Basis Reduction and the Complexity of Branch-and-Bound

Abstract

The classical branch-and-bound algorithm for the integer feasibility problem (0.1) Findx ∈ Q ∩ Z n , withhas exponential worst case complexity. We prove that it is surprisingly efficient on reformulations of (0.1), in which the columns of the constraint matrix are short and near orthogonal, i.e., a reduced basis of the generated lattice: when the entries of A are from {1, . . . , M } for a large enough M , branch-and-bound solves almost all reformulated instances at the root node. For all A matrices we prove an upper bound on the width of the reformulations along the last unit vector.Our results generalize the results of Furst and Kannan on the solvability of subset sum problems; also, we prove them via branch-and-bound, an algorithm traditionally considered inefficient from the theoretical point of view.We use two main tools: first, we find a bound on the size of the branch-and-bound tree based on the norms of the Gram-Schmidt vectors of the constraint matrix. Second, building on the ideas of Furst and Kannan, we bound the number of integral matrices for which the shortest nonzero vectors of certain lattices are long.We explore practical aspects of these results. We compute numerical values of M which guarantee that 90 and 99 percent of the reformulated problems solve at the root: these turn out to be surprisingly small when the problem size is moderate. We also confirm with a computational study that random integer programs become easier, as the coefficients grow.

Basis Reduction Methods

“…Here we describe an analysis of the reformulation methods based on the paper of Pataki et al [10], without assuming any structure on the matrix A in (29). Interestingly, we will find that ordinary B&B solves the reformulation of the majority of the instances without any branching.…”

“…Proof Sketch We outline a proof of the first statement, and refer the reader to Pataki et al [10] for details, and the proof of the second. For convenience, we shall write (A; I) for the matrix obtained by stacking A on top of I, and the meaning of ( 1 ; 2 ) and (w 1 ; w 2 ) will be analogous.…”

“…We refer the reader to Lemma 2.2 in Pataki et al [10] for details. Using this result, it follows that when M is as given in the first statement of the theorem, the fraction of bad matrices is at most 1/2 n , and this completes the proof.…”

“…The first is an optimization version introduced in Cornuéjols and Dawande [41], which attempts to minimize Ax − b 1 . The second is a relaxed version studied in Pataki et al [10], namely,…”

“…A recent result of Pataki et al [10] shows that when branch-and-bound is applied to the reformulation of bounded integer programs, the majority of instances get solved without generating any subproblems, that is, at the root node. This result may seem counterintuitive; however, it fits in nicely with the previous work on ''low density'' subset sum problems, which have been widely used in cryptography.…”