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Abstract. We consider the NP-hard problem of finding a minimum norm vector in n-dimensional real or complex Euclidean space, subject to m concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an O(m 2) approximation in the real case and an O(m) approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank 1 (namely, outer products of some given so-called steering vectors) and the phase spread of the entries of these steering vectors are bounded away from π/2, we establish a certain “constant factor ” approximation (depending on the phase spread but independent of m and n) for both the SDP relaxation and a convex QP restriction of the original NP-hard problem. Finally, we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints. We show that an SDP relaxation for this nonconvex QP provides an O(1 / ln(m)) approximation, which is analogous to a result of Nemirovski et al. [Math. Program., 86 (1999), pp. 463–473] for the real case. Key words. semidefinite programming relaxation, nonconvex quadratic optimization, approximation bound

We discuss approximation algorithms for the coloring problem and the maximum independent set problem in 3-uniform hypergraphs. An algorithm for coloring 3-uniform 2-colorable hypergraphs in ~ O(n 1=5 ) colors is presented, improving previously known results. Also, for every xed > 1=2, we describe an algorithm that, given a 3-uniform hypergraph H on n vertices with an independent set of size n, nds an independent set of size ~ 3196 n; n 6 3 )). For certain values of we are able to improve this using the Local Ratio Approach. The results are obtained through Semidenite Programming relaxations of these optimization problems. 1

We obtain improved semidefinite programming based approximation algorithms for all the natural maximum bisection problems of graphs. Among the problems considered are: MAX n/ -BISECTION -- partition the vertices of the graph into two sets of equal size such that the total weight of edges connecting vertices from different sides is maximized; MAX n/2-VERTEX-COVER -- find a set containing half of the vertices such that the total weight of edges touching this set is maximized; MAX n/2-DENSE-SUBGRAPH -- find a set containing half of the vertices such that the total weight of edges connecting two vertices from this set is maximized; and MAX n/2-UnCUT -- partition the vertices into two sets of equal size such that the total weight of edges that do not cross the cut is maximized. We also consider the directed versions of these problems, such as MAX n/2-DIRECTED-BISECTION and MAX n/2-DIRECTED-UnCUT. These results can be used to obtain improved approximation algorithms for the unbalanced versions of the partition problems mentioned above, where we want to partition the graph into two sets of size k and n - k, where k is not necessarily n/2 . Our results improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others. All these results may be viewed as extensions of the MAX CUT algorithm of Goemans and Williamson, and the MAX 2-SAT and MAX DI-CUT algorithms of Feige and Goemans.

Given a collection of sets of 2-D views of 3-D objects and a similarity measure between them, we present a method for summarizing the sets using a small subset called a bounded canonical set (BCS), whose members best represent the members of the original set. This means that members of the BCS are as dissimilar from each other as possible, while at the same time being as similar as possible to the nonBCS members. This paper will extend our earlier work on computing canonical sets [2] in several ways: by omitting the need for a multi-objective optimization, by allowing the imposition of cardinality constraints, and by introducing a total similarity function. We evaluate the applicability of BCS to view selection in a view-based object recognition environment.

We present a complete analysis of the MAX 2-SAT and MAX DI-CUT approximation algorithms of Feige and Goemans using various analytical and computational tools. By fine-tuning the rotation functions used we obtain minutely improved performance ratios for these problems. The rotation functions used for getting these improvements are essentially optimal as the performance ratios obtained using them almost completely match upper bounds that we obtain on the performance ratios that can be achieved using any rotation function. We also discuss possibilities of getting improved approximation algorithms for these problems.

We obtain the following new coloring results: ffl A 3-colorable graph on n vertices with maximum degree \Delta can be colored, in polynomial time, using O((\Delta log \Delta) 1=3 \Delta log n) colors. This slightly improves an O((\Delta 1=3 log 1=2 \Delta) \Delta log n) bound given by Karger, Motwani and Sudan. More generally, k-colorable graphs with maximum degree \Delta can be colored, in polynomial time, using O((\Delta 1\Gamma2=k log 1=k \Delta) \Delta log n) colors.

We obtain the following new coloring results: A 3-colorable graph on n vertices with maximum degree can be colored, in polynomial time, using O(( log ) log n) colors. This slightly improves an O(( ) log n) bound given by Karger, Motwani and Sudan. More generally, k-colorable graphs with maximum degree can be colored, in polynomial time, using O(( 1=k ) log n) colors.

Abstract. A common approach to the image matching problem is representing images as sets of features in some feature space followed by establishing correspondences among the features. Previous work by Huttenlocher and Ullman [1] shows how a similarity transformation- rotation, translation, and scaling- between two images may be determined assuming that three corresponding image points are known. While robust, such methods suffer from computational inefficiencies for general feature sets. We describe a method whereby the feature sets may be summarized using the Stable Bounded Canonical Set (SBCS), thus allowing the efficient computation of point correspondences between large feature sets. We use a notion of stability to influence the set summarization such that stable image features are preferred. Fig. 1. A) Blob and ridge feature extraction with centroids of blobs and ridges denoted, B) Stable Bounded Canonical Set (SBCS) construction, C) Determine transformation, D) Outline shows transformation determined from SBCS. 1