Coping with Np-hardness: Approximating Minimum Bisection and Heuristics for Maximum Clique 2 Approximating Minimum Bisection 13 1.2 Approximation Algorithms

Many important optimization problems are known to be NP-hard. That is, unless P = NP, there is no polynomial time algorithm that optimally solves these problems on every input instance. We study algorithmic ways for \coping" with NP-hard optimization problems. One possible approach for coping with the NP-hardness is to relax the requirement for exact solution, and devise approximation algorithms, i.e. e cient algorithms that produce a solution that is guaranteed to be nearly optimal. In the last decade, our understanding of many NP-hard optimization problems was greatly improved, both from the direction of approximation algorithms and from the direction of hardness of approximation. However, there is still a large gap in our understanding of the approximability of several fundamental problems. A notable example is the minimum bisection problem, that requires to nd in a graph a minimum-cost cut that partitions the vertices into two equal-size sets. This problem has applications both in theory and in practice. The seminal work of Leighton and Rao (1988) was largely motivated by this problem, and led to algorithms with approximation ratio O(logn) for several related problems. However, prior to our work no sublinear (in n) approximation ratio was known for this problem, and its approximability is a famous open problem. We signi cantly improve the known approximation ratio for minimum bisection. Our algorithms achieve an approximation ratio O(log2 n), which is \in the same ballpark" as the current approximation ratios for many related problems. Another approach for coping with the NP-hardness is to relax the requirement for worst-case analysis, and consider instead heuristic algorithms that are successful on average-case input instances. One main di culty in providing rigorous analysis of heuristics lies in realistically modeling average-case instances. Consider for example the hidden clique problem. In a random model for the problem, a random graph on n vertices is chosen (i.e. Gn;1=2) and then a clique of size k is randomly placed in the graph, and the goal is to nd the planted clique in the graph. A semi-random model may extend this random model by allowing an adversary to remove any edge that is not inside the planted clique. We devise for the hidden clique problem a heuristic that is based on the Lov asz theta function, a well-known semide nite programming relaxation of maximum clique. Our heuristic is successful in the semi-random model when k (pn). In contrast, previous heuristics have similar success in the random model but fail in the semi-random model. We also study relaxations that are stronger than the Lov asz theta function, namely those obtained by the \lift-and-project" method of Lov asz and Schrijver (1991). We show that on a random graph Gn;1=2 the value of these stronger relaxations is comparable to the theta function, and hence they do not extend our heuristic mentioned above to a planted clique of smaller size k = o(pn). This thesis is based on the following papers. 1. U. Feige and R. Krauthgamer. Finding and certifying a large hidden clique in a semirandom graph. Random Structures Algorithms, 16(2):195{208, 2000. 2. U. Feige, R. Krauthgamer, and K. Nissim. Approximating the minimum bisection size. In 32nd Annual ACM Symposium on Theory of Computing, pages 530{536, May 2000. 3. U. Feige and R. Krauthgamer. A polylogarithmic approximation of the minimum bisection. In 41st Annual IEEE Symposium on Foundations of Computer Science, pages 105{115, November 2000. 4. U. Feige and R. Krauthgamer. The probable value of the Lov asz-Schrijver relaxations for maximum independent set. Manuscript, April 2001. Declaration. The author, Robert Krauthgamer, declares that this thesis summarizes his independent work under the supervision of Prof. Uriel Feige, with the exception of Section 2.6, whose results were obtained jointly with Kobbi Nissim.