Topics in basis reduction and integer programming

MUSTAFA KEMAL TURAL: Topics in Basis Reduction and Integer P ogramming (Under the direction of Gábor Pataki) A basis reduction algorithm computes a reduced basis of a lat tice consisting of short and nearly orthogonal vectors. The best known basis reduction method i s due to Lenstra, Lenstra and Lovász (LLL): their algorithm has been extensively used in cryptog raphy, experimental mathematics and integer programming. Lenstra used the LLL basis reduction algorith m o show that the integer programming problem can be solved in polynomial time when the number of va riables is fixed. In this thesis, we study some topics in basis reduction and in teger programming. We make the following contributions. We unify the fundamental inequalities in an LLL reduced basi s, which express the shortness and near orthogonality of the basis. We analyze two recent integer programming reformulation te chniques which also rely on basis reduction. The reformulation methods are easy to describe. Th y are also successful in practice in solving several classes of hard integer programs. First, we analyze the reformulation techniques on bounded k napsack problems. The only analyses so far are for knapsack problems with a constraint vector hav ing a certain decomposable structure. Here we do not assume any a priori structure on the constraint vect or. We then analyze the reformulation techniques on bounded int ger programs. We show that if the coefficients of the constraint matrix are drawn from a suffici ently large interval, then branch and bound creates at most one node at each level if applied to the reform ulated instances. On the practical side, we give some numerical values as to how large the numbers should be to make sure that for90 and99 percent of the reformulated instances, the number of subpro blems that need to be enumerated by branch and bound is at most one at each level. Th ese values turned out to be surprisingly small when the problem size is moderate. We also analyze the solvability of the “majority” of the low d ensity subset sum problems using the method of branch and bound when the coefficients are chosen fr om a large interval. iii Acknowledgements I would never have been able to finish my dissertation without the guidance of my committee members, help from friends and support from my family and wife. First of all, I would like to express my deepest gratitude to m y advisor Dr. Gábor Pataki for his invaluable guidance, advice and inspiration during my grad uate study. I would like to thank Dr. David Rubin for carefully going through my dissertation and sugge sting corrections. I would also like to thank my other dissertation committee members Dr. Shu Lu, Dr. Scot t Pr van and Dr. Jon Tolle. I want to thank Dr. Serhan Ziya for his support throughout my g raduate study. I also want to thank Dr. Edcard Carlstein who has been a remarkable teaching advi sor. Dr. Edcard Carlstein was always there to listen and to give advice. I am deeply indebted to my parents, Makbule Tural and Veysi Tu ral. Their love and support provided me the energy to attain my study. Finally, I would like to thank my wife Dilber for her continua l support, patience and understanding. The completion of this dissertation would not have been possible without her endless love and unconditional support.