In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games

We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instance---a collection of clauses each of length at most three---is satisfiable, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary MAX 3SAT instances. Our algorithm uses semidefinite programming and may be seen as a sequel to the MAXCUT algorithm of Goemans and Williamson and the MAX 2SAT algorithm of Feige and Goemans. Though the algorithm itself is fairly simple, its analysis is quite complicated as it involves the computation of volumes of spherical tetrahedra. Hastad has recently shown that, assuming P 6= NP , no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7=8, even when restricted to satisfiable instances of the problem. Our algorithm is therefore optimal in this sense. We also describe a method of obtaining direct semi...

We describe a few applications of semide nite programming in combinatorial optimization.

An instance of MAX 3CSP is a collection of m clauses of the form f i (z i1 ; z i2 ; z i3 ), where the z ij 's are literals, or constants, from the set f0; 1; x 1 ; : : : ; xn ; x 1 ; : : : ; xng, and the f i 's are arbitrary Boolean functions depending on (at most) three variables. Each clause has a non-negative weight w i associated with it. A solution to the instance is an assignment of 0-1 values to the variables x 1 ; : : : ; xn that maximizes P n i=1 w i f i (z i1 ; z i2 ; z i3 ), the total weight of the satisfied clauses. The MAX 3CSP problem is clearly a generalization of the MAX 3SAT problem. (In an instance of the MAX 3SAT problem f i (z i1 ; z i2 ; z i3 ) = z i1 z i2 z i3 for every i.) Karloff and Zwick have recently obtained a 8 -approximation algorithm for MAX 3SAT. Their algorithm is based on a new semidefinite relaxation of the problem. Hastad showed that no polynomial time algorithm can achieve a better performance ratio, unless P=NP. Here we use similar techniques to obtain a -approximation algorithm for MAX 3CSP. The performance ratio of this algorithm is also optimal, as follows again from the work of Hastad. We also obtain better performance ratios for several special cases of the problem. Our results include: 2 -approximation algorithm for MAX 3AND, the problem in which each clause is of the form z i1 z i2 z i3 . This result is optimal and it implies the result for MAX 3CSP.

We present a tool, outward rotations, for enhancing the performance of several semidefinite programming based approximation algorithms. Using outward rotations, we obtain an approximation algorithm for MAX CUT that, in many interesting cases, performs better than the algorithm of Goemans and Williamson. We also obtain an improved approximation algorithm for MAX NAE-f3g-SAT. Finally, we provide some evidence that outward rotations can also be used to obtain improved approximation algorithms for MAX NAE-SAT and MAX SAT. 1 Introduction MAX CUT is perhaps the simplest and most natural APX-complete constraint satisfaction problem (see, e.g., [AL97]). There are various simple ways of obtaining a performance guarantee of 1/2 for the problem. One of them, for example, is just choosing a random cut. No performance guarantee better than 1/2 was known for the problem until Goemans and Williamson [GW95], in a major breakthrough, used semidefinite programming to obtain an approximation algorithm ...

Recent advances in information retrieval over hyperlinked corpora have convincingly demonstrated that links carry less noisy information than text. We investigate the feasibility of applying link-based methods in new applications domains. The specific application we consider is to partition authors into opposite camps within a given topic in the context of newsgroups. A typical newsgroup posting consists of one or more quoted lines from another posting followed by the opinion of the author. This social behavior gives rise to a network in which the vertices are individuals and the links represent "responded-to" relationships. An interesting characteristic of many newsgroups is that people more frequently respond to a message when they disagree than when they agree. This behavior is in sharp contrast to the WWW link graph, where linkage is an indicator of agreement or common interest. By analyzing the graph structure of the responses, we are able to effectively classify people into opposite camps. In contrast, methods based on statistical analysis of text yield low accuracy on such datasets because the vocabulary used by the two sides tends to be largely identical, and many newsgroup postings consist of relatively few words of text.

MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least ff \Delta sdp, where ff ' 0:87856 and sdp is the value of the semidefinite program, which is an upper bound on opt, the number of edges in the maximum cut. This manuscript shows the following results: 1. The integrality ratio of the semidefinite program is ff. The previously known bound on the integrality ratio was roughly 0:8845. 2. In the presence of the so called "triangle constraints", the integrality ratio is no better than roughly 0:891. The previously known bound was above ...

Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue µ of any non-bipartite graph on n vertices with diameter D and maximum degree ∆ satisfies µ � − ∆ + 1 (D+1)n. This improves previous estimates and is tight up to a constant factor. The second result is the determination of the precise approximation guarantee of the max cut algorithm of Goemans and Williamson for graphs G =(V,E) in which the size of the max cut is at least A|E|, for all A between 0.845 and 1. This extends a result of Karloff. 1.

We analyze the addition of a simple local improvement step to various known randomized approximation algorithms. Let α ∼ 0.87856 denote the best approximation ratio currently known for the Max Cut problem on general graphs [GW95]. We consider a semidefinite relaxation of the Max Cut problem, round it using the random hyperplane rounding technique of ([GW95]), and then add a local improvement step. We show that for graphs of degree at most Δ, our algorithm achieves an approximation ratio of at least α+ε, where ε>0 is a constant that depends only on Δ. In particular, using computer assisted analysis, we show that for graphs of maximal degree 3, our algorithm obtains an approximation ratio of at least 0.921, and for 3-regular graphs, the approximation ratio is at least 0.924. We note that for the semidefinite relaxation of Max Cut used in [GW95], the integrality gap is at least 1/0.884, even for 2-regular graphs.

Abstract not found