On Tree-Partition-Width

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has tree-partition-width at most $24k\Delta$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum degree $\Delta$, and tree-partition-width at least (\eighth-\epsilon)k\Delta. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$ without the restriction that $k\geq3$

Popularity versus Similarity in Growing Networks

Popularity is attractive -- this is the formula underlying preferential attachment, a popular explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections that nodes have follows power laws observed in many real networks. Preferential attachment has been directly validated for some real networks, including the Internet. Preferential attachment can also be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks, or duplication. Here we show that popularity is just one dimension of attractiveness. Another dimension is similarity. We develop a framework where new connections, instead of preferring popular nodes, optimize certain trade-offs between popularity and similarity. The framework admits a geometric interpretation, in which popularity preference emerges from local optimization. As opposed to preferential attachment, the optimization framework accurately describes large-scale evolution of technological (Internet), social (web of trust), and biological (E.coli metabolic) networks, predicting the probability of new links in them with a remarkable precision. The developed framework can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon

Tree-Partitions with Small Bounded Degree Trees

A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size $O(k\Delta)$. We prove the same result with the extra property that the underlying tree has maximum degree $O(\Delta)$ and $O(|V(G)|/k)$ vertices

Making Networks Robust to Component Failures

In this thesis, we consider instances of component failure in the Internet and in networked cyber-physical systems, such as the communication network used by the modern electric power grid (termed the smart grid). We design algorithms that make these networks more robust to various component failures, including failed routers, failures of links connecting routers, and failed sensors. This thesis divides into three parts: recovery from malicious or misconfigured nodes injecting false information into a distributed system (e.g., the Internet), placing smart grid sensors to provide measurement error detection, and fast recovery from link failures in a smart grid communication network. First, we consider the problem of malicious or misconfigured nodes that inject and spread incorrect state throughout a distributed system. Such false state can degrade the performance of a distributed system or render it unusable. For example, in the case of network routing algorithms, false state corresponding to a node incorrectly declaring a cost of 0 to all destinations (maliciously or due to misconfiguration) can quickly spread through the network. This causes other nodes to (incorrectly) route via the misconfigured node, resulting in suboptimal routing and network congestion. We propose three algorithms for efficient recovery in such scenarios and evaluate their efficacy. The last two parts of this thesis consider robustness in the context of the electric power grid. We study the use and placement of a sensor, called a Phasor Measurement Unit (PMU), currently being deployed in electric power grids worldwide. PMUs provide voltage and current measurements at a sampling rate orders of magnitude higher than the status quo. As a result, PMUs can both drastically improve existing power grid operations and enable an entirely new set of applications, such as the reliable integration of renewable energy resources. However, PMU applications require correct (addressed in thesis part 2) and timely(covered in thesis part 3) PMU data. Without these guarantees, smart grid operators and applications may make incorrect decisions and take corresponding (incorrect) actions. The second part of this thesis addresses PMU measurement errors, which have been observed in practice. We formulate a set of PMU placement problems that aim to satisfy two constraints: place PMUs near each other to allow for measurement error detection and use the minimal number of PMUs to infer the state of the maximum number of system buses and transmission lines. For each PMU placement problem, we prove it is NP-Complete, propose a simple greedy approximation algorithm, and evaluate our greedy solutions. In the last part of this thesis, we design algorithms for fast recovery from link failures in a smart grid communication network. We propose, design, and evaluate solutions to all three aspects of link failure recovery: (a) link failure detection, (b) algorithms for pre-computing backup multicast trees, and (c) fast backup tree installation. To address (a), we design link-failure detection and reporting mechanisms that use OpenFlow to detect link failures when and where they occur inside the network. OpenFlow is an open source framework that cleanly separates the control and data planes for use in network management and control. For part (b), we formulate a new problem, Multicast Recycling, that pre-computes backup multicast trees that aim to minimize control plane signaling overhead. We prove Multicast Recycling is at least NP-hard and present a corresponding approximation algorithm. Lastly, two control plane algorithms are proposed that signal data plane switches to install pre-computed backup trees. An optimized version of each installation algorithm is designed that finds a near minimum set of forwarding rules by sharing forwarding rules across multicast groups. This optimization reduces backup tree install time and associated control state. We implement these algorithms using the POX open-source OpenFlow controller and evaluate them using the Mininet emulator, quantifying control plane signaling and installation time

Product structure of graph classes with strongly sublinear separators

We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class $\mathcal{G}$ admits $O(n^{1-\epsilon})$ separators, then for any fixed $\delta\in(0,\epsilon)$ every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-depth and a complete graph of size $O(n^{1-\epsilon+\delta})$. This result holds with $\delta=0$ if we allow $H$ to have tree-depth $O(\log\log n)$. Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on $\delta$ in our results and the above $\text{td}(H)\in O(\log\log n)$ bound are both best possible. We prove that $n$-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth $t$ and a complete graph of size $O(n^{1/t})$, which is best possible. Finally, we investigate the conjecture that for any hereditary graph class $\mathcal{G}$ that admits $O(n^{1-\epsilon})$ separators, every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-width and a complete graph of size $O(n^{1-\epsilon})$. We prove this for various classes $\mathcal{G}$ of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial Expansion Classes" which had an error, added section "Lower Bounds", and added a new author; v4: minor revisions and corrections

Synthesis, structure and power of systolic computations

AbstractA variety of problems related to systolic architectures, systems, models and computations are discussed. The emphases are on theoretical problems of a broader interest. Main motivations and interesting/important applications are also presented. The first part is devoted to problems related to synthesis, transformations and simulations of systolic systems and architectures. In the second part, the power and structure of tree and linear array computations are studied in detail. The goal is to survey main research directions, problems, methods and techniques in not too formal a way