A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we determi...

Branch-and-cut methods are exact algorithms for integer programming problems. They consist of a combination of a cutting plane method with a branch-and-bound algorithm. These methods work by solving a sequence of linear programming relaxations of the integer programming problem. Cutting plane methods improve the relaxation of the problem to more closely approximate the integer programming probl...

In this paper we give a fast algorithm to generate all partitions of a positive integer n. Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. It is known the fact that the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. Using tree structures for storin...

In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also giv...

We present new exact and heuristic algorithms for the prizecollecting Steiner tree problem. The exact algorithm rst reduces the size of the input graph while preserving equivalence of the optimal solutions and then uses mixed integer linear programming to solve the resulting instance. For our heuristic, we reduce the size of the instance graph further, but without guaranteeing the equivalence o...

Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a deep result, conjectured forty years ago by Hoffman, and proved seventeen years later by Estes. This short paper provides an independent and eleme...

In this paper, we present two Integer Programming formulations for the k-Cardinality Tree Problem. The first is a multiflow formulation while the second uses a lifting of the Miller-Tucker-Zemlin constraints. Based on our computational experience, we suggest a two-phase exact solution approach that combines two different solution techniques, each one exploring one of the proposed formulations.

We describe a multistage, stochastic, mixed-integer programming model for planning capacity expansion of production facilities. A scenario tree represents uncertainty in the model; a general mixed-integer program defines the operational submodel at each scenario-tree node, and capacity-expansion decisions link the stages. We apply “variable splitting” to two model variants, and solve those vari...

For a given set X ⊆ Z of integer points, we investigate the smallest number of facets of any polyhedron whose set of integer points is conv(X) ∩ Z. This quantity, which we call the relaxation complexity of X, corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that gets along without auxiliary variables. We show that the use...

100 Integer Multiplication with Over ow Detection or Saturation Michael J. Schulte, Pablo I. Balzola, Ahmet Akkas, and Robert W. Brocato Abstract|High-speed multiplication is frequently used in general-purpose and application-speci c computer systems. These systems often support integer multiplication, where two n-bit integers are multiplied to produce a 2n-bit product. To prevent growth in wor...