In this paper we study the existence of homomorphisms @math using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number @math for which there exists an assignment of unit vectors @math to its vertices such that @math when @math . Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for @math the Kneser graph @math and the @math -Kneser graph @math are cores, and furthermore, that for @math there exists a homomorphism @math if and only if @math divides @math . In terms of new applications, we show that the even-weight component of the distance @math -graph of the @math -cube @math is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms @math when @math . Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence's list of strongly regular graphs and found that at least 84 are cores.
In order for our approach to work, we must be able to identify UVC graphs and, moreover, find their unique vector colorings. This is the problem we consider in the first part of this work. UVC graphs are interesting in their own right. They were first introduced in @cite_0 to construct tractable instances of the graph realization problem. In the same work UVC graphs were used to construct uniquely colorable graphs. UVC graphs are also closely related to the notion of universal completability (equivalently, the universal rigidity of apex graphs). This in turn is relevant to the low-rank matrix completion problem @cite_20 .
@cite_0 @cite_20
2962861404 2167758305 2067912884 1969366022
An embedding (i p_i R ^d ) of the vertices of a graph G is called universally completable if the following holds: For any other embedding (i q_i R ^ k ) satisfying (q_i^ T q_j = p_i^ T p_j ) for (i = j ) and i adjacent to j, there exists an isometry mapping (q_i ) to (p_i ) for all ( i V(G) ). The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of G, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on ( Z _2^n (n 5) ) show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovasz theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and q-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable. The low-rank matrix completion problem is a fundamental problem with many important applications. Recently, [4],[13] and [5] obtained the first non-trivial theoretical results for the problem assuming that the observed entries are sampled uniformly at random. Unfortunately, most real-world datasets do not satisfy this assumption, but instead exhibit power-law distributed samples. In this paper, we propose a graph theoretic approach to matrix completion that solves the problem for more realistic sampling models. Our method is simpler to analyze than previous methods with the analysis reducing to computing the threshold for complete cascades in random graphs, a problem of independent interest. By analyzing the graph theoretic problem, we show that our method achieves exact recovery when the observed entries are sampled from the Chung-Lu-Vu model, which can generate power-law distributed graphs. We also hypothesize that our algorithm solves the matrix completion problem from an optimal number of entries for the popular preferential attachment model and provide strong empirical evidence for the claim. Furthermore, our method is easy to implement and is substantially faster than existing methods. We demonstrate the effectiveness of our method on random instances where the low-rank matrix is sampled according to the prevalent random graph models for complex networks and present promising preliminary results on the Netflix challenge dataset. We address the problems of contour detection, bottom-up grouping and semantic segmentation using RGB-D data. We focus on the challenging setting of cluttered indoor scenes, and evaluate our approach on the recently introduced NYU-Depth V2 (NYUD2) dataset [27]. We propose algorithms for object boundary detection and hierarchical segmentation that generalize the gPb-ucm approach of [2] by making effective use of depth information. We show that our system can label each contour with its type (depth, normal or albedo). We also propose a generic method for long-range amodal completion of surfaces and show its effectiveness in grouping. We then turn to the problem of semantic segmentation and propose a simple approach that classifies super pixels into the 40 dominant object categories in NYUD2. We use both generic and class-specific features to encode the appearance and geometry of objects. We also show how our approach can be used for scene classification, and how this contextual information in turn improves object recognition. In all of these tasks, we report significant improvements over the state-of-the-art. In this paper, we address the problems of contour detection, bottom-up grouping, object detection and semantic segmentation on RGB-D data. We focus on the challenging setting of cluttered indoor scenes, and evaluate our approach on the recently introduced NYU-Depth V2 (NYUD2) dataset (, ECCV, 2012). We propose algorithms for object boundary detection and hierarchical segmentation that generalize the @math gPb-ucm approach of (TPAMI, 2011) by making effective use of depth information. We show that our system can label each contour with its type (depth, normal or albedo). We also propose a generic method for long-range amodal completion of surfaces and show its effectiveness in grouping. We train RGB-D object detectors by analyzing and computing histogram of oriented gradients on the depth image and using them with deformable part models (, TPAMI, 2010). We observe that this simple strategy for training object detectors significantly outperforms more complicated models in the literature. We then turn to the problem of semantic segmentation for which we propose an approach that classifies superpixels into the dominant object categories in the NYUD2 dataset. We design generic and class-specific features to encode the appearance and geometry of objects. We also show that additional features computed from RGB-D object detectors and scene classifiers further improves semantic segmentation accuracy. In all of these tasks, we report significant improvements over the state-of-the-art.