Space-eecient Approximation Algorithms for Maxcut and Coloring Semideenite Programs

The essential part of the best known approximation algorithm for graph MAXCUT is approximately solving MAXCUT's semideenite relaxation. For a graph with n nodes and m edges, previous work on solving its semidef-inite relaxation for MAXCUT requires space ~ O(n 2). Under the assumption of exact arithmetic, we show how an approximate solution can be found in space O(m+n 1:5), where O(m) comes from the input; and therefore reduce the space required by the best known approximation algorithm for graph MAXCUT. Using the above space-eecient algorithm as a subroutine, we show an approximate solution for COLORING's semideenite relaxation can be found in space O(m) + ~ O(n 1:5). This reduces not only the space required by the best known approximation algorithm for graph COLORING, but also the space required by the only known polynomial-time algorithm for nding a maximum clique in a perfect graph.