Research Report and Proposal

when I joined the Max Planck Institut as a post-doctoral fellow, I have been working at Semideenite Programming and its applications to Approximation Problems. The techniques from Semideenite Programming have proved useful in the design of good approximation algorithms as is evinced by the ground breaking paper of Goemans and Williamson on Max Cut and Max Sat 13]. Since then several researchers have used semideenite Programming to give good approximation algorithms for Max Bisection and Max k-Cut 11], approximately coloring 3-colorable graphs 15] and several other problems. The power of Semideenite Programming is not fully understood, although it is known to yield better relaxations than Linear Programming in wide variety of situations. To understand the nature of Semideenite Programming better, I have looked at it both from the persepctive of decision problems and approximation algorithms. For instance, it is exactly because of semideenite programming that one can get the classical result of Loavsz 17] which says that determining the clique number (or chromatic number) of perfect graphs can be done in polynomial time. In my recent work 2] with Roman Bacik, we have shown in a precise sense that semideenite relaxations are better than linear relaxations under some very general situations. In a sense, this is a generalization of the work of 17] and 12]. With my co-author Ramesh Hariharan, I have been able to correctly de-randomize the Max CUT algorithm of Goemans and Williamson, which has a fundamental aw in it. I have also been working on understanding the power and limitations of Semideenite Programming in coloring 3-colorable graphs and in general k-colorable graphs. The recent paper by 15] shows that Semideenite Programming yields a randomized O(n 0:25 log n) approximation algorithm for coloring 3-colorable graphs. Before this, the best known algorithm is by Avrim Blum, which colors 3-colorable graphs in O(n 3=8 log O(1) n). They also show that such techniques cannot yield an algorithm which achieves better than (n 0:01). I have shown that this lower bound can be improved to (n 0:05). Szegedy 22] also has the same result albeit using diierent methods. I have also been able to show 1