Incremental Algorithms for Lattice Problems

In this short note we give incremental algorithms for the following lattice problems: finding a basis of a lattice, computing the successive minima, and determining the orthogonal decomposition. We prove an upper bound for the number of update steps for every insertion order. For the determination of the orthogonal decomposition we efficiently implement an argument due to Kneser.Comment: 5 page

On the Lattice Isomorphism Problem

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem running in time n^{O(n)} times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.Comment: 23 pages, SODA 201

Quantum Algorithm for Computing the Period Lattice of an Infrastructure

We present a quantum algorithm for computing the period lattice of infrastructures of fixed dimension. The algorithm applies to infrastructures that satisfy certain conditions. The latter are always fulfilled for infrastructures obtained from global fields, i.e., algebraic number fields and function fields with finite constant fields. The first of our main contributions is an exponentially better method for sampling approximations of vectors of the dual lattice of the period lattice than the methods outlined in the works of Hallgren and Schmidt and Vollmer. This new method improves the success probability by a factor of at least 2^{n^2-1} where n is the dimension. The second main contribution is a rigorous and complete proof that the running time of the algorithm is polynomial in the logarithm of the determinant of the period lattice and exponential in n. The third contribution is the determination of an explicit lower bound on the success probability of our algorithm which greatly improves on the bounds given in the above works. The exponential scaling seems inevitable because the best currently known methods for carrying out fundamental arithmetic operations in infrastructures obtained from algebraic number fields take exponential time. In contrast, the problem of computing the period lattice of infrastructures arising from function fields can be solved without the exponential dependence on the dimension n since this problem reduces efficiently to the abelian hidden subgroup problem. This is also true for other important computational problems in algebraic geometry. The running time of the best classical algorithms for infrastructures arising from global fields increases subexponentially with the determinant of the period lattice.Comment: 52 pages, 4 figure

A primal Barvinok algorithm based on irrational decompositions

We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with low index. We give computational results that show that the new algorithms are faster than the existing algorithms by a large factor.Comment: v3: New all-primal algorithm. v4: Extended introduction, updated computational results. To appear in SIAM Journal on Discrete Mathematic

A generalization of the integer linear infeasibility problem

Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is closely related to non-squarefree elements of Markov bases for sampling contingency tables with given marginals. To study a family of systems with no integer solution, we focus on a commutative semigroup generated by a finite subset of $\Z^d$ and its saturation. An element in the difference of the semigroup and its saturation is called a ``hole''. We show the necessary and sufficient conditions for the finiteness of the set of holes. Also we define fundamental holes and saturation points of a commutative semigroup. Then, we show the simultaneous finiteness of the set of holes, the set of non-saturation points, and the set of generators for saturation points. We apply our results to some three- and four-way contingency tables. Then we will discuss the time complexities of our algorithms.Comment: This paper has been published in Discrete Optimization, Volume 5, Issue 1 (2008) p36-5