Extensions and limits to vertex sparsification

Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, E[subscript H]) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there exist flow sparsifiers that simultaneously preserve the congestion of all multicommodity flows within an O(log k / log log k)-factor where |K| = k. This bound improves to O(1) if G excludes any fixed minor. This is a strengthening of previous results, which consider the problem of finding a graph H = (K, E[subscript H]) (a cut sparsifier) that approximately preserves the value of minimum cuts separating any partition of the terminals. Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers). Thereby, we immediately improve all approximation ratios derived using vertex sparsification in [14]. We also prove an Ω(log log k) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. The proof of this theorem relies on a technique (which we refer to as oblivious dual certifcates) for proving super-constant congestion lower bounds against many multicommodity flows at once. Our result implies that approximation algorithms for multicommodity flow-type problems designed by a black box reduction to a "uniform" case on k nodes (see [14] for examples) must incur a super-constant cost in the approximation ratio

On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter. The second part of our work deals with providing a lower bound to Courcelle's famous theorem, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Using our results from the first part of our work we establish a strong lower bound for tractability of MSO on classes of colored graphs

An Algorithm for the Graph Crossing Number Problem

We study the Minimum Crossing Number problem: given an $n$-vertex graph $G$, the goal is to find a drawing of $G$ in the plane with minimum number of edge crossings. This is one of the central problems in topological graph theory, that has been studied extensively over the past three decades. The first non-trivial efficient algorithm for the problem, due to Leighton and Rao, achieved an $O(n\log^4n)$-approximation for bounded degree graphs. This algorithm has since been improved by poly-logarithmic factors, with the best current approximation ratio standing on O(n \poly(d) \log^{3/2}n) for graphs with maximum degree $d$. In contrast, only APX-hardness is known on the negative side. In this paper we present an efficient randomized algorithm to find a drawing of any $n$-vertex graph $G$ in the plane with O(OPT^{10}\cdot \poly(d \log n)) crossings, where $OPT$ is the number of crossings in the optimal solution, and $d$ is the maximum vertex degree in $G$. This result implies an \tilde{O}(n^{9/10} \poly(d))-approximation for Minimum Crossing Number, thus breaking the long-standing $\tilde{O}(n)$-approximation barrier for bounded-degree graphs

Decomposition of sequential and concurrent models

Le macchine a stati finiti (FSM), sistemi di transizioni (TS) e le reti di Petri (PN) sono importanti modelli formali per la progettazione di sistemi. Un problema fodamentale è la conversione da un modello all'altro. Questa tesi esplora il mondo delle reti di Petri e della decomposizione di sistemi di transizioni. Per quanto riguarda la decomposizione dei sistemi di transizioni, la teoria delle regioni rappresenta la colonna portante dell'intero processo di decomposizione, mirato soprattutto a decomposizioni che utilizzano due sottoclassi delle reti di Petri: macchine a stati e reti di Petri a scelta libera. Nella tesi si dimostra che una proprietà chiamata ``chiusura rispetto all'eccitazione" (excitation-closure) è sufficiente per produrre un insieme di reti di Petri la cui sincronizzazione è bisimile al sistema di transizioni (o rete di Petri di partenza, se la decomposizione parte da una rete di Petri), dimostrando costruttivamente l'esistenza di una bisimulazione. Inoltre, è stato implementato un software che esegue la decomposizione dei sistemi di transizioni, per rafforzare i risultati teorici con dati sperimentali sistematici. Nella seconda parte della dissertazione si analizza un nuovo modello chiamato MSFSM, che rappresenta un insieme di FSM sincronizzate da due primitive specifiche (Wait State - Stato d'Attesa e Transition Barrier - Barriera di Transizione). Tale modello trova un utilizzo significativo nella sintesi di circuiti sincroni a partire da reti di Petri a scelta libera. In particolare vengono identificati degli errori nell'approccio originale, fornendo delle correzioni.Finite State Machines (FSMs), transition systems (TSs) and Petri nets (PNs) are important models of computation ubiquitous in formal methods for modeling systems. Important problems involve the transition from one model to another. This thesis explores Petri nets, transition systems and Finite State Machines decomposition and optimization. The first part addresses decomposition of transition systems and Petri nets, based on the theory of regions, representing them by means of restricted PNs, e.g., State Machines (SMs) and Free-choice Petri nets (FCPNs). We show that the property called ``excitation-closure" is sufficient to produce a set of synchronized Petri nets bisimilar to the original transition system or to the initial Petri net (if the decomposition starts from a PN), proving by construction the existence of a bisimulation. Furthermore, we implemented a software performing the decomposition of transition systems, and reported extensive experiments. The second part of the dissertation discusses Multiple Synchronized Finite State Machines (MSFSMs) specifying a set of FSMs synchronized by specific primitives: Wait State and Transition Barrier. It introduces a method for converting Petri nets into synchronous circuits using MSFSM, identifies errors in the initial approach, and provides corrections

Algorithms for Graph Connectivity and Cut Problems - Connectivity Augmentation, All-Pairs Minimum Cut, and Cut-Based Clustering

We address a collection of related connectivity and cut problems in simple graphs that reach from the augmentation of planar graphs to be k-regular and c-connected to new data structures representing minimum separating cuts and algorithms that smoothly maintain Gomory-Hu trees in evolving graphs, and finally to an analysis of the cut-based clustering approach of Flake et al. and its adaption to dynamic scenarios

Doctor of Philosophy

dissertationRecent breakthroughs in silicon photonics technology are enabling the integration of optical devices into silicon-based semiconductor processes. Photonics technology enables high-speed, high-bandwidth, and high-fidelity communications on the chip-scale-an important development in an increasingly communications-oriented semiconductor world. Significant developments in silicon photonic manufacturing and integration are also enabling investigations into applications beyond that of traditional telecom: sensing, filtering, signal processing, quantum technology-and even optical computing. In effect, we are now seeing a convergence of communications and computation, where the traditional roles of optics and microelectronics are becoming blurred. As the applications for opto-electronic integrated circuits (OEICs) are developed, and manufacturing capabilities expand, design support is necessary to fully exploit the potential of this optics technology. Such design support for moving beyond custom-design to automated synthesis and optimization is not well developed. Scalability requires abstractions, which in turn enables and requires the use of optimization algorithms and design methodology flows. Design automation represents an opportunity to take OEIC design to a larger scale, facilitating design-space exploration, and laying the foundation for current and future optical applications-thus fully realizing the potential of this technology. This dissertation proposes design automation for integrated optic system design. Using a buildingblock model for optical devices, we provide an EDA-inspired design flow and methodologies for optical design automation. Underlying these flows and methodologies are new supporting techniques in behavioral and physical synthesis, as well as device-resynthesis techniques for thermal-aware system integration. We also provide modeling for optical devices and determine optimization and constraint parameters that guide the automation techniques. Our techniques and methodologies are then applied to the design and optimization of optical circuits and devices. Experimental results are analyzed to evaluate their efficacy. We conclude with discussions on the contributions and limitations of the approaches in the context of optical design automation, and describe the tremendous opportunities for future research in design automation for integrated optics

Enabling Resilience in Cyber-Physical-Human Water Infrastructures

Rapid urbanization and growth in urban populations have forced community-scale infrastructures (e.g., water, power and natural gas distribution systems, and transportation networks) to operate at their limits. Aging (and failing) infrastructures around the world are becoming increasingly vulnerable to operational degradation, extreme weather, natural disasters and cyber attacks/failures. These trends have wide-ranging socioeconomic consequences and raise public safety concerns. In this thesis, we introduce the notion of cyber-physical-human infrastructures (CPHIs) - smart community-scale infrastructures that bridge technologies with physical infrastructures and people. CPHIs are highly dynamic stochastic systems characterized by complex physical models that exhibit regionwide variability and uncertainty under disruptions. Failures in these distributed settings tend to be difficult to predict and estimate, and expensive to repair. Real-time fault identification is crucial to ensure continuity of lifeline services to customers at adequate levels of quality. Emerging smart community technologies have the potential to transform our failing infrastructures into robust and resilient future CPHIs.In this thesis, we explore one such CPHI - community water infrastructures. Current urban water infrastructures, that are decades (sometimes over a 100 years) old, encompass diverse geophysical regimes. Water stress concerns include the scarcity of supply and an increase in demand due to urbanization. Deterioration and damage to the infrastructure can disrupt water service; contamination events can result in economic and public health consequences. Unfortunately, little investment has gone into modernizing this key lifeline.To enhance the resilience of water systems, we propose an integrated middleware framework for quick and accurate identification of failures in complex water networks that exhibit uncertain behavior. Our proposed approach integrates IoT-based sensing, domain-specific models and simulations with machine learning methods to identify failures (pipe breaks, contamination events). The composition of techniques results in cost-accuracy-latency tradeoffs in fault identification, inherent in CPHIs due to the constraints imposed by cyber components, physical mechanics and human operators. Three key resilience problems are addressed in this thesis; isolation of multiple faults under a small number of failures, state estimation of the water systems under extreme events such as earthquakes, and contaminant source identification in water networks using human-in-the-loop based sensing. By working with real world water agencies (WSSC, DC and LADWP, LA), we first develop an understanding of operations of water CPHI systems. We design and implement a sensor-simulation-data integration framework AquaSCALE, and apply it to localize multiple concurrent pipe failures. We use a mixture of infrastructure measurements (i.e., historical and live water pressure/flow), environmental data (i.e., weather) and human inputs (i.e., twitter feeds), combined and enhanced with the domain model and supervised learning techniques to locate multiple failures at fine levels of granularity (individual pipeline level) with detection time reduced by orders of magnitude (from hours/days to minutes). We next consider the resilience of water infrastructures under extreme events (i.e., earthquakes) - the challenge here is the lack of apriori knowledge and the increased number and severity of damages to infrastructures. We present a graphical model based approach for efficient online state estimation, where the offline graph factorization partitions a given network into disjoint subgraphs, and the belief propagation based inference is executed on-the-fly in a distributed manner on those subgraphs. Our proposed approach can isolate 80% broken pipes and 99% loss-of-service to end-users during an earthquake.Finally, we address issues of water quality - today this is a human-in-the-loop process where operators need to gather water samples for lab tests. We incorporate the necessary abstractions with event processing methods into a workflow, which iteratively selects and refines the set of potential failure points via human-driven grab sampling. Our approach utilizes Hidden Markov Model based representations for event inference, along with reinforcement learning methods for further refining event locations and reducing the cost of human efforts.The proposed techniques are integrated into a middleware architecture, which enables components to communicate/collaborate with one another. We validate our approaches through a prototype implementation with multiple real-world water networks, supply-demand patterns from water utilities and policies set by the U.S. EPA. While our focus here is on water infrastructures in a community, the developed end-to-end solution is applicable to other infrastructures and community services which operate in disruptive and resource-constrained environments

Vertex sparsification and universal rounding algorithms

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 125-129).Suppose we are given a gigantic communication network, but are only interested in a small number of nodes (clients). There are many routing problems we could be asked to solve for our clients. Is there a much smaller network - that we could write down on a sheet of paper and put in our pocket - that approximately preserves all the relevant communication properties of the original network? As we will demonstrate, the answer to this question is YES, and we call this smaller network a vertex sparsifier. In fact, if we are asked to solve a sequence of optimization problems characterized by cuts or flows, we can compute a good vertex sparsifier ONCE and discard the original network. We can run our algorithms (or approximation algorithms) on the vertex sparsifier as a proxy - and still recover approximately optimal solutions in the original network. This novel pattern saves both space (because the network we store is much smaller) and time (because our algorithms run on a much smaller graph). Additionally, we apply these ideas to obtain a master theorem for graph partitioning problems - as long as the integrality gap of a standard linear programming relaxation is bounded on trees, then the integrality gap is at most a logarithmic factor larger for general networks. This result implies optimal bounds for many well studied graph partitioning problems as a special case, and even yields optimal bounds for more challenging problems that had not been studied before. Morally, these results are all based on the idea that even though the structure of optimal solutions can be quite complicated, these solution values can be approximated by crude (even linear) functions.by Ankur Moitra.Ph.D