No abstract

A physical map has been constructed of the human genome containing 15,086 sequence-tagged sites (STSs), with an average spacing of 199 kilobases. The project involved assembly of a radiation hybrid map of the human genome containing 6193 loci and incorporated a genetic linkage map of the human genome containing 5264 loci. This information was combined with the results of STS-content screening of 10,850 loci against a yeast artificial chromosome library to produce an integrated map, anchored by the radiation hybrid and genetic maps. The map provides radiation hybrid coverage of 99 percent and physical coverage of 94 percent of the human genome. The map also represents an early step in an international project to generate a transcript map of the human genome, with more than 3235 expressed sequences localized. The STSs in the map provide a scaffold for initiating large-scale sequencing of the human genome.

Discrete Applied Mathematics 123 (2002) 75-102. doi:10.1016/S0166-218X(01)00338-92016-03-04T18:46:49

In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 be a given feasible solution. The solution x 0 may or may not be an optimal solution of P with respect to the cost vector c. The inverse optimization problem is to perturb the cost vector c to d so that x 0 is an optimal solution of P with respect to d and d − c p is minimum, where d − c p is some selected L p norm. In this paper, we consider the inverse linear programming problem under L 1 norm (where d − c p = i∈J w j d j − c j , with J denoting the index set of variables x j and w j denoting the weight of the variable j) and under L norm (where d − c p = max j∈J w j d j − c j. We prove the following results: (i) If the problem P is a linear programming problem, then its inverse problem under the L 1 as well as L norm is also a linear programming problem. (ii) If the problem P is a shortest path, assignment or minimum cut problem, then its inverse problem under the L 1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problem P is a minimum cost flow problem, then its inverse problem under the L 1 norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problem P is a minimum cost flow problem, then its inverse problem under the L norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problem P is polynomially solvable for linear cost functions, then inverse versions of P under the L 1 and L norms are also polynomially solvable.

In this paper, we present improved polynomial time algorithms for the max flow problem defined on sparse networks with n nodes and m arcs. We show how to solve the max flow problem in O(nm + m 31/16 log 2 n) time. In the case that m = O(n 1.06 ), this improves upon the best previous algorithm due to King, Rao, and Tarjan, who solved the max flow problem in O(nm log m/(n log n) n) time. This establishes that the max flow problem is solvable in O(nm) time for all values of n and m. In the case that m = O(n), we improve the running time to O(n 2 / log n).