On bounding the bandwidth of graphs with symmetry

We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite programming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. This fixing results in several smaller subproblems that need to be solved to obtain the new bound. In order to efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we develop a heuristic approach based on the well-known reverse Cuthill-McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs under consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices

Semi-Definite Relaxations for Minimum Bandwidth and other Vertex-Ordering problems

We present simple semi-definite programming relaxations for the NP-hard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain approximation guarantees for both of these vertex-ordering problems

Semi-Definite Relaxations for Minimum Bandwidth and other Vertex-Ordering problems

We present simple semi-definite programming relaxations for the NP-hard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain new approximation guarantees for both of these vertex-ordering problems. 1 Introduction Let the vertices of an undirected graph be ordered 1; 2; : : : ; n. We define the dilation of an edge (i; j) as the difference ji \Gamma jj, i.e., the length of the edge when the vertices of the graph are placed on the line in the order 1; 2; : : : ; n. Given a graph G = (V; E), we consider the following two problems: 1. Minimum Bandwidth : find an ordering that minimizes the maximum dilation among all the edges, i.e., minimizes max e2E dilation(e): 2. Minimum-length Linear Ordering : find an ordering that minimizes the length of the ordering where length is defined as: s X e2E dilation(e) 2 : That is, the squared length is the sum of the squares of dilation..

On semidefinite programming bounds for graph bandwidth

In this paper, we propose two new lower bounds on graph bandwidth and cyclic bandwidth based on semidefinite programming (SDP) relaxations of the quadratic assignment problem. We compare the new bounds with two other SDP bounds reported in [A. Blum, G. Konjevod, R. Ravi, and S. Vempala, Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems, Theoret. Comput. Sci. 235(1) (2000), pp. 25–42; J. Povh and F. Rendl, A copositive programming approach to graph partitioning, SIAM J. Optim. 18(1) (2007), pp. 223–241]

Algorithms for string and graph layout

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 121-125).Many graph optimization problems can be viewed as graph layout problems. A layout of a graph is a geometric arrangement of the vertices subject to given constraints. For example, the vertices of a graph can be arranged on a line or a circle, on a two- or three-dimensional lattice, etc. The goal is usually to place all the vertices so as to optimize some specified objective function. We develop combinatorial methods as well as models based on linear and semidefinite programming for graph layout problems. We apply these techniques to some well-known optimization problems. In particular, we give improved approximation algorithms for the string folding problem on the two- and three-dimensional square lattices. This combinatorial graph problem is motivated by the protein folding problem, which is central in computational biology. We then present a new semidefinite programming formulation for the linear ordering problem (also known as the maximum acyclic subgraph problem) and show that it provides an improved bound on the value of an optimal solution for random graphs. This is the first relaxation that improves on the trivial "all edges" bound for random graphs.by Alantha Newman.Ph.D