Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.Comment: conference version in Symposium on Theory of Computing (1994

Linear Programming for Large-Scale Markov Decision Problems

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose two techniques, one based on stochastic convex optimization, and one based on constraint sampling. In both cases, we give bounds that show that the performance of our algorithms approaches the best achievable by any policy in the comparison class. Most importantly, these results depend on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithms in a queuing application.Comment: 27 pages, 3 figure

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Dynamic Legislative Policy Making

We prove existence of stationary Markov perfect equilibria in an infinite-horizon model of legislative policy making in which the policy outcome in one period determines the status quo in the next. We allow for a multidimensional policy space and arbitrary smooth stage utilities. We prove that all such equilibria are essentially in pure strategies and that proposal strategies are differentiable almost everywhere. We establish upper hemicontinuity of the equilibrium correspondence, and we derive conditions under which each equilibrium of our model determines a unique invariant distribution characterizing long run policy outcomes. We illustrate the equilibria of the model in a numerical example of policy making in a single dimension, and we discuss extensions of our approach to accommodate much of the institutional structure observed in real-world politics.

Joint Hypergraph Learning and Sparse Regression for Feature Selection

In this paper, we propose a unified framework for improved structure estimation and feature selection. Most existing graph-based feature selection methods utilise a static representation of the structure of the available data based on the Laplacian matrix of a simple graph. Here on the other hand, we perform data structure learning and feature selection simultaneously. To improve the estimation of the manifold representing the structure of the selected features, we use a higher order description of the neighbour- hood structures present in the available data using hypergraph learning. This allows those features which participate in the most significant higher order relations to be se- lected, and the remainder discarded, through a sparsification process. We formulate a single objective function to capture and regularise the hypergraph weight estimation and feature selection processes. Finally, we present an optimization algorithm to re- cover the hyper graph weights and a sparse set of feature selection indicators. This process offers a number of advantages. First, by adjusting the hypergraph weights, we preserve high-order neighborhood relations reflected in the original data, which cannot be modeled by a simple graph. Moreover, our objective function captures the global discriminative structure of the features in the data. Comprehensive experiments on 9 benchmark data sets show that our method achieves statistically significant improve- ment over state-of-art feature selection methods, supporting the effectiveness of the proposed method

Localization and security algorithms for wireless sensor networks and the usage of signals of opportunity

In this dissertation we consider the problem of localization of wireless devices in environments and applications where GPS (Global Positioning System) is not a viable option. The _x000C_rst part of the dissertation studies a novel positioning system based on narrowband radio frequency (RF) signals of opportunity, and develops near optimum estimation algorithms for localization of a mobile receiver. It is assumed that a reference receiver (RR) with known position is available to aid with the positioning of the mobile receiver (MR). The new positioning system is reminiscent of GPS and involves two similar estimation problems. The _x000C_rst is localization using estimates of time-di_x000B_erence of arrival (TDOA). The second is TDOA estimation based on the received narrowband signals at the RR and the MR. In both cases near optimum estimation algorithms are developed in the sense of maximum likelihood estimation (MLE) under some mild assumptions, and both algorithms compute approximate MLEs in the form of a weighted least-squares (WLS) solution. The proposed positioning system is illustrated with simulation studies based on FM radio signals. The numerical results show that the position errors are comparable to those of other positioning systems, including GPS. Next, we present a novel algorithm for localization of wireless sensor networks (WSNs) called distributed randomized gradient descent (DRGD), and prove that in the case of noise-free distance measurements, the algorithm converges and provides the true location of the nodes. For noisy distance measurements, the convergence properties of DRGD are discussed and an error bound on the location estimation error is obtained. In contrast to several recently proposed methods, DRGD does not require that blind nodes be contained in the convex hull of the anchor nodes, and can accurately localize the network with only a few anchors. Performance of DRGD is evaluated through extensive simulations and compared with three other algorithms, namely the relaxation-based second order cone programming (SOCP), the simulated annealing (SA), and the semi-de_x000C_nite programing (SDP) procedures. Similar to DRGD, SOCP and SA are distributed algorithms, whereas SDP is centralized. The results show that DRGD successfully localizes the nodes in all the cases, whereas in many cases SOCP and SA fail. We also present a modi_x000C_cation of DRGD for mobile WSNs and demonstrate the e_x000E_cacy of DRGD for localization of mobile networks with several simulation results. We then extend this method for secure localization in the presence of outlier distance measurements or distance spoo_x000C_ng attacks. In this case we present a centralized algorithm to estimate the position of the nodes in WSNs, where outlier distance measurements may be present

Routing and search on large scale networks

In this thesis, we address two seemingly unrelated problems, namely routing in large wireless ad hoc networks and comparison based search in image databases. However, the underlying problem is in essence similar and we can use the same strategy to attack those two problems. In both cases, the intrinsic complexity of the problem is in some sense low, and we can exploit this fact to design efficient algorithms. A wireless ad hoc network is a communication network consisting of wireless devices such as for instance laptops or cell phones. The network does not have any fixed infrastructure, and hence nodes which cannot communicate directly over the wireless medium must use intermediate nodes as relays. This immediately raises the question of how to select the relay nodes. Ideally, one would like to find a path from the source to the destination which is as short as possible. The length of the found path, also called route, typically depends on how much signaling traffic is generated in order to establish the route. This is the fundamental trade-off that we will investigate in this thesis. As mentioned above, we try and exploit the fact that the communication network is intrinsically low-dimensional, or in other words has low complexity. We show that this is indeed the case for a large class of models and that we can design efficient algorithms for routing that use this property. Low dimensionality implies that we can well embed the network in a low-dimensional space, or build simple hierarchical decompositions of the network. We use both those techniques to design routing algorithms. Comparison based search in image databases is a new problem that can be defined as follows. Given a large database of images, can a human user retrieve an image which he has in mind, or at least an image similar to that image, without going sequentially through all images? More precisely, we ask whether we can search a database of images only by making comparisons between images. As a case in point, we ask whether we can find a query image q only by asking questions of the type "does image q look more like image A or image B"? The analogous to signaling traffic for wireless networks would here be the questions we can ask human users in a learning phase anterior to the search. In other words, we would like to ask as few questions as possible to pre-process and prepare the database, while guaranteeing a certain quality of the results obtained in the search phase. As the underlying image space is not necessarily metric, this raises new questions on how to search spaces for which only rank information can be obtained. The rank of A with respect to B is k, if A is B's kth nearest neighbor. In this setup, low-dimensionality is analogous to the homogeneity of the image space. As we will see, the homogeneity can be captured by properties of the rank relationships. In turn, homogeneous spaces can be well decomposed hierarchically using comparisons. Further, it allows us to design good hash functions. To design efficient algorithms for these two problems, we can apply the same techniques mutatis mutandis. In both cases, we relied on the intuition that the problem has a low intrinsic complexity, and that we can exploit this fact. Our results come in the form of simulation results and asymptotic bounds