Tadpole--an off-line router for the NuMesh system

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1997.Includes bibliographical references (p. 59-60).by Patrick Joseph LoPresti.M.S

Adding Multiple Cost Constraints to Combinatorial Optimization Problems, with Applications to Multicommodity Flows

Minimum cost multicommodity flow is an instance of a simpler problem (multicommodity flow) to which a cost constraint has been added. In this paper we present a general scheme for solving a large class of such "cost-added" problems---even if more than one cost is added. One of the main applications of this method is a new deterministic algorithm for approximately solving the minimumcost multicommodity flow problem. Our algorithm finds a (1 + ffl) approximation to the minimum cost flow in ~ O(ffl \Gamma3 kmn) time, where k is the number of commodities, m is the number of edges, and n is the number vertices in the input problem. This improves the previous best deterministic bounds of O(ffl \Gamma4 kmn 2 ) [9] and ~ O(ffl \Gamma2 k 2 m 2 ) [15] by factors of n=ffl and fflkm=n respectively. In fact, it even dominates the best randomized bound of ~ O(ffl \Gamma2 km 2 ) [15]. The algorithm presented in this paper efficiently solves several other interesting generali..

Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system

A conic linear system is a system of the form¶¶(FP d )Ax = b ¶ x ∈ C X ,¶¶where A:X ? Y is a linear operator between n - and m -dimensional linear spaces X and Y , b ∈ Y , and C X ⊂X is a closed convex cone. The data for the system is d =( A,b ). This system is “well-posed” to the extent that (small) changes in the data d =( A,b ) do not alter the status of the system (the system remains feasible or not). Renegar defined the “distance to ill-posedness,”ρ( d ), to be the smallest change in the data Δ d =(Δ A ,Δ b ) needed to create a data instance d +Δ d that is “ill-posed,” i.e., that lies in the intersection of the closures of the sets of feasible and infeasible instances d ′ =( A ′ , b ′ ) of (FP (·) ). Renegar also defined the condition number ?( d ) of the data instance d as the scale-invariant reciprocal of ρ( d ) : ?( d )= .¶In this paper we develop an elementary algorithm that computes a solution of (FP d ) when it is feasible, or demonstrates that (FP d ) has no solution by computing a solution of the alternative system. The algorithm is based on a generalization of von Neumann’s algorithm for solving linear inequalities. The number of iterations of the algorithm is essentially bounded by¶¶ O (  ?( d ) 2 ln(?( d )))¶¶where the constant depends only on the properties of the cone C X and is independent of data d . Each iteration of the algorithm performs a small number of matrix-vector and vector-vector multiplications (that take full advantage of the sparsity of the original data) plus a small number of other operations involving the cone C X . The algorithm is “elementary” in the sense that it performs only a few relatively simple computations at each iteration.¶The solution of the system (FP d ) generated by the algorithm has the property of being “reliable” in the sense that the distance from to the boundary of the cone C X , dist( ,∂ C X ), and the size of the solution, ∥ ∥, satisfy the following inequalities:¶¶∥ ∥≤ c 1 ?( d ),dist( ,∂ C X )≥ c 2 , and ≤ c 3 ?( d ),¶¶where c 1 , c 2 , c 3 are constants that depend only on properties of the cone C X and are independent of the data d (with analogous results for the alternative system when the system (FP d ) is infeasible).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42344/1/10107-88-3-451_00880451.pd