Space-Efficient Approximation Algorithms for MAXCUT and COLORING Semidefinite Programs

The essential part of the best known approximation algorithm for graph MAXCUT is approximately solving MAXCUT’s semidefinite relaxation. For a graph with n nodes and m edges, previous work on solving its semidefinite relaxation for MAXCUT requires space Õ(n). Under the assumption of exact arithmetic, we show how an approximate solution can be found in space O(m + n), 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-efficient algorithm as a subroutine, we show an approximate solution for COLORING’s semidefinite relaxation can be found in space O(m)+ Õ(n). 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 finding a maximum clique in a perfect graph.