integer programming problem with a fixed number of constraints, which actually only solve the integer feasibility problem.

Fast integer programming in fixed dimension

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Most frequently asked questions

1. What are the contributions in "Fast integer programming in fixed dimension" ?

2. What is the simplest way to reduce a lattice basis?

3. What is the simplest way to perform the LLL-algorithm?

4. What is the greatest common divisor of two integers?

5. What is the shortest vector problem for A, U and k?

6. What is the simplest way to solve the integer programming problem?

7. What is the linear complexity of the parametric shortest vector problem?

8. What is the simplest way to solve a lattice basis?

9. What is the definition of a two-layer simplex?

10. What is the shortest vector problem in fixed dimension n?

11. What is the optimum of the integer programming problem?

12. What is the optimal solution of IP over polytopesPi?

13. What is the value of the potential of an integral lattice basis?

14. What is the optimum to the integer programming problem?

15. What is the optimum of the optimization problem for a two-layer simplex?

16. What is the reason why Clarkson used a random sampling algorithm?

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References

Computers and Intractability: A Guide to the Theory of NP-Completeness

Johnson: Computers and Intractability-A Guide to the Theory of NP-Completeness

Lecture Notes in Computer Science 2382

Johnson: computers and intractability: a guide to the theory of np- completeness (freeman

The Art in Computer Programming

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