Combinatorial Geometry of Graph Partitioning - I

The c-Balanced Separator problem is a graph-partitioning problem in which given a graph G, one aims to find a cut of minimum size such that both the sides of the cut have at least cn vertices. In this paper, we present new directions of progress in the c-Balanced Separator problem. More specifically, we propose a family of mathematical programs, that depend upon a parameter p > 0, and is an extension of the uniform version of the SDPs proposed by Goemans and Linial for this problem. In fact for the case, when p = 1, if one can solve this program in polynomial time then simply using the Goemans-Williamson’s randomized rounding algorithm for Max Cut [11] will give an O(1)-factor approximation algorithm for c-Balanced Separator improving the best known approximation factor of O( √ logn) due to Arora, Rao and Vazirani [4]. This family of programs is not convex but one can transform them into so called concave programs in which one optimizes a concave function over a convex feasible set. It is well known that the optima of such programs lie at one of the extreme points of the feasible set [26]. Our main contribution is a combinatorial characterization of some extreme points of the feasible set of the mathematical program, for p = 1 case, which to the best of our knowledge is the first of its kind. We further demonstrate how this characterization can be used to solve the program in a restricted setting. Non-convex programs have recently been investigated by Bhaskara and Vijayaraghvan [6] in which they design algorithms for approximating Matrix pnorms although their algorithmic techniques are analytical in nature. It is important to note that the properties of concave programs allows one to apply techniques due to Hoffman [18] or Tuy et al [26] to solve such problems with arbitrary accuracy that, for special forms of concave programs, converge in polynomial time. A significant portion of this work was done when the author was a B-Tech, M-Tech dual degree student at IIT-Kanpur, India ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 184 (2010)