On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})$-approximation algorithm. When $\alpha \geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log^{1-\epsilon}n}$-approximation algorithm, improving upon the $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption)

Greedy δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy δ-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most δ variables of the problem. (A simple example is Vertex Cover, with δ=2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems. © 2012 Springer Science+Business Media, LLC

On the complexity of privacy-preserving complex event processing

Complex Event Processing (CEP) Systems are stream processing systems that monitor incoming event streams in search of user-specified event patterns. While CEP systems have been adopted in a variety of applications, the privacy implications of event pattern reporting mechanisms have yet to be studied — a stark contrast to the significant amount of attention that has been devoted to privacy for relational systems. In this paper we present a privacy problem that arises when the system must support desired patterns (those that should be reported if detected) and private patterns (those that should not be revealed). We formalize this problem, which we term privacy-preserving, utility maximizing CEP (PP-CEP), and analyze its complexity under various assumptions. Our results show that this is a rich problem to study and shed some light on the difficulty of developing algorithms that preserve utility without compromis-ing privacy

Greedy δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy δ-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most δ variables of the problem. (A simple example is Vertex Cover, with δ=2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems. © 2012 Springer Science+Business Media, LLC

Greedy Δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy Δ-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most Δ variables of the problem. (A simple example is VERTEX COVER, with Δ = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems

Greedy D-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems

Scalable optimization algorithms for recommender systems

Recommender systems have now gained significant popularity and been widely used in many e-commerce applications. Predicting user preferences is a key step to providing high quality recommendations. In practice, however, suggestions made to users must not only consider user preferences in isolation; a good recommendation engine also needs to account for certain constraints. For instance, an online video rental that suggests multimedia items (e.g., DVDs) to its customers should consider the availability of DVDs in stock to reduce customer waiting times for accepted recommendations. Moreover, every user should receive a small but sufficient number of suggestions that the user is likely to be interested in. This thesis aims to develop and implement scalable optimization algorithms that can be used (but are not restricted) to generate recommendations satisfying certain objectives and constraints like the ones above. State-of-the-art approaches lack efficiency and/or scalability in coping with large real-world instances, which may involve millions of users and items. First, we study large-scale matrix completion in the context of collaborative filtering in recommender systems. For such problems, we propose a set of novel shared-nothing algorithms which are designed to run on a small cluster of commodity nodes and outperform alternative approaches in terms of efficiency, scalability, and memory footprint. Next, we view our recommendation task as a generalized matching problem, and propose the first distributed solution for solving such problems at scale. Our algorithm is designed to run on a small cluster of commodity nodes (or in a MapReduce environment) and has strong approximation guarantees. Our matching algorithm relies on linear programming. To this end, we present an efficient distributed approximation algorithm for mixed packing-covering linear programs, a simple but expressive subclass of linear programs. Our approximation algorithm requires a poly-logarithmic number of passes over the input, is simple, and well-suited for parallel processing on GPUs, in shared-memory architectures, as well as on a small cluster of commodity nodes.Empfehlungssysteme haben eine beachtliche Popularität erreicht und werden in zahlreichen E-Commerce Anwendungen eingesetzt. Entscheidend für die Generierung hochqualitativer Empfehlungen ist die Vorhersage von Nutzerpräferenzen. Jedoch sollten in der Praxis nicht nur Vorschläge auf Basis von Nutzerpräferenzen gegeben werden, sondern gute Empfehlungssysteme müssen auch bestimmte Nebenbedingungen berücksichtigen. Zum Beispiel sollten online Videoverleihfirmen, welche ihren Kunden multimediale Produkte (z.B. DVDs) vorschlagen, die Verfügbarkeit von vorrätigen DVDs beachten, um die Wartezeit der Kunden für angenommene Empfehlungen zu reduzieren. Darüber hinaus sollte jeder Kunde eine kleine, aber ausreichende Anzahl an Vorschlägen erhalten, an denen er interessiert sein könnte. Diese Arbeit strebt an skalierbare Optimierungsalgorithmen zu entwickeln und zu implementieren, die (unter anderem) eingesetzt werden können Empfehlungen zu generieren, welche weitere Zielvorgaben und Restriktionen einhalten. Derzeit existierenden Ansätzen mangelt es an Effizienz und/oder Skalierbarkeit im Umgang mit sehr großen, durchaus realen Datensätzen von, beispielsweise Millionen von Nutzern und Produkten. Zunächst analysieren wir die Vervollständigung großskalierter Matrizen im Kontext von kollaborativen Filtern in Empfehlungssystemen. Für diese Probleme schlagen wir verschiedene neue, verteilte Algorithmen vor, welche konzipiert sind auf einer kleinen Anzahl von gängigen Rechnern zu laufen. Zudem können sie alternative Ansätze hinsichtlich der Effizienz, Skalierbarkeit und benötigten Speicherkapazität überragen. Als Nächstes haben wir die Empfehlungsproblematik als ein generalisiertes Zuordnungsproblem betrachtet und schlagen daher die erste verteilte Lösung für großskalierte Zuordnungsprobleme vor. Unser Algorithmus funktioniert auf einer kleinen Gruppe von gängigen Rechnern (oder in einem MapReduce-Programmierungsmodel) und erzielt gute Approximationsgarantien. Unser Zuordnungsalgorithmus beruht auf linearer Programmierung. Daher präsentieren wir einen effizienten, verteilten Approximationsalgorithmus für vermischte lineare Packungs- und Überdeckungsprobleme, eine einfache aber expressive Unterklasse der linearen Programmierung. Unser Algorithmus benötigt eine polylogarithmische Anzahl an Scans der Eingabedaten. Zudem ist er einfach und sehr gut geeignet für eine parallele Verarbeitung mithilfe von Grafikprozessoren, unter einer gemeinsam genutzten Speicherarchitektur sowie auf einer kleinen Gruppe von gängigen Rechnern

Linear Programming Tools and Approximation Algorithms for Combinatorial Optimization

We study techniques, approximation algorithms, structural properties and lower bounds related to applications of linear programs in combinatorial optimization. The following "Steiner tree problem" is central: given a graph with a distinguished subset of required vertices, and costs for each edge, find a minimum-cost subgraph that connects the required vertices. We also investigate the areas of network design, multicommodity flows, and packing/covering integer programs. All of these problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio. Overall, we show some new techniques that enhance the already-substantial corpus of LP-based approximation methods, and we also look for limitations of these techniques. The first half of the thesis deals with linear programming relaxations for the Steiner tree problem. The crux of our work deals with hypergraphic relaxations obtained via the well-known full component decomposition of Steiner trees; explicitly, in this view the fundamental building blocks are not edges, but hyperedges containing two or more required vertices. We introduce a new hypergraphic LP based on partitions. We show the new LP has the same value as several previously-studied hypergraphic ones; when no Steiner nodes are adjacent, we show that the value of the well-known bidirected cut relaxation is also the same. A new partition uncrossing technique is used to demonstrate these equivalences, and to show that extreme points of the new LP are well-structured. We improve the best known integrality gap on these LPs in some special cases. We show that several approximation algorithms from the literature on Steiner trees can be re-interpreted through linear programs, in particular our hypergraphic relaxation yields a new view of the Robins-Zelikovsky 1.55-approximation algorithm for the Steiner tree problem. The second half of the thesis deals with a variety of fundamental problems in combinatorial optimization. We show how to apply the iterated LP relaxation framework to the problem of multicommodity integral flow in a tree, to get an approximation ratio that is asymptotically optimal in terms of the minimum capacity. Iterated relaxation gives an infeasible solution, so we need to finesse it back to feasibility without losing too much value. Iterated LP relaxation similarly gives an O(k^2)-approximation algorithm for packing integer programs with at most k occurrences of each variable; new LP rounding techniques give a k-approximation algorithm for covering integer programs with at most k variable per constraint. We study extreme points of the standard LP relaxation for the traveling salesperson problem and show that they can be much more complex than was previously known. The k-edge-connected spanning multi-subgraph problem has the same LP and we prove a lower bound and conjecture an upper bound on the approximability of variants of this problem. Finally, we show that for packing/covering integer programs with a bounded number of constraints, for any epsilon > 0, there is an LP with integrality gap at most 1 + epsilon