A note on Probably Certifiably Correct algorithms

Many optimization problems of interest are known to be intractable, and while there are often heuristics that are known to work on typical instances, it is usually not easy to determine a posteriori whether the optimal solution was found. In this short note, we discuss algorithms that not only solve the problem on typical instances, but also provide a posteriori certificates of optimality, probably certifiably correct (PCC) algorithms. As an illustrative example, we present a fast PCC algorithm for minimum bisection under the stochastic block model and briefly discuss other examples

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Efficient Semidefinite Spectral Clustering via Lagrange Duality

We propose an efficient approach to semidefinite spectral clustering (SSC), which addresses the Frobenius normalization with the positive semidefinite (p.s.d.) constraint for spectral clustering. Compared with the original Frobenius norm approximation based algorithm, the proposed algorithm can more accurately find the closest doubly stochastic approximation to the affinity matrix by considering the p.s.d. constraint. In this paper, SSC is formulated as a semidefinite programming (SDP) problem. In order to solve the high computational complexity of SDP, we present a dual algorithm based on the Lagrange dual formalization. Two versions of the proposed algorithm are proffered: one with less memory usage and the other with faster convergence rate. The proposed algorithm has much lower time complexity than that of the standard interior-point based SDP solvers. Experimental results on both UCI data sets and real-world image data sets demonstrate that 1) compared with the state-of-the-art spectral clustering methods, the proposed algorithm achieves better clustering performance; and 2) our algorithm is much more efficient and can solve larger-scale SSC problems than those standard interior-point SDP solvers.Comment: 13 page

Efficient graph cuts for unsupervised image segmentation using probabilistic sampling and SVD-based approximation

The application of graph theoretic methods to unsupervised image partitioning has been a very active field of research recently. For weighted graphs encoding the (dis)similarity structure of locally extracted image features, unsupervised segmentations of images into coherent structures can be computed in terms of extremal cuts of the underlying graphs. In this context, we focus on the normalized cut criterion and a related recent convex approach based on semidefinite programming. As both methods soon become computationally demanding with increasing graph size, an important question is how the computations can be accelerated. To this end, we study an SVD approximation method in this paper which has been introduced in a different clustering context. We apply this method, which is based on probabilistic sampling, to both segmentation approaches and compare it with the Nyström extension suggested for the normalized cut. Numerical results confirm that by means of the sampling-based SVD approximation technique, reliable segmentations can be computed with a fraction (less than 5%) of the original computational cost

Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving $(SDP)$ that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm

On a registration-based approach to sensor network localization

We consider a registration-based approach for localizing sensor networks from range measurements. This is based on the assumption that one can find overlapping cliques spanning the network. That is, for each sensor, one can identify geometric neighbors for which all inter-sensor ranges are known. Such cliques can be efficiently localized using multidimensional scaling. However, since each clique is localized in some local coordinate system, we are required to register them in a global coordinate system. In other words, our approach is based on transforming the localization problem into a problem of registration. In this context, the main contributions are as follows. First, we describe an efficient method for partitioning the network into overlapping cliques. Second, we study the problem of registering the localized cliques, and formulate a necessary rigidity condition for uniquely recovering the global sensor coordinates. In particular, we present a method for efficiently testing rigidity, and a proposal for augmenting the partitioned network to enforce rigidity. A recently proposed semidefinite relaxation of global registration is used for registering the cliques. We present simulation results on random and structured sensor networks to demonstrate that the proposed method compares favourably with state-of-the-art methods in terms of run-time, accuracy, and scalability

Towards an SDP-based Approach to Spectral Methods: A Nearly-Linear-Time Algorithm for Graph Partitioning and Decomposition

In this paper, we consider the following graph partitioning problem: The input is an undirected graph $G=(V,E),$ a balance parameter $b \in (0,1/2]$ and a target conductance value $\gamma \in (0,1).$ The output is a cut which, if non-empty, is of conductance at most $O(f),$ for some function $f(G, \gamma),$ and which is either balanced or well correlated with all cuts of conductance at most $\gamma.$ Spielman and Teng gave an $\tilde{O}(|E|/\gamma^{2})$-time algorithm for $f= \sqrt{\gamma \log^{3}|V|}$ and used it to decompose graphs into a collection of near-expanders. We present a new spectral algorithm for this problem which runs in time $\tilde{O}(|E|/\gamma)$ for $f=\sqrt{\gamma}.$ Our result yields the first nearly-linear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primal-dual semidefinite-programming SDP approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than $\gamma,$ somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale's framework to bound the running time. Our result contains technical ingredients that may be of independent interest.Comment: To appear in SODA 201