Topics in basis reduction and integer programming Public Deposited

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  • March 22, 2019
  • Tural, Mustafa Kemal
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
  • A basis reduction algorithm computes a reduced basis of a lattice consisting of short and nearly orthogonal vectors. The best known basis reduction method is due to Lenstra, Lenstra and Lovász (LLL): their algorithm has been extensively used in cryptography, experimental mathematics and integer programming. Lenstra used the LLL basis reduction algorithm to show that the integer programming problem can be solved in polynomial time when the number of variables is fixed. In this thesis, we study some topics in basis reduction and integer programming. We make the following contributions. We unify the fundamental inequalities in an LLL reduced basis, which express the shortness and near orthogonality of the basis. We analyze two recent integer programming reformulation techniques which also rely on basis reduction. The reformulation methods are easy to describe. They are also successful in practice in solving several classes of hard integer programs. First, we analyze the reformulation techniques on bounded knapsack problems. The only analyses so far are for knapsack problems with a constraint vector having a certain decomposable structure. Here we do not assume any a priori structure on the constraint vector. We then analyze the reformulation techniques on bounded integer programs. We show that if the coefficients of the constraint matrix are drawn from a sufficiently large interval, then branch and bound creates at most one node at each level if applied to the reformulated instances. On the practical side, we give some numerical values as to how large the numbers should be to make sure that for 90 and 99 percent of the reformulated instances, the number of subproblems that need to be enumerated by branch and bound is at most one at each level. These values turned out to be surprisingly small when the problem size is moderate. We also analyze the solvability of the ``majority of the low density subset sum problems using the method of branch and bound when the coefficients are chosen from a large interval.
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  • Lu, Shu
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  • University of North Carolina at Chapel Hill
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