Approximation of Dense-n/2-Subgraph and the Complement of Min-Bisection

We consider the DENSE-n/2-SUBGRAPH problem, i.e., determine a block of half number nodes from a weighted graph such that the sum of the edge weights, within the subgraph induced by the block, is maximized. We prove that a strengthened semidefinite relaxation with a mixed rounding technique yields a .586 approximations of the problem. The previous best-known results for approximating this problem are:25 using a simple coin-toss randomization, .48 using a semidefinite relaxation, .5 using a linear programming relaxation or another semidefinite relaxation. In fact, a un-strengthened SDP relaxation provably yields no more than:5 approximation. We also consider the complement of the graph MIN-BISECTION problem, i.e., partitioning the nodes into two blocks of equal cardinality so as to maximize the weights of non-crossing edges. We present a .602 approximation of the complement of MIN-BISECTION

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