Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Robust convex optimisation techniques for autonomous vehicle vision-based navigation

This thesis investigates new convex optimisation techniques for motion and pose estimation. Numerous computer vision problems can be formulated as optimisation problems. These optimisation problems are generally solved via linear techniques using the singular value decomposition or iterative methods under an L2 norm minimisation. Linear techniques have the advantage of offering a closed-form solution that is simple to implement. The quantity being minimised is, however, not geometrically or statistically meaningful. Conversely, L2 algorithms rely on iterative estimation, where a cost function is minimised using algorithms such as Levenberg-Marquardt, Gauss-Newton, gradient descent or conjugate gradient. The cost functions involved are geometrically interpretable and can statistically be optimal under an assumption of Gaussian noise. However, in addition to their sensitivity to initial conditions, these algorithms are often slow and bear a high probability of getting trapped in a local minimum or producing infeasible solutions, even for small noise levels. In light of the above, in this thesis we focus on developing new techniques for finding solutions via a convex optimisation framework that are globally optimal. Presently convex optimisation techniques in motion estimation have revealed enormous advantages. Indeed, convex optimisation ensures getting a global minimum, and the cost function is geometrically meaningful. Moreover, robust optimisation is a recent approach for optimisation under uncertain data. In recent years the need to cope with uncertain data has become especially acute, particularly where real-world applications are concerned. In such circumstances, robust optimisation aims to recover an optimal solution whose feasibility must be guaranteed for any realisation of the uncertain data. Although many researchers avoid uncertainty due to the added complexity in constructing a robust optimisation model and to lack of knowledge as to the nature of these uncertainties, and especially their propagation, in this thesis robust convex optimisation, while estimating the uncertainties at every step is investigated for the motion estimation problem. First, a solution using convex optimisation coupled to the recursive least squares (RLS) algorithm and the robust H filter is developed for motion estimation. In another solution, uncertainties and their propagation are incorporated in a robust L convex optimisation framework for monocular visual motion estimation. In this solution, robust least squares is combined with a second order cone program (SOCP). A technique to improve the accuracy and the robustness of the fundamental matrix is also investigated in this thesis. This technique uses the covariance intersection approach to fuse feature location uncertainties, which leads to more consistent motion estimates. Loop-closure detection is crucial in improving the robustness of navigation algorithms. In practice, after long navigation in an unknown environment, detecting that a vehicle is in a location it has previously visited gives the opportunity to increase the accuracy and consistency of the estimate. In this context, we have developed an efficient appearance-based method for visual loop-closure detection based on the combination of a Gaussian mixture model with the KD-tree data structure. Deploying this technique for loop-closure detection, a robust L convex posegraph optimisation solution for unmanned aerial vehicle (UAVs) monocular motion estimation is introduced as well. In the literature, most proposed solutions formulate the pose-graph optimisation as a least-squares problem by minimising a cost function using iterative methods. In this work, robust convex optimisation under the L norm is adopted, which efficiently corrects the UAV’s pose after loop-closure detection. To round out the work in this thesis, a system for cooperative monocular visual motion estimation with multiple aerial vehicles is proposed. The cooperative motion estimation employs state-of-the-art approaches for optimisation, individual motion estimation and registration. Three-view geometry algorithms in a convex optimisation framework are deployed on board the monocular vision system for each vehicle. In addition, vehicle-to-vehicle relative pose estimation is performed with a novel robust registration solution in a global optimisation framework. In parallel, and as a complementary solution for the relative pose, a robust non-linear H solution is designed as well to fuse measurements from the UAVs’ on-board inertial sensors with the visual estimates. The suggested contributions have been exhaustively evaluated over a number of real-image data experiments in the laboratory using monocular vision systems and range imaging devices. In this thesis, we propose several solutions towards the goal of robust visual motion estimation using convex optimisation. We show that the convex optimisation framework may be extended to include uncertainty information, to achieve robust and optimal solutions. We observed that convex optimisation is a practical and very appealing alternative to linear techniques and iterative methods

Campaigning Via LPs: Solving Blotto and Beyond

The competition between the Republican and the Democrat nominees in the U.S presidential election is known as Colonel Blotto in game theory. In the classical Colonel Blotto game -- introduced by Borel in 1921 -- two colonels simultaneously distribute their troops across multiple battlefields. The outcome of each battlefield is determined by a winner-take-all rule, independently of other battlefields. In the original formulation, the goal of each colonel is to win as many battlefields as possible. The Colonel Blotto game and its extensions have been used in a wide range of applications from political campaigns (exemplified by the U.S presidential election) to marketing campaigns, from (innovative) technology competitions, to sports competitions. For almost a century, there have been persistent efforts for finding the optimal strategies of the Colonel Blotto game, however it was left unanswered whether the optimal strategies are polynomially tractable. In this thesis, we present several algorithms for solving Blotto games in polynomial time and will discuss their applications in practice

Agent Interactions In Decentralized Environments

The decentralized Markov decision process (Dec-POMDP) is a powerful formal model for studying multiagent problems where cooperative, coordinated action is optimal, but each agent acts based on local data alone. Unfortunately, it is known that Dec-POMDPs are fundamentally intractable: they are NEXP-complete in the worst case, and have been empirically observed to be beyond feasible optimal solution. To get around these obstacles, researchers have focused on special classes of the general Dec-POMDP problem, restricting the degree to which agent actions can interact with one another. In some cases, it has been proven that these sorts of structured forms of interaction can in fact reduce worst-case complexity. Where formal proofs have been lacking, empirical observations suggest that this may also be true for other cases, although less is known precisely. This thesis unifies a range of this existing work, extending analysis to establish novel complexity results for some popular restricted-interaction models. We also establish some new results concerning cases for which reduced complexity has been proven, showing correspondences between basic structural features and the potential for dimensionality reduction when employing mathematical programming techniques. As our new complexity results establish that worst-case intractability is more widespread than previously known, we look to new ways of analyzing the potential average-case difficulty of Dec-POMDP instances. As this would be extremely difficult using the tools of traditional complexity theory, we take a more empirical approach. In so doing, we identify new analytical measures that apply to all Dec-POMDPs, whatever their structure. These measures allow us to identify problems that are potentially easier to solve on average, and validate this claim empirically. As we show, the performance of well-known optimal dynamic programming methods correlates with our new measure of difficulty. Finally, we explore the approximate case, showing that our measure works well as a predictor of difficulty there, too, and provides a means of setting algorithm parameters to achieve far more efficient performance

Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum