Cuts, Zonotopes and Arrangements

where Q is an n × n rational symmetric matrix, is a classical NP-hard combinatorial optimization problem. It is well known that the weighted max-cut problem can be considered as a special case. In fact, there are simple polynomial reductions between the weighted max-cut problem and the 01QP. This intereting result, due to [13], will be reviewed in subsection 2. The 01QP problem remains NP-hard even when Q is positive definite or when it is indefinite of rank two [12]. If a linear term is added to the objective function, the problem remains NP-hard even when Q is negative definite. Well known polynomial cases are, for instance, when (a) the matrix Q is of rank one, in which case the solution can be found by inspection, (b) Q has nonnegative off-diagonal elements [16], and (c) the graph underlying the associated max-cut problem is series parallel [3]. Recently, a new polynomially solvable case [1] has been found. This case, that we call the fixedrank convex (FRC) case, is when Q is positive semidefinite and of fixed rank d. This result, which will be reviewed in subsection 3, reduces the search space of 2 0/1 feasible solutions to that of O(nd−1) restricted 0/1 solutions by a geometric transformation. More precisely these restricted solutions are the extreme points of a d-dimensional affine image of the unit n-dimensional hypercube. This object is known as a zonotope. One main purpose of the present article is to investigate the practical impact of this reduction and applicable algorithms. In particular, we propose a more efficient modification of the reverse search algorithm [2] for generating all extreme points of a given zonotope in Rd , and we study the performance of a straightforward implementation using the C language. Some of the largest instances we could solve on standard Unix workstations are randomly generated rank 3 cases with n = 250 and rank 4 cases with n = 70. It should be observed that randomly generated cases are in some sense hardest for the enumeration of extreme points, since they are known to attain (in probability one) the maximum size of output. While we are not aware of any specific applications of low rank FRC 01QP problems, we strongly hope that the exact solutions our algorithm can compute will be useful. For example, one can test