Node-Weighted Prize Collecting Steiner Tree and Applications

The Steiner Tree problem has appeared in the Karp's list of the first 21 NP-hard problems and is well known as one of the most fundamental problems in Network Design area. We study the Node-Weighted version of the Prize Collecting Steiner Tree problem. In this problem, we are given a simple graph with a cost and penalty value associated with each node. Our goal is to find a subtree T of the graph minimizing the cost of the nodes in T plus penalty of the nodes not in T. By a reduction from set cover problem it can be easily shown that the problem cannot be approximated in polynomial time within factor of (1-o(1))ln n unless NP has quasi-polynomial time algorithms, where n is the number of vertices of the graph. Moss and Rabani claimed an O(log n)-approximation algorithm for the problem using a Primal-Dual approach in their STOC'01 paper \cite{moss2001}. We show that their algorithm is incorrect by providing a counter example in which there is an O(n) gap between the dual solution constructed by their algorithm and the optimal solution. Further, evidence is given that their algorithm probably does not have a simple fix. We propose a new algorithm which is more involved and introduces novel ideas in primal dual approach for network design problems. Also, our algorithm is a Lagrangian Multiplier Preserving algorithm and we show how this property can be utilized to design an O(log n)-approximation algorithm for the Node-Weighted Quota Steiner Tree problem using the Lagrangian Relaxation method. We also show an application of the Node Weighted Quota Steiner Tree problem in designing algorithm with better approximation factor for Technology Diffusion problem, a problem proposed by Goldberg and Liu in \cite{goldberg2012} (SODA 2013). In Technology Diffusion, we are given a graph G and a threshold θ(v) associated with each vertex v and we are seeking a set of initial nodes called the seed set. Technology Diffusion is a dynamic process defined over time in which each vertex is either active or inactive. The vertices in the seed set are initially activated and each other vertex v gets activated whenever there are at least θ(v) active nodes connected to v through other active nodes. The Technology Diffusion problem asks to find the minimum seed set activating all nodes. Goldberg and Liu gave an O(rllog n)-approximation algorithm for the problem where r and l are the diameter of G and the number of distinct threshold values, respectively. We improve the approximation factor to O(min{r,l}log n) by establishing a close connection between the problem and the Node Weighted Quota Steiner Tree problem

Design and analysis of an FPGA-based, multi-processor HW-SW system for SCC applications

The last 30 years have seen an increase in the complexity of embedded systems from a collection of simple circuits to systems consisting of multiple processors managing a wide variety of devices. This ever increasing complexity frequently requires that high assurance, fail-safe and secure design techniques be applied to protect against possible failures and breaches. To facilitate the implementation of these embedded systems in an efficient way, the FPGA industry recently created new families of devices. New features added to these devices include anti-tamper monitoring, bit stream encryption, and optimized routing architectures for physical and functional logic partition isolation. These devices have high capacities and are capable of implementing processors using their reprogrammable logic structures. This allows for an unprecedented level of hardware and software interaction within a single FPGA chip. High assurance and fail-safe systems can now be implemented within the reconfigurable hardware fabric of an FPGA, enabling these systems to maintain flexibility and achieve high performance while providing a high level of data security. The objective of this thesis was to design and analyze an FPGA-based system containing two isolated, softcore Nios processors that share data through two crypto-engines. FPGA-based single-chip cryptographic (SCC) techniques were employed to ensure proper component isolation when the design is placed on a device supporting the appropriate security primitives. Each crypto-engine is an implementation of the Advanced Encryption Standard (AES), operating in Galois/Counter Mode (GCM) for both encryption and authentication. The features of the microprocessors and architectures of the AES crypto-engines were varied with the goal of determining combinations which best target high performance, minimal hardware usage, or a combination of the two

On the Integrality Gap of Directed Steiner Tree Problem

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V, and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only achieve a guarantee of O(|X|^ε) for any constant ε > 0. Furthermore, the integrality gap of a natural LP relaxation can be as bad as Ω(√|X|).  We demonstrate that l rounds of the Sherali-Adams hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from Ω( k) to O(l · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2l rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(l · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap bound in the standard LP relaxation, complementing the fact that the gap can be as large as Ω(√k) in graphs with 4 layers. Finally, we consider quasi-bipartite instances of Directed Steiner Tree meaning no edge in E connects two Steiner nodes V − (X ∪ {r}). By a simple reduction from Set Cover, it is still NP-hard to approximate quasi-bipartite instances within a ratio better than O(log|X|). We present a polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree. Our approach also bounds the integrality gap of the natural LP relaxation by the same quantity. A novel feature of our algorithm is that it is based on the primal-dual framework, which typically does not result in good approximations for network design problems in directed graphs

Online node-weighted steiner forest and extensions via disk paintings

We give the first polynomial-time online algorithm for the node-weighted Steiner forest problem with a poly-logarithmic competitive ratio. The competitive ratio of our algorithm is optimal up to a logarithmic factor. For the special case of graphs with an excluded fixed minor (e.g., planar graphs), we obtain a logarithmic competitive ratio, which is optimal up to a constant, using a different online algorithm. Both these results are obtained as special cases of generic results for a large class of problems that can be encoded as online {0, 1}-proper functions. Our results are obtained by using a new framework for online network design problems that we call disk paintings. The central idea in this technique is to amortize the cost of primal updates to a set of carefully selected mutually disjoint fixed-radius dual disks centered at a subset of terminals. We hope that this framework will be useful for other online network design problems

Primal-Dual Techniques for Online Algorithms and Mechanisms

An offline algorithm is one that knows the entire input in advance. An online algorithm, however, processes its input in a serial fashion. In contrast to offline algorithms, an online algorithm works in a local fashion and has to make irrevocable decisions without having the entire input. Online algorithms are often not optimal since their irrevocable decisions may turn out to be inefficient after receiving the rest of the input. For a given online problem, the goal is to design algorithms which are competitive against the offline optimal solutions. In a classical offline scenario, it is often common to see a dual analysis of problems that can be formulated as a linear or convex program. Primal-dual and dual-fitting techniques have been successfully applied to many such problems. Unfortunately, the usual tricks come short in an online setting since an online algorithm should make decisions without knowing even the whole program. In this thesis, we study the competitive analysis of fundamental problems in the literature such as different variants of online matching and online Steiner connectivity, via online dual techniques. Although there are many generic tools for solving an optimization problem in the offline paradigm, in comparison, much less is known for tackling online problems. The main focus of this work is to design generic techniques for solving integral linear optimization problems where the solution space is restricted via a set of linear constraints. A general family of these problems are online packing/covering problems. Our work shows that for several seemingly unrelated problems, primal-dual techniques can be successfully applied as a unifying approach for analyzing these problems. We believe this leads to generic algorithmic frameworks for solving online problems. In the first part of the thesis, we show the effectiveness of our techniques in the stochastic settings and their applications in Bayesian mechanism design. In particular, we introduce new techniques for solving a fundamental linear optimization problem, namely, the stochastic generalized assignment problem (GAP). This packing problem generalizes various problems such as online matching, ad allocation, bin packing, etc. We furthermore show applications of such results in the mechanism design by introducing Prophet Secretary, a novel Bayesian model for online auctions. In the second part of the thesis, we focus on the covering problems. We develop the framework of "Disk Painting" for a general class of network design problems that can be characterized by proper functions. This class generalizes the node-weighted and edge-weighted variants of several well-known Steiner connectivity problems. We furthermore design a generic technique for solving the prize-collecting variants of these problems when there exists a dual analysis for the non-prize-collecting counterparts. Hence, we solve the online prize-collecting variants of several network design problems for the first time. Finally we focus on designing techniques for online problems with mixed packing/covering constraints. We initiate the study of degree-bounded graph optimization problems in the online setting by designing an online algorithm with a tight competitive ratio for the degree-bounded Steiner forest problem. We hope these techniques establishes a starting point for the analysis of the important class of online degree-bounded optimization on graphs

Bed properties and three-dimensional topography from radar at Rutford Ice Stream, West Antarctica

Outlet glaciers and ice streams of the Antarctic Ice Sheet provide dis-charge pathways, transporting >90% of the continents ice into the oceans. Elongated landforms beneath fast flowing ice streams form as a result of ice-bed interactions. Understanding their link to ice flow dynamics will better inform subglacial processes and allow these processes to be correctly implemented in predictive numerical flow models, thus improving predictions of future contributions to sea level rise. In this thesis, a section of the bed of Rutford Ice Stream (West Antarctica), containing numerous elongated subglacial landforms, was analysed using a suite of 2D and 3D radar data with repeat surveys. Bed properties vary spatially over a 100 m scale and imply the pat-tern of inferred basal motion in this area is more complex, and basal sliding dominated areas are more extensive, than previously assumed. Local erosion rates are high (1 m/a), indicating a mobile bed, whereas most of the bed shows no temporal change, implying stability of the basal environment. Observations of landforms shortening, and previ-ous observations of landforms extending, highlight that landforms are a non-static part of the bed. Isolated landforms appear to consist of a more rigid sediment at their upstream end with softer sediment down-stream. Some landforms contain a water body (up to 10 km length) along their crest. 3D processed data reveal a so far unseen moat (de-pression) around one landform. Dimensions of the upstream part of the moat are comparable to dimensions of the upstream end of the landform (<50 m height, <300 m width). Observations suggest land-forms are depositional features, while the moat was likely eroded. The radar and other data analysed provide detailed landform and moat ar-chitecture, at a resolution comparable to digital elevation models of deglaciated terrain, and together with interpreted properties give a solid basis for testing existing landform formation theories

Application of advanced on-board processing concepts to future satellite communications systems

An initial definition of on-board processing requirements for an advanced satellite communications system to service domestic markets in the 1990's is presented. An exemplar system architecture with both RF on-board switching and demodulation/remodulation baseband processing was used to identify important issues related to system implementation, cost, and technology development

Microstimulation and multicellular analysis: A neural interfacing system for spatiotemporal stimulation

Willfully controlling the focus of an extracellular stimulus remains a significant challenge in the development of neural prosthetics and therapeutic devices. In part, this challenge is due to the vast set of complex interactions between the electric fields induced by the microelectrodes and the complex morphologies and dynamics of the neural tissue. Overcoming such issues to produce methodologies for targeted neural stimulation requires a system that is capable of (1) delivering precise, localized stimuli a function of the stimulating electrodes and (2) recording the locations and magnitudes of the resulting evoked responses a function of the cell geometry and membrane dynamics. In order to improve stimulus delivery, we developed microfabrication technologies that could specify the electrode geometry and electrical properties. Specifically, we developed a closed-loop electroplating strategy to monitor and control the morphology of surface coatings during deposition, and we implemented pulse-plating techniques as a means to produce robust, resilient microelectrodes that could withstand rigorous handling and harsh environments. In order to evaluate the responses evoked by these stimulating electrodes, we developed microscopy techniques and signal processing algorithms that could automatically identify and evaluate the electrical response of each individual neuron. Finally, by applying this simultaneous stimulation and optical recording system to the study of dissociated cortical cultures in multielectode arrays, we could evaluate the efficacy of excitatory and inhibitory waveforms. Although we found that the proximity of the electrode is a poor predictor of individual neural excitation thresholds, we have shown that it is possible to use inhibitory waveforms to globally reduce excitability in the vicinity of the electrode. Thus, the developed system was able to provide very high resolution insight into the complex set of interactions between the stimulating electrodes and populations of individual neurons.Ph.D.Committee Chair: Stephen P. DeWeerth; Committee Member: Bruce Wheeler; Committee Member: Michelle LaPlaca; Committee Member: Robert Lee; Committee Member: Steve Potte