We consider streaming principal component analysis when the stochastic data-generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja’s algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments.

We consider line outages in the transmission network of the power grid, and specifically those caused by natural disasters or large-scale physical attacks. In such networks, an outage of a line may lead to overload on other lines, thereby leading to their outage. Such a cascade may have devastating effects not only on the power grid but also on the interconnected communication networks. We study a model of such failures and show that it differs from other models used to analyze cascades (e.g., epidemic/percolation-based models). Inspired by methods developed for network-survivability analysis, we show how to identify the most vulnerable locations in the network. We also perform extensive numerical experiments with real grid data to estimate the effects of geographically correlated outages and briefly discuss mitigation methods. The developed techniques can indicate potential locations for grid monitoring, and hence, will have impact on the deployment of the smart-grid networking infrastructure.

We consider an optimization problem of the form where P ⊆ ℝn is a polyhedron defined by m inequalities and Q is general and the μh ∊ ℝn and the rh quantities are given. In the case |S| = 1, |K| = 0 and m = 0 one obtains the classical trust-region subproblem; a strongly NP-hard problem which has been the focus of much interest because of applications to combinatorial optimization and nonlinear programming. We prove that for each fixed pair |S| and |K| our problem can be solved in polynomial time provided that either (1) |K| > 0 and the number of faces of P that intersect ∩h{x ∊ ℝn : ‖x – μh‖ ≤ rh, 1 ≤ j ≤ p} is polynomially bounded, or (2) |K| = 0 and m is bounded.

We adapt a method proposed by Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152] to design an algorithm that computes $\epsilon$-optimal solutions to fractional packing problems by solving $O(\epsilon^{-1}\sqrt{Kn\ln(m)})$ separable convex quadratic programs, where n is the number of variables, m is the number of constraints, and K is the maximum number of nonzero elements in any constraint. We show that the quadratic program can be approximated to any degree of accuracy by an appropriately defined piecewise-linear program. For the special case of the maximum concurrent flow problem on a graph $G = (V,E)$ with rational capacities and demands, we obtain an algorithm that computes an $\epsilon$-optimal flow by solving shortest path problems, i.e., problems in which the number of shortest paths computed grows as $O(\epsilon^{-1} \log(\epsilon^{-1}))$ in $\epsilon$ and polynomially in the size of the problem. In contrast, previous algorithms required $\Omega(\epsilon^{-2})$ iterations. We also describe extensions to the maximum multicommodity flow problem, the pure covering problem, and mixed packing-covering problem.

We adapt a method proposed by Nesterov [16] to design an algorithm that computes ε-optimal solutions to fractional packing problems by solving O*(ε-1 √Kn) separable convex quadratic programs, where K is the maximum number of non-zeros per row and n is the number of variables. We also show that the quadratic program can be approximated to any degree of accuracy by an appropriately defined piecewise-linear program. For the special case of the maximum concurrent flow problem on a graph G =(V,E) with rational capacities and demands we obtain an algorithm that computes an Ε-optimal flow by solving O*(ε-1 K3/2|E| √|V| (log 1/ε+ LU + LD)) shortest path problems, where K is the number of commodities, and LU, LD are, respectively, the number of bits needed to store the capacities and demands. We also show that the complexity of computing a maximum multicommodity flow is O*(1/εlog2(1/ε)). In contrast, previous algorithms required Ω(ε-2) iterations.

We study a capacity expansion problem that arises in telecommunication network design. Given a capacitated network and a traffic demand matrix, the objective is to add capacity to the edges, in multiples of various modularities, and route traffic, so that the overall cost is minimized. We study the polyhedral structure of a mixed-integer formulation of the problem and develop a cutting-plane algorithm using facet defining inequalities. The algorithm produces an extended formulation providing both a vary good lower bound and a starting point for branch and bound. The overall algorithm appears effective when applied to problem instances using real-life data.

Abstract We consider a combinatorial problem arising in the design and operation of lightwave networks. Nodes in such networks are equipped with tunable transmitters and receivers and communication occurs when the frequency of some transmitter is the same as that of a receiver. This technology enables us to update the network topology to respond to changes in traffic patterns. There are two main optimization problems related to this network structure, one being the design of a target graph more suitable to (future) traffic conditions, and the other being the problem of transforming the current network to this target network. This paper discusses the second problem, i.e., the transition phase when the modifications on the current graph are made through a sequence of intermediate connection networks. In particular, we move from one graph to another by swapping two independent edges in the current graph for two other independent edges not in the current graph, so that the union forms a four‐cycle. We characterize the sequence requiring the minimum number of intermediate graphs together with the necessary and sufficient conditions for the existence of such a sequence. We also develop upper and lower bounds on the length of a shortest sequence by formulating an integer program and solving its continuous relaxation to optimality. We also give an efficient algorithm for the case when the intermediate graphs are required to be connected. © 1994 by John Wiley & Sons, Inc.

Let G be a graph with weights on the edges, S a subset of vertices, and k an integer. The problem of computing a minimum-weight subset of edges that meets all the cuts of cardinality $ \leqslant k$ that separate pairs of vertices in S is considered. This problem is motivated by issues in network survivability. Assuming $|S| = 2$, it is shown that although this problem is NP-hard, it can be solved in linear time for each fixed value of k. Furthermore, if $|S| > 2$, the problem is NP-hard even for small values of k but can be solved in linear time for each fixed k and $|S|$.

This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists. In particular, we show that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph. Using some recent results of Goodman, Pollack, and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, which achieves the minimum number of crossings from among all such drawings. While this result has no bearing on the P versus NP question, it is fairly negative with regard to applications. We also study the problem of drawing a graph with polygonal edges, to achieve the (unrestricted) minimum number of crossings. Here we obtain a tight bound on the smallest number of breakpoints which are required in the polygonal lines.

The problem of estimating the number of 1’s in a square 0-1 matrix with certain forbidden configurations is considered, and nearly tight bounds are provided. This is motivated by a problem in computational geometry.

For any fixed surface S and fixed integer $k \geqq 0$, a linear-time algorithm is presented that tests whether selected vertices of a graph drawn on S can be covered with k or fewer faces.

The problem of finding a minimum-weight k-connected spanning subgraph of a complete graph, assuming that the edge weights satisfy the triangle inequality, is studied. It is shown that the class of minimum-weight k-edge connected spanning subgraphs can be restricted to those subgraphs which, in addition to the connectivity requirements, satisfy the following two conditions: (I) Every vertex has degree k or $k + 1$; (II) Removing any $1, 2, \cdots ,$ or k edges does not leave the resulting connected components all k-edge connected. For the k-vertex connected case, the parallel result is obtained with “k-edge” replaced by “k-vertex,” with the added technical restriction that $| V |\geqq 2k$ for condition (I) to hold. This generalizes recent work of Monma, Munson, and Pulleyblank for the case $k = 2$.

Abstract In this paper we study the problem of finding a minimum‐weight collection of edges in a planar graph which separates a given set of vertices from the outer face. This problem has two variants: either a given embedding is specified, or the best possible embedding is to be found. We present polynomial‐time algorithms for each case. We show how to use these results to recognize a special case of the steiner tree problem in graphs which is polynomially solvable. A closely related problem is shown to be NP ‐complete.

The pair $(G,D)$ consisting of a planar graph $G = (V,E)$ with n vertices together with a subset of d special vertices $D \subseteq V$ is called k-planar if there is an embedding of G in the plane so that at most k faces of G are required to cover all of the vertices in D. Checking 1-planarity can be done in linear-time since it reduces to a problem of checking planarity of a related graph. We present an algorithm which given a graph G and a value k either determines that G is not k-planar or generates an appropriate embedding and associated minimum cover in $O(c^k n)$ time, where c is a constant. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, $d = \theta (n)$, and all facial cycles have bounded length. These results provide a polynomial-time recognition algorithm for special cases of Steiner tree problems in graphs which are solvable in polynomial time.

We consider the problem of designing reliable networks at low cost, and show that for several standard and nonstandard models, 0-1 effects occur.