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.
We consider a continuous multi-facility location problem where the demanding entities are regions in the plane instead of points. Each region may consist of a finite or an infinite number of points. The service point of a station can be anywhere in the region that is assigned to it. We do not allow fractional assignments, that is, each region is assigned to exactly one facility. The problem we consider can be stated as follows: given m demand regions in the plane, find the locations of q facilities and allocate regions to the facilities so as to minimize the sum of squares of the maximum Euclidean distances of the demand regions to the facility locations they are assigned to. We assume that the regions are closed polygons as any region can be approximated within any desired accuracy with a polygon.We first propose mathematical programming formulations of single and multiple facility location problems. The single facility location problem is formulated as a second order cone program (SOCP) which can be solved in polynomial time. The multiple facility location problem is formulated as a mixed integer SOCP. This formulation is weak and does not even solve medium-size problems. We therefore propose heuristics to solve larger instances of the problem. We develop three heuristics that work when the regions are polygons. When the demand regions are rectangles with sides parallel to coordinate axes, a special heuristic is developed. We compare our heuristics in terms of both solution quality and computational time.
, 90 pages The notions of betweenness centrality (BC) and its extension group betweenness centrality (GBC) are widely used in social network analyses. We introduce variants of them; namely, the k-step BC and k-step GBC. The k-step GBC of a group of vertices in a network is a measure of the likelihood that at least one group member will get the information communicated between a randomly chosen pair of vertices through a randomly chosen shortest path within the first k steps of the start of the communication. The k-step GBC of a single vertex is the k-step BC of that vertex. The introduced centrality measures may find uses in applications where it is important or critical to obtain the information within a fixed time of the start of the communication. For the introduced centrality measures, we propose an algorithm that can compute successively the k-step GBC of several groups of vertices. Moreover, we propose a mixed integer programming formulation to compute the group that has the highest k-step GBC value and a heuristic approach to compute a group of vertices whose k-step GBC value is high. The performances of the proposed algorithms are evaluated through computational experiments on real and randomly generated networks.
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