Donna Llewellyn

Local optimization on graphs

Abstract The complexity of finding local optima is an open problem for many neighborhood structures. We show how to derive close lower and upper bounds on the minimum number of function evaluations needed to find a local optimum in an arbitrary graph. When these bounding techniques are applied to the hypercube, the results give insights into the class PLS and the gap between the average and worst-case behavior of local search.

A primal dual integer programming algorithm

Abstract We introduce the framework for a primal dual integer programming algorithm. We prove convergence, and discuss some special cases.

Dividing and conquering the square

Abstract A local minimum of a matrix is a cell whose value is smaller than those of its four adjacent cells. For an n × n square matrix, we find a local minimum with at most 2.554n queries, and prove a lower bound of 2n queries required by any method. For a different neighborhood corresponding to the eight possible moves of a chess king, we prove upper and lower bounds of3n + O(log n) and 2n, respectively.

Matching 2-lattice polyhedra: finding a maximum vector

Abstract Matching 2-lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, etc. In this paper we develop a polynomial-time extreme point algorithm for finding a maximum cardinality vector in a matching 2-lattice polyhedron.

Two-lattice polyhedra: duality and extreme points

Abstract Two-lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhedron is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum ‘cardinality’ vectors in the polyhedron. This characterization is at the heart of our extreme point algorithm (Chang et al., ISyE Technical Report No. J-94-05, ISyE, Georgia Institute of Technology, Atlanta, GA 30332) for finding a maximum cardinality vector in a matching 2-lattice polyhedron.