Temperature-aware processor frequency assignment for MPSoCs using convex optimization The increasing processing capability of Multi-Processor Systemson-Chips (MPSoCs) is leading to an increase in chip power dissipation, which in turn leads to significant increase in chip temperature. An important challenge facing the MPSoC designers is to achieve the highest performance system operation that satisfies the temperature and power consumption constraints. The frequency of operation of the different processors and the application workload assignment play a critical role in determining the performance, power consumption and temperature profile of the MPSoC. In this paper, we propose novel convex optimization based methods that solve this important problem of temperature-aware processor frequency assignment, such that the total system performance is maximized and the temperature and power constraints are met. We perform experiments on several realistic SoC benchmarks using a cycle-accurate FPGA-based thermal emulation platform, which show that the systems designed using our methods meet the temperature and power consumption requirements at all time instances, while achieving maximum performance. Business analysts forecast a 200 billion dollar market for mediarich, mobile System-on-Chip (SoC) terminals in the near future [1] . These forthcoming SoCs will be composed in future technology nodes of multiple homogeneous and/or heterogeneous processors, namely Multi-Processor System-on-Chip (MPSoC) platforms.. However, the semiconductor industry is still facing several technological challenges to build these systems. In the near future, power dissipation and consequent thermal implications will be often the dominant restriction to MPSoC performance and a significant element in overall cost. In fact, prohibitively expensive packaging, heat sinks and additional cooling solutions at the physical level will be necessary to handle the processor temperatures in next technology nodes [2] . Localized temperature increase can produce transient reduction in overall system performance, unreliable timing delay variations or even permanent damages in the devices [3] . To achieve a reliable and efficient system operation, the MPSoCs need to operate below a maximum temperature value and power budget. Determining the operating frequencies of the different processors of the MPSoCs, such that the performance of the system is maximized, while satisfying the temperature and power budget constraints, is a challenging task.. In this paper, we present novel design methods based on convex optimization to solve this critical problem of determining the operating frequencies of the processors, such that the system performance is maximized and the temperature and power budget constraints are satisfied at all time instances of operation. We integrate accurate thermal models of MPSoCs, which have been validated against 3D finite-element analysis for estimating the heat flow of the chip (presented in [29] ), in a mathematical optimization framework. We present methods that can be applied to two different types of systems: one that has support for Dynamic Voltage and Frequency Scaling (DVFS), where each processor's frequency can be varied over time, and the other where no DVFS is applied. As both types of MPSoCs exist today, we address the design of both systems.. We validate the solutions produced by the methods by integrating them onto a cycle-accurate FPGA-based thermal emulation platform [28] and perform experiments on several realistic SoC benchmarks. We extract the real temperature measurements from the platform, which show that the system always operates below the maximum temperature constraint. We also present ways in which our methods can be used to perform design space exploration stud-ies, such as detailed trade-offs between performance and temperature constraints for different working conditions, which can be used very early in the system design flow.. We have the following realistic assumptions for the design process: (1) We assume that the goal of our optimization methods is to maximize the total amount of work done by the processors. During system design, this problem is commonly encountered in practice. As an example, when designing chip multi-processors, an important performance goal is to maximize the total amount of instructions that can be executed in certain time [27] . This workload maximization objective translates to maximizing the sum of the frequencies of the different processors over time, as a faster processor can execute more instructions in a given time interval. We also solve a problem variation, where the application workload is fixed, and the objective is to minimize system power consumption. (2) Once the processor frequencies are set based on our methods, we assume that the actual application tasks are mapped onto the processors with compiler or OS support, as done widely in MPSoCs that have multiple processing elements [27] . Forthcoming MPSoCs can provide a very high processing power for embedded devices. However, one of the major design challenges is that the power dissipation of MPSoCs is rapidly increasing, which leads to large increase in chip temperature. This can severely affect system performance, and even produce permanent damages in the device in extreme cases. Achieving a highperformance operation, satisfying temperature and power constraints is critical for MPSoCs. In this paper, we have presented novel design approaches to perform optimal thermal-aware frequency assignment in MPSoCs, based on convex optimization. Our methods guarantee that the chip temperature remains below the maximum threshold at all time instances of operation. Our experimental results show that, even in very complex MPSoCs, our approach enables to setup optimal frequencies for each processor, as well as provides designers with a consistent framework to perform studies of trade-offs between performance and temperature very early in the design flow. The performance-power optimization of processors using DVFS policies has received considerable attention [4] - [6] . However, most of these works do not consider temperature constraints of the system. In the temperature-aware design domain, two important subproblems have been considered: modeling the thermal behavior of the system and designing methods to control processor performance and power consumption to meet the temperature constraints. Several recent works have addressed the issue of on-chip thermal modeling [8] - [16] . At the physical level, various methods can be used to model the heat transfer in the substrate. Finite-difference time domain [8] , finite element [10] , model reduction [11] , random walk [12, 13] and Green-function [14] based algorithms have been applied for on-chip thermal analysis. At the architectural level, [15] presents a thermal/power model for super-scalar architectures. The work presented in [16] investigates the impact of temperature and voltage variations across the die of embedded cores.. Based on these and other similar models, Dynamic Thermal Management (DTM) techniques have been suggested [17] - [20] . In [17] , the use of the feedback control theory is proposed as a way to implement adaptive techniques in the processor architecture. In [18] , a predictive frame-based DTM algorithm is presented. The authors in [20] perform extensive studies on empirical DPM techniques for thermal management. Their results show that DVFS can be very inefficient if the invocation time is not set appropriately. In [21] - [24] , temperature-aware floorplanning is used to place circuit blocks, such that an even thermal profile is obtained. System-level solutions have been defined to reduce the temperature in MPSoCs using different scheduling mechanisms [25, 26] . Finally, in [27] , a detailed review of thermal management techniques for multi-core architectures is presented.. Most of these existing thermal management techniques are based on monitoring and tuning of processor frequencies that do not result in optimal solutions. Moreover, they do not provide a guarantee that the temperature constraints will be satisfied at all instances of operation, which is critical for achieving system reliability.

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Combining constructive and equational geometric constraint-solving techniques In the past few years, there has been a strong trend towards developing parametric, computer-aided design systems based on geometric constraint solving. An effective way to capture the design intent in these systems is to define relationships between geometric and technological variables. In general, geometric constraint solving including functional relationships requires a general approach and appropriate techniques to achieve the expected functional capabilities. This work reports on a hybrid method that combines two geometric constraint solving techniques: constructive and equational. The hybrid solver has the capability of managing functional relationships between dimension variables and variables representing conditions external to the geometric problem. The hybrid solver is described as a rewriting system and is shown to be correct. In design and manufacturing applications, users of computer-aided design systems are interested in defining functional relationships between dimension parameters, since such relationships express design intent very flexibly. Some relationships between parameters can be implemented by structuring the sketch appropriately [Fudos 1993 ]. Moreover, simple functional relationships are the content of certain geometry theorems, such as the theorems of proportionality and many other classical results. Such theo-rems can be added to the constraint solver to extend its analysis capabilities. But in general, geometric constraint solving including functional relationships between dimension parameters requires a more general approach and requires appropriate techniques and tools to achieve those functional capabilities that users expect.. For surveys of the literature on geometric constraint solving see, for example, Anderl and Mendgen [1995] , Fudos [1993] , Henderson [1996] , and Hsu [1996] . Briefly, the problem of integrating functional relationships and geometric constraints has been attempted by mapping both problems into a common representational domain. In particular, we can map both problems to a system of nonlinear equations and then solve the system. The technique used to solve the system of equations can be based on graph decomposition, [Ait-Aoudia et al. 1993; Lamure and Michelucci 1998 ]. The approach reported in Dufourd et al. [1997] and Mathis et al. [1998] sets up a formal system of equations that is solved in a second interpretative phase using a Newton-Raphson method. Geometric constraint problems considered by the authors do not include functional relationships between dimension parameters.. Some authors analyze the equations using propagation techniques [Henderson 1996; Hsu 1996; Rosendahl and Berling 1998 ]. Broadly speaking, this mapping approach risks losing domain-specific information associated with the geometric constraint problem. Moreover, the decomposition and analysis techniques assume that the resulting equations are independent, and this assumption may be violated on the geometry side where overconstrained problems may arise [Bouma et al. 1995; Joan-Arinyo and SotoRiera 1997b] .. This work reports on a technique we have developed to deal with geometric constraint solving problems that include functional relationships defined between dimension variables and variables representing properties external to the geometric problem [Soto-Riera 1998] . The approach we propose in this article combines native constructive geometric constraint solving techniques with algebraic equation solving without mapping one problem domain to the other. Rather, we propose combining a constructive, rule-based geometric constraint solver with a solver for functional relationships. Both solvers should remain largely independent of each other and proceed incrementally in parallel, with progress of each solver being posted to the other as the problem solution unfolds. This approach improves the work reported in Hoffmann and Joan-Arinyo [1997] in two ways. The hybrid solver is organized in a uniform way as a rewriting system. As a result, there is no need of a specific algorithm to control the information flow in the solver. In addition, we introduce a new concept in the analysis of the functional relationships that we call restriction of a bigraph. This makes it possible to subsume in just one rule all the rules dealing with equational analysis. Furthermore, it permits extending the solver scope to consider certain classes of problems involving geometric elements with more than two degrees of freedom, for instance, circular arcs and circles with unknown radii, without extending the repertoire of rules.. The contents of the article are organized as follows. Section 2 provides basic definitions. Section 3 briefly describes the constructive solver. In Section 4 first we recall some fundamental concepts about bigraphs in connection with equation system solving that we use. Then we present our approach to equation analysis, and define the hybrid solver as a rewriting system. Section 5 develops a simple case study to illustrate how our approach works. The correctness is proved in Section 6. Finally, Section 7 summarizes the work. We have presented a hybrid technique for solving geometric constraint problems with the capability of managing problems involving functional relationships between dimensions and external variables. The method combines constructive and equational approaches and is especially effective for constructive geometric constraint solvers that use the rewriting rule paradigm.. Our technique works on two sets of data, the geometric constraint data and the symbolic equation data. Geometric data are represented by a set of constraint sets. Symbolic equations are represented by a bigraph. The information flow between these two structures is managed by a new rewriting rule: the -reduction rule. This rule valuates symbolic constraints and external variables by solving a subset of constraint equations from the bigraph and adds the resulting valuated constraints to the set of constraint sets.. It has been shown that when the constraint problem is not overconstrained, the method is correct. That is, if there is a sequence of rewriting steps that reduces the constraint graph to a single CD set and the bigraph Constraint Solving Techniques • to the empty graph, then the method finds such a sequence. Since the rules can be applied in any order, strategies to conveniently optimize the analysis process can be devised.

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Deeper model endgame analysis A reference model of Fallible Endgame Play has been implemented and exercised with the chessengine WILHELM. Past experiments have demonstrated the value of the model and the robustness of decisions based on it: experiments agree well with a Markov Model theory. Here, the reference model is exercised on the well-known endgame KBBKN. In previous papers [6, 7] , a reference model of fallible endgame play has been defined in terms of a spectrum of Reference Endgame Players (REPs) R c . The REPs are defined as choosing their moves stochastically from an endgame table (EGT), using only the values and depths of successor positions.. Here, we survey and compare existing experimental and theoretical results, and report on the latest findings with the familiar, complex endgame KBBKN. In Section 2, we revisit the basic concepts and theory of the REP model, while in Section 3, we describe the REP implementation in WILHELM [1] . In Sections 4-7, we review past experiments, compare experiment and theory, and introduce the KBBKN results. Section 8 summarises and notes some questions arising from this work.

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Reliability and performance-based design by artificial neural network Whilst conventional approach in structural design is based on reliability-calibrated factored design formula, performance-based design customizes a solution to the specific circumstance. In this work, an artificial neural network approach is employed to determine implicit limit state functions for reliability evaluations in performance-based design and to optimally evaluate a set of design variables under specified performance criteria and corresponding desired reliability levels in design. Case examples are shown for reliability design. Through the establishment of the response and reliability databases, for specified target reliabilities, structural response computations are integrated with the evaluation of design parameters and design can be accomplished. By employing this methodology, with the same performance requirements, pertinent design parameters can be altered in order to evaluate feasible design alternatives, to explore the usage of various structural materials and to define required material quality control. Recent catastrophic structural failures due to earthquakes and cyclones have uncovered the limitations of contemporary design practices and revealed the requirements for novel concepts and methodologies for structural performance evaluation and design. One of the key issues in evaluation and design process is a proper consideration and handling of the large uncertainty in both the loadings and the complex structural behaviour, in particular for those within the non-linear range (Wen, 2001 ).. With the above background, it is strongly believed that the engineering profession entails new paradigms in order to furnish the best potential service to society compatible to the advent of the computer era. In fact, the previous transition from Allowable Stress Design to Partial Factors Method (PFM) has contributed partly to this development. Although PFM can be regarded semi-probabilistic with respect to load and resistance factors, it is still deterministic from the designer's point of view. The recent advancement and development of computer technology provides a good chance of the transition from the PFM to a structural reliability assessment concept based on a fully probabilistic approach. Besides, nowadays the emergence and wide popularity of powerful desktop computer with high operating speed renders the development of efficient customized design within reach.. In previous research on performance-based design, the probability of failure is usually evaluated by importance sampling simulation around a design point acquired through an approximate analysis using a first order response surface and FORM (Melchers, 1999) . The evaluation of the performance function is usually performed through a local interpolation procedure. It may be difficult to develop a global response surface over the entire variable domain in many problems, especially in non-linear structural dynamics. For instance, it may not be possible to represent peaks and valleys representing resonance at different frequencies with sufficient accuracy. As a developing and promising technology, the capability of an artificial neural network to cope with uncertainty in complex situations, has been seized upon for wide ranging applications in recent years (Rumelhart, Hinton & Williams, 1986; Garrett, 1994; Chau & Cheng, 2002; Chau, 2004a Chau, , 2004b . With the recent advances in artificial intelligence (AI) technology, it is possible to develop an approximating limit state function or response surface, which can fulfil the above purpose, with a neural network approach.. This work delineates an approach to reliability estimation, which integrates a response surface methodology and artificial neural network. The major contribution of this study is mainly on the application of artificial neural network on the inverse performance-based design to determine design parameters that satisfy reliability constraints, which has not been attempted previously by other researchers. In this work, an artificial neural network approach is employed to determine implicit performance functions for reliability evaluations in performance-based design and to optimally evaluate design variables under specified performance criteria and pertinent target reliability levels in inverse performance-based design. It is proven that, with the learning process, a neural network can be applied to make a good fitness on a whole area approximation in simulation of actual limit state functions. Through the inverse performancebased design approach, while the performance requirements are maintained the same, pertinent design parameters can be altered in order to evaluate feasible design alternatives, explore the usage of various structural materials and define required material quality control. Although the traditional codified design approach may still be prevalent at present and remain a useful format of satisfying minimum requirements, the performance-based design approach has the capability to assist in evaluating decisions or innovations. It is believed that the efficiency in evaluating these design parameters is of principal significance for the ultimate general adoption of performance-based design in solving day-to-day practical structural design problems. In this work, two back-propagation neural networks are employed. It firstly delineates an approach to reliability estimation, which integrates a response surface methodology and artificial neural network. The central step of the methodology is to represent the performance or limit state function, through discrete data corresponding to a set of variable vectors, previously acquired from compiled deterministic and algorithmic analyses. As such, the compilation and gleaning of structural response data is undertaken independently of the reliability computation. Whilst the former step may be computationally intensive, the latter step only entails a pattern recognition process under AI technology with less computational effort. This renders the possibility of a fast evaluation of reliability and efficient implementation of performance-based design. A virtually global surface is matched to structural response data for various combinations of the design variables. Whilst the implementation of performance-based approaches is based on an explicit specification of criteria, employing a structural response analysis for the computation of the performance requirements and reliability analysis to determine the attained exceedence probability in each limit state, in solving practical design problems, an inverse performance-based approach is more convenient.. Under this approach, the performance criteria and corresponding target reliability levels are given and a set of design parameters to optimally satisfy the criteria with the desired reliabilities are to be determined. The design variables can also be evaluated directly by the second neural network with pattern recognition between the attained reliabilities and the target specifications for all limit states. This renders the development of useful and pragmatic performance-based design software which is of fundamental significance for its general acceptance in real engineering problems.

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Leveraging Search and Content Exploration by Exploiting Context in Folksonomy Systems With the advent of Web 2.0 tagging became a popular feature in social media systems. People tag diverse kinds of content, e.g. products at Amazon, music at Last.fm, images at Flickr, etc. In the last years several researchers analyzed the impact of tags on information retrieval. Most works focussed on tags only and ignored context information. In this article we present context-aware approaches for learning semantics and improve personalized information retrieval in tagging systems.We investigate how explorative search, initialized by clicking on tags, can be enhanced with automatically produced context information so that search results better fit to the actual information needs of the users. We introduce the SocialHITS algorithm and present an experiment where we compare different algorithms for ranking users, tags, and resources in a contextualized way.We showcase our approaches in the domain of images and present the TagMe! system that enables users to explore and tag Flickr pictures. In TagMe! we further demonstrate how advanced context information can easily be generated: TagMe! allows users to attach tag assignments to a specific area within an image and to categorize tag assignments. In our corresponding evaluation we show that those additional facets of tag assignments gain valuable semantics, which can be applied to improve existing search and ranking algorithms significantly. During the last decade, the tagging paradigm attracted much attention in the Web community. More and more Web systems allow their users to annotate content with freely chosen keywords (tags). The tagging feature helps users to organize content for future retrieval (Marlow et al. 2006b ). Resource sharing systems like Delicious 1 , Flickr 2 , or Last.fm 3 would not work without the users, who assign tags to the shared bookmarks, images, and music respectively, because tag assignments are used as information source to provide diverse features such as recommendation, search, or exploration features. For example, tag clouds, which depict the popularity of tags within the system, intuitively allow users to explore a repository of tag-annotated resources, just by clicking on tags.. Beside search algorithms that simply detect resources directly annotated with the search tag, more advanced algorithms are available that exploit the full structure of the folksonomy (Vander Wal 2007) . A folksonomy is basically a collection of all tag assignments (user-tag-resource bindings) in the system. It can be modeled as a graph which makes it possible to apply graph-based search and ranking algorithms according to the paradigm of PageRank (Page et al. 1998) . Such ranking algorithms like FolkRank (Hotho et al. 2006c) , which is based on PageRank and applicable to folksonomies, not only allow to rank resources but also tags and users. This feature expands the scope of applications to tag recommendations, user/expert search, etc.. Hence, ranking algorithms play a central role in a multitude of applications, however all ranking algorithms have to face the problem of ambiguity. For example, the tag "java" might be assigned to resources related to programming or the island of Indonesia. Another problem is caused by tags that are re-used on various occasions with different (though implicit) meaning. For instance, the tag "to-read" might be added by a same user at different times to scientific papers that are relevant for a research work or to websites that explain what to see in some location the user would like to visit on holidays. If the tag "to-read" would be used in a query, likely the ranking algorithm outcome would not satisfy the user because such algorithms lack the means to contextualize the ranking. Correspondingly, for broad tags like "music" or "web", which are assigned to a huge amount of resources, it is difficult to compute a ranking that fits to the actual desires of the user.. In this article we examine how the problems, mentioned above, can be solved by exploiting context information that is either embedded in the folksonomy or constructed from the user interactions (clicks). We present a lightweight approach for contextualized search in folksonomy systems without requiring the systems to do extensive user modeling and without any prerequisites for the user. We do so by proposing general strategies that rely on the context information in a way that is orthogonal to the ranking algorithm that is used. We model context by the notion of tag clouds (list of weighted tags), e.g., it can be formed by the tag cloud of a resource, from which the user initiates a search activity. So, for example, if a user has navigated to an image in Flickr showing the Indonesian island Java and thereafter clicks on the tag "java" to explore further photos of the island, then it is beneficial to consider also the other tags of the image (e.g. "indonesia", etc.) to adapt the outcome of the search to the user's actual needs.. The interesting novelty of our proposal is that we do not only restrict the context information to the profile of the user, who initiates the query, but also present strategies that consider content the user is currently browsing. In addition to the contextual browsing we analyze the benefits of embedding context information directly into the folksonomy. Therefore, we present a model that enables systems to attach context information explicitly to particular tag assignments. For our investigations we implemented TagMe! (Abel et al. 2009c ), a tagging and exploration interface for Flickr pictures, that introduces three types of context: (i) spatial information describing to which part of a resource a tag assignment belongs to, (ii) categories for organizing tag assignments, and (iii) URIs that describe the semantic meaning of a tag assignment. Our experiments reveal that the exploitation of such context information has a significant impact on the search performance.. In this article we introduce different approaches for leveraging context in folksonomy systems. The main contributions can be summarized as follows.. • We introduce a lightweight approach that models context by means of tag clouds and allows the adaption of search results, produced by arbitrary ranking algorithms, to the actual user needs.. • We propose a new folksonomy model for embedding arbitrary context informa-tion into folksonomies and show that the consideration of context significantly improves existing ranking algorithms.. • We present new ranking algorithms such as SocialHITS, a novel ranking algorithm, which adapts the HITS algorithm (Kleinberg 1999) to folksonomies, as well as FolkRank-based algorithms and prove that there are situations in which our algorithms significantly perform better than existing ranking algorithms.. In our experiments, we not only evaluate the quality of ranking resources and tags, but also-and this constitutes the originality of our experiments-measure the performance of ranking user entities. Further, we showcase our approaches in two different systems: In the TagMe! system (Abel et al. 2009c ), a tagging and exploration front-end for Flickr, as well as in the GroupMe! system (Abel et al. 2007 ), a social bookmarking system that allows users to organize their bookmarks in thematic groups.. The article is organized as follows. In the next section we will discuss previous work and provide further motivation for the work presented in this article. In Section 2 we present our notion of context in folksonomy systems. We define a formal context folksonomy model and outline our approach to contextualize rankings. In Section 3 we discuss several ranking algorithms that exploit context information in folksonomy systems. We evaluate our approaches with respect to different applications: In Section 4 we evaluate our approaches for adapting search to the actual desires of individual users and in Section 5 we examine how context embedded in the folksonomy can help to improve search. We end our paper with conclusions in Section 6. Section 4 is based on work that we presented in (Abel et al. 2009a ). Ranking in folksonomies is currently an important research topic. In this article we proposed different approaches for improving ranking approaches by exploiting context information. We introduced a model for integrating context information into folksonomies and presented an approach that allows the adaption of rankings to the actual context of a user independently from the underlying ranking algorithm. We presented different strategies that are able to construct context from user interactions (clicks) by the notion of tag clouds even if no previous knowledge about the user is available. Furthermore, we introduced FolkRank-based algorithms for exploiting context information and SocialHITS, a new ranking algorithm for folksonomy systems. We showed that SocialHITS significantly improves the HITSbased approach proposed in (Wu et al. 2006) .. We analyzed the performance of SocialHITS and other folksonomy-based ranking algorithms for the task of contextualizing rankings while considering different types of context and revealed that those strategies, which do not require any previous knowledge about the user, perform significantly better than tag-based user modeling. For example, by considering the the tag cloud of a resource the user has just visited we are able to adapt the ranking of a subsequent search activity to the user's current context. A remarkable feature of our evaluation was that we also measured the ranking performance with respect to the task of ranking users, which is new in the field of research on folksonomies and further promises high impact on the future of social networking. Here, we identified SocialHITS as one of the most promising ranking algorithms. Our evaluations further reveal that the exploitation of context embedded in folksonomies, e.g. categories and URIs attached to individual tag assignments, are beneficial to learning relationships between tags and help to improve search significantly. In addition to our extensive evaluations we showcase our approaches in the TagMe! system, a tagging and exploration interface for Flickr pictures. We present the results according to the following structure. We first try to evaluate the performance of the newly introduced SocialHITS algorithm, independently from the used context strategy. Afterwards we overview our core results that allow us to answer the questions raised at the beginning of this section. In Subsection 4.4.3 we analyze the performance of the strategies when they have to rank (a) user and (b) resource entities. We will particularly investigate the ability of the algorithms to rank users, because this has not been studied extensively in previous work yet. Our result analysis finishes with a summary regarding the performance of the different context models, which are used to adapt the rankings to the actual context of a user. We tested the statistical significance of all following results with a two-tailed t-Test and a significance level of α = 0.05. The null hypothesis H0 is that some strategy s 1 is as good as another strategy s 2 , while H1 states that s 1 is better than s 2 .. In summary, the exploitation of context embedded in the folksonomy is beneficial for ranking resources. While the size and position of the area helps to improve the precision particularly at the top of the resource rankings, the DBpedia and category context successfully contribute to improve the recall. And by combining the different context types we are able to improve the ranking performance of FolkRank significantly. For research carried out in the field of tagging systems the understanding of folksonomies (Vander Wal 2007), which evolve over time when users assign tags to resources, is a matter of particular interest. Formal folksonomy models have been proposed in (Hotho et al. 2006a , Mika 2005 ) and usually interpret a folksonomy as collection of tag assignments possibly enriched with context information like time (Halpin et al. 2007) or characteristics of the setting, in which a tag assignment was made (Abel et al. 2007) . Further, there exist models that try to incorporate the tagging behavior of users (Dellschaft and Staab 2008, Halpin et al. 2007) . We extend those folksonomy models with context information that can be attached to individual tag assignments. For example, we follow the MOAT (Passant and Laublet 2008) approach and attach DBpedia URIs (Auer et al. 2007) , which refer to the structured Wikipedia data, to clearly define the meaning of tag assignments.. Given traditional folksonomy models there are a number of research fields including but not limited to the design of search algorithms (Bao et al. 2007 , Hotho et al. 2006c , computing recommendations (Byde et al. 2007 , Jäschke et al. 2007 , Sigurbjörnsson and van Zwol 2008 , deducing semantics from tags (Hotho et al. 2006b , Rattenbury et al. 2007 , or user modeling , Li et al. 2008 , Michlmayr and Cayzer 2007 . Here, ranking algorithms are indispensable as they allow for the ordering of search results, recommendations, etc. A fundamental assumption of research in the field of folksonomy systems is that tags assigned by the users to resources describe the content of the resources very well. This assumption is proved in (Li et al. 2008) , where the authors compared the actual content of Web pages with tags assigned to these pages in the Delicious system.. In (Abel et al. 2009b ) we compared different ranking algorithms, FolkRank (Hotho et al. 2006c) , GFolkRank (Abel et al. 2009b) , GRank (Abel et al. 2009b) , SocialPageRank, and SocialSimRank (Bao et al. 2007) , with regards to their ability to rank resources and tags. We discovered that those algorithms which utilize the entirety of the information in a folksonomy (including context information attached to the tag assignments) performed best. Our previous findings motivate the work presented in this article. We propose strategies enabling the integration of context information independently from both the used ranking algorithm and the underlying folksonomy model. From a more technical perspective, our strategies for contextualizing search result rankings are based on query expansion (Voorhees 1994) . Instead of applying co-occurrence based techniques (Kim and Choi 1999) or using dictionaries, such as WordNet, we follow the approach of (Chirita et al. 2007 ) and utilize context information to expand queries and contextualize rankings.. Previous work related to ranking in folksonomies mainly focusses on ranking of resources (Abel et al. 2009b , Bao et al. 2007 , Hotho et al. 2006c or tags (Abel et al. 2008, Sigurbjörnsson and van Zwol 2008) . In this paper we go beyond state of the art and evaluate folksonomy-based algorithms with respect to their performance when ranking user entities, so to allow the identification of users with certain interests or expertise. This capability, though not yet exploited sufficiently in existing social networking services, like Facebook 1 or LinkedIn 2 , is of tremendous interest to research in social networking and has many practical applications. While social networking systems require the users to input their interests, competencies, or relations to other users explicitly, tagging systems, on the contrary, capture such information automatically and allow social networks to be constructed implicitly (Nauerz and Groh 2008) . However, the retrieval of user entities has not been studied sufficiently in the field of folksonomy systems. Hence, we evaluate only ranking algorithms, which also allow the ranking of users. Based on Kleinberg's Hyperlink-Induced Topic Search (HITS) algorithm (Kleinberg 1999) and ideas presented in (Wu et al. 2006) we propose SocialHITS, as the notion of authorities and hubs appears to be appropriate for user entities, in particular.

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Spanners for bounded tree-length graphs This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most δ, i.e., the tree-length δ graphs. For such graphs we construct additive 2δ-spanners with O(δn+n log n) edges, and additive 4δ-spanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of tree-length δ for which every multiplicative δ-spanner (and thus every additive (δ − 1)-spanner) requires Ω (n 1+1/Θ(δ) ) edges. Let G be an unweighted connected graph with n vertices. A subgraph H of G is an (s, r )-spanner if d H (u, v) s · d G (u, v) + r for all pair of vertices u, v of G. An (s, 0)-spanner is also termed multiplicative s-spanner, and an (1, r )-spanner is termed additive r -spanner. An (s, r )-spanner is also an (s + r, 0)-spanner (in particular, an additive r -spanner is a multiplicative (r + 1)-spanner), but the reverse is false in general.. The main objective is to construct for a graph an (s, r )-spanner with few edges. There are many applications of spanners, for example, the complexity of a lot of distributed algorithms depends on the number of messages, itself depending on the number of edges [3, 4] . Sparse spanners occur also in the efficiency of compact routing schemes [5] . Unfortunately, given an arbitrary graph G and three integers s, r and m, determine whether G admits an (s, r )-spanner with m or fewer edges, is NP-complete [6] , even if we restrict r = 0 (see also [7] [8] [9] [10] [11] for complexity issue). Best known results on (s, r )-spanners for general graphs are summarized in the following table.. (s, r )-spanner Edges Reference. (1, 2) Θ(n 3/2 ) [12-14] 1 (1, 6) O(n 4/3 ). O(kn 1+1/k ) [14] , k 4 even (k − 2 + , 2k − 2 − ) O( −1 kn 1+1/k ) [14] , k 3 odd, > 0 (k − 1 + , 2k − 4 − ) O( −1 kn 1+1/k ) [14] , k 4 even, > 0 (1 + , β( , k)) O(β( , k)n 1+1/k ) [14] , 3 k 2. An interesting question still left open is to know whether every graph has an additive (2k −2)-spanner with O(n 1+1/k ) edges, for k > 2. In the affirmative, this would generalize the result of [13, 14] (k = 2) and implies also the observation of [16] . Another interesting question is whether the O(n 1+1/k ) edge bound for multiplicative (2k − 1)-spanner is tight or not. This bound directly relies on an 1963 Erdös Conjecture [20] on the existence of graphs with Ω (n 1+1/k ) edges and girth at least 2k + 2. This has been proved only for k = 1, 2, 3 and k = 5 [21] . Better bounds can be achieved if we restrict spanners to be trees [22] , or if particular classes of graphs are considered like: planar graphs [11] and more structured graphs (e.g., see [23] and [24] for a survey). Among them, the class of chordal graphs is of particular interests [7, 6, 25] . A graph is k-chordal if its induced cycles are of length at most k. Chordal graphs coincide with 3-chordal. Here below are summarized the best constructions for k-chordal graphs. In this paper we showed that any n-vertex graph of tree-length δ admits an additive 2δ-spanner with O(δn +n log n) edges and an additive 4δ-spanner with O(δn) edges. These results provided new upper bounds for chordal graphs (δ = 1). We have also shown a lower bound which says that there are graphs of tree-length δ for which every multiplicative δ-spanner (and thus every additive (δ − 1)-spanner) must have Ω (n 1+1/Θ(δ) ) edges.. We conclude this paper with few open problems.. (1) Do the tree-length δ graphs admit additive 2δ-spanners with O(n log n) edges (δ independent number of edges)? This was the case for the k-chordal graphs [2] . (2) What about additive 4δ-spanners with O(n) edges? Again, this was the case for the k-chordal graphs [25] . (3) Do the tree-length δ graphs admit sparse additive δ-spanners? (4) Given an arbitrary graph G, what is the complexity of constructing a tree-decomposition of G into bags of smallest diameter δ?

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Example-based image color and tone style enhancement Color and tone adjustments are among the most frequent image enhancement operations. We define a color and tone style as a set of explicit or implicit rules governing color and tone adjustments. Our goal in this paper is to learn implicit color and tone adjustment rules from examples. That is, given a set of examples, each of which is a pair of corresponding images before and after adjustments, we would like to discover the underlying mathematical relationships optimally connecting the color and tone of corresponding pixels in all image pairs. We formally define tone and color adjustment rules as mappings, and propose to approximate complicated spatially varying nonlinear mappings in a piecewise manner. The reason behind this is that a very complicated mapping can still be locally approximated with a low-order polynomial model. Parameters within such low-order models are trained using data extracted from example image pairs. We successfully apply our framework in two scenarios, low-quality photo enhancement by transferring the style of a high-end camera, and photo enhancement using styles learned from photographers and designers. With the prevalence of digital cameras, there have been increasing research and practical interests in digital image enhancement. Tone and color adjustments are among the most frequent operations. While such adjustments often need to be determined on an individual basis, there exist many scenarios where tonal and color adjustments follow common implicit rules. For example, photographers often carefully tune the temperature and tint of existing colors in a photograph to convey specific impressions. For a specific impression, the types of adjustments are usually consistent across different photographs. As another example, it is well-known that photographs taken by different digital cameras have varying degrees of tone and color discrepancies. This is because each type of camera has its own built-in radiance and color response curves. We define a tone and color style as a set of explicit or implicit rules or curves governing tonal and color adjustments.. Manually adjusting the tone and color of a photograph to achieve a desired style is often tedious and labor-intensive. However, if tone and color styles can be formulated mathematically in a digital form, they can be automatically and easily applied to novel input images to make them look more appealing. Unfortunately, the rules governing a tone and color style are most often not explicitly available. For instance, it is typically very hard for a photographer to mathematically summarize the rules he uses to achieve a certain impression; and it also involves much work to calibrate the radiance and color response curves of a camera especially considering the fact that color response curves need to cover the entire visible spectrum. Therefore, our goal in this paper is to learn implicit tone and color adjustment rules from examples. That is, given a number of examples, each of which is a pair of corresponding images before and after adjustments, we would like to discover the underlying mathematical relationships optimally connecting the tone and color of corresponding pixels in all image pairs.. For the following reasons, it is challenging to learn tone and color adjustment rules from examples. First, the relationships we need to identify are buried in noisy data available from example image pairs. We need to rely on machine learning and data mining techniques to discover hidden patterns and relationships. Second, the relationships are likely to be highly nonlinear and spatially varying. This is because camera radiance and color response curves are nonlinear, and the adjustment rules used by photographers most likely vary according to tone and color characteristics of local image regions. Third, there may exist other factors in the relationships in addition to the tone and color of individual pixels. For instance, the amount of adjustment may depend on neighborhood statistics which correlate with the material and texture properties of underlying object surfaces.. In this paper, we develop a learning based method to achieve our goal. We formally define tone and color adjustment rules as mappings, and propose to approximate complicated spatially varying nonlinear mappings in a piecewise manner. The reason behind this is that a very complicated mapping can still be locally approximated with a linear or low-order polynomial model. Thus, to tackle the complexity and nonlinearity, we adopt a divide-and-conquer strategy by first dividing a feature space into a number of subspaces and then approximating the tone and color mappings within each subspace with a low-order model. Parameters within such low-order models are trained using data extracted from example image pairs. In theory, as long as the mappings exist, they can be approximated to a high precision if the feature space is subdivided into a sufficient number of subspaces.. More specifically, we have the following contributions in this paper. First, we develop a systematic framework for learning complicated tone and color styles from examples. The original feature space is divided into subspaces using binary classification trees. Every leaf node in a classification tree corresponds to a subspace, where a distinct pair of local color and gradient mappings are trained. Second, we identify effective low-order parametric models for local color and gradient mappings. The local gradient mappings are particularly effective for contrast enhancement. Effective techniques have also been developed for luminance channel optimization and spatially coherent color mapping. Third, we apply this framework in two scenarios. In the first scenario, we learn the picture style of a high-end digital camera and use that style to enhance photographs taken by low-end cameras. In the second scenario, we learn styles of images manually enhanced by photographers, and the learned styles can be successfully applied to novel input photographs as well. Our example-based framework requires a training stage during which color and tone mappings are learned from example image pairs. Once these mappings have been obtained, they can be applied to new images to enhance a certain color and tone style. These training and application phases are illustrated in Figures 2 and 3 Our basic idea is to learn hidden relationships between the color and tone of registered pixels. To effectively learn a style transformation, we propose to train color mappings using corresponding pixels in the image pairs. In general, such mappings are highly nonlinear and spatially varying. Finding a single closed-form parametric model for such mappings is an elusive task. We decide to represent these mappings in a piecewise manner.. We extract local statistical features from neighborhoods centered at every pixel in both images of every training pair. The features may include the average and standard deviation of colors within a neighborhood as well as gradient information. Let F be a multidimensional feature space where feature vectors extracted from training images reside. We build a binary feature space partition tree that divides F into n subspaces (leaf nodes), F1, F2, . . . , Fn, so that F = F1 ∪ F2 ∪ F3 ∪ . . . ∪ Fn. All feature vectors contained in each leaf node define one feature subspace, and we train a local color mapping specifically tailored for that feature subspace. All subspace color mappings constitute our final piecewise approximation of the overall nonlinear color mapping.. Since luminance gradients along edges can significantly affect the overall image contrast, we further build a second binary feature space partition tree related to luminance gradients, and train local mappings for luminance gradients along edges.. The output of this training stage consists of two binary space partition trees: one for color mapping, and the other for luminance gradient mapping (Figure 2 ). In this paper, we have developed a method for learning color and tone styles from examples. This method is capable of discovering underlying mathematical relationships in color and tone between corresponding image pairs. We have successfully applied our framework in two scenarios, low-quality photo enhancement by transferring the style of a high-end camera, and photo enhancement using styles learned from photographers and designers.. Limitations. Like most other data-driven techniques, our proposed method requires a training stage where data-driven models are optimized. The performance of the trained data-driven models relies on the quality and quantity of the training data. Fortu- nately, this is not a serious problem with images. As a typical image already has at least hundreds of thousands of pixels with different colors and intensities, the total number of images in a training dataset does not need to be very large. Our color and gradient mappings can already produce reasonably good results given a dozen or so representative training image pairs. Another issue is that we require pixel-level image registration within every training image pair, especially when the two images in a pair are taken by different cameras. Such a level of precision is not easy to achieve with existing image registration algorithms. As a result, we had to rely on user-provided hard constraints to improve automatic registration results. Providing such constraints requires a certain amount of manual work. [Piti and Kokaram 2007] . Our enhanced result is based on the first style in Figure 13 . Our work has been inspired by previous work on image enhancement, including automatic methods for exposure correction, contrast enhancement and color correction as well as interactive techniques for tonal and color adjustments.. Inspired by a model of the lightness and color perception of human vision, Jobson et al. [1997] proposed a multiscale center/surround retinex that achieves simultaneous dynamic range compression, color consistency, and lightness rendition. It is essentially a multiscale framework that simultaneously performs local and global color and contrast adjustments. Bae et al. [2006] performs automatic local and global tonal adjustments to an input image to make its local and global contrasts similar to those of a reference image. Lischinski et al. [2006] introduced an interactive local tone adjustment method based on a sparse set of scribbles. Cohen-Or et al. [2006] performs color adjustment according to harmonization rules. Shapira et al. [2009] proposed an interesting method to edit image appearances interactively using Gaussian mixture models (GMMs). Wang et al. [2010] presents a method for adjusting the color composition of an image to match a predefined color theme. All these techniques directly perform tonal or color adjustments to individual input images without learning parametric models for such adjustments.. Gradient domain image editing techniques and systems have been presented in [Fattal et al. 2002; Perez et al. 2003 ]. Fattal et al. in- troduces an effective method for high dynamic range compression by strategically reducing the magnitude of image gradients. Perez et al. presents a framework for gradient domain image editing by altering and re-integrating the original gradient field. An important difference between these existing techniques and our approach in this paper is that our approach identifies the connections between gradients from photos with two different styles and model such connections as parametric gradient mappings, while the above techniques only perform gradient domain adjustments to individual input images without training parametric models.. Much work has been performed on image color transfer [Reinhard et al. 2001; Chang et al. 2005; Piti and Kokaram 2007] . Typically, such a method performs a statistical analysis to impose the color characteristics of a reference image onto another source image. Though much progress has been made, the quality of the results still relies on the compatibility between source and reference images as pixelwise correspondences between them do not exist. Based on techniques in image search, cosegmentation and color transfer, Dale et al. [2009] developed a multi-purpose image enhancement framework that leverages a large database of images. Color transfer results are used as training data for learning local parametric mappings. A significant difference between such color transfer based techniques and our work is that color transfer only transfers colors from one or multiple reference images to the source image and any mappings obtained during this process only work for a specific source image while our work in this paper attempts to train a set of tone and color mappings that are generally applicable to a class of images so that users do not need to choose a reference image for each source image. Siddiqui and Bouman [2008] developed a multi-stage color correction algorithm for enhancing digital images obtained from low quality imaging devices such as cell phone cameras. In this framework, a pixel color is classified into a category using a Gaussian mixture model and local color correction is performed using an affine transform specifically tailored for that category. The parameters in the Gaussian mixture model and affine transforms are learned in an offline training procedure. The differences between our work and [Siddiqui and Bouman 2008] are summarized as follows. First, we perform color correction and contrast enhancement simultaneously in the same framework while [Siddiqui and Bouman 2008] only performs color correction. Second, in our work, color adjustments are performed using quadratic models instead of affine models. Figure 4 demonstrates that quadratic models outperform affine models. Third, more importantly, the image and pixel categories adopted in [Siddiqui and Bouman 2008] are only suitable for imaging devices but not for photographers' stylistic adjustments while the pixel categories in our method are automatically extracted from training data using hierarchical clustering.. Global image enhancement techniques and pipelines have been presented in [Battiato et al. 2004; Kang et al. 2010] . In [Kang et al. 2010] , global parameters for image enhancement are trained using personalized enhancement examples. In contrast, our method trains local parametric models instead of global parameters.

O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O


A survey on self-healing systems: approaches and systems Abstract Present large-scale information technology environments are complex, heterogeneous compositions often affected by unpredictable behavior and poor manageability. This fostered substantial research on designs and techniques that enhance these systems with an autonomous behavior. In this survey, we focus on the selfhealing branch of the research and give an overview of the current existing approaches. The survey is introduced by an outline of the origins of self-healing. Based on the principles of autonomic computing and self-adapting system research, we identify selfhealing systems' fundamental principles. The extracted principles support our analysis of the collected approaches. In a final discussion, we summarize the approaches' common and individual characteristics. A comprehensive tabular overview of the researched material concludes the survey.

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Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP Abstract-We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem.We present (2 − )-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier. Prize-collecting problems involve situations where there are various demand points that desire to be "served" by some structure and we must find the structure of lowest cost to accomplish this. However, if some of the demand points are too expensive to serve, then we can refuse to serve them and instead pay a penalty. The two most famous prize-collecting problems are the prize-collecting Steiner tree (PCST) and prize-collecting traveling salesman (PCTSP) problems. In both cases, we are given a connected graph G = (V, E), a non-negative cost c e for each edge e ∈ E, denoting the cost to purchase that edge, and a non-negative penalty (sometimes called a prize) π v for each node v ∈ V . In PCST, we must select some tree T spanning a set V (T ) of nodes so as to minimize the combined cost where V (T ) = V − V (T ), and the notation c(T ) and π(V (T )) follows the usual convention that when x is a vector and S is a subset of its indices, x(S) = i∈S x i . That is, we aim to minimize the total cost of the tree plus the penalties of the nodes not spanned. Although T denotes a set of edges, in a slight abuse of notation we will often write T to denote the set V (T ) of nodes. Since we will never take the complement of an edge set, this should cause no ambiguity. PCTSP is the same, except that the set of edges should form a cycle instead of a tree. We also study a variant of PCTSP, the prize-collecting stroll (PCS) problem, in which the edges should form a stroll (i.e., path). When π v = ∞ for all v ∈ V , PCTSP simplifies to the ordinary TSP because we have no choice but to visit all nodes, and PCS simplifies to PATH-TSP, the path version of TSP. Similarly, if each penalty is 0 or ∞, PCST simplifies to an instance of the ordinary Steiner tree problem; in this case, the nodes {v : π v = ∞} are called terminals (which the Steiner tree must connect) and the nodes {v : π v = 0} are called Steiner nodes (which the tree may use if it helps). These are all notorious NPhard problems, so PCST, PCTSP, and PCS are NP-hard as well. Thus, we are interested in obtaining approximation algorithms for them.. In PCTSP and PCS, one typically assumes that the graph is complete and the edge costs satisfy the triangle inequality. Otherwise the problem is inapproximable, since determining whether the edges of cost zero form a Hamiltonian cycle or Hamiltonian path is NP-hard. These assumptions open the door to algorithms that use "shortcutting." In PCST, these extra assumptions have no effect.. In the rooted version of these problems, there is a specified root node r that must be spanned. Since the rooted and unrooted versions are reducible to each other, 1 we consider only the rooted version for the rest of the paper.. The first approximation algorithms for the PCST and PCTSP problems were given by Bienstock et al. [7] , although the PCTSP had been introduced earlier by Balas [5] . Bienstock et al. achieved a factor of 3 for PCST and 2.5 for PCTSP by rounding the optimal solution to a linear programming (LP) relaxation. Later, Goemans and Williamson [18] constructed primal-dual algorithms using the same LP relaxation to obtain a 2-approximation for both problems, building on work of Agrawal, Klein and Ravi [1] . Chaudhuri et al. modified the Goemans-Williamson algorithm to achieve a 2-approximation for PCS [8] . In a 1998 conference talk covering unpublished work [15] , Goemans mentioned (exactly) two open problems: resolving whether the integrality gap of the Held-Karp LP relaxation for TSP is exactly 4 3 and improving the PCTSP approximation ratio below 2. The conference version of the Goemans-Williamson paper [17] appeared in 1992; in the 17 years since, there have been no improved approximation algorithms invented for either PCST or PCTSP, until now. This paper gives (2 − )-approximation algorithms for both problems, and for PCS as well.. By contrast, improvements for the original Steiner tree problem came fast and furiously after Zelikovsky's announcement, in 1990, of an 11/6-approximation for Steiner tree ( [28] , [29] ), the first algorithm to beat the naive 2-approximation. Berman and Ramaiyer's improvement of the 11/6 = 1.833... bound of Zelikovsky to 1.746 appeared in 1992 [6] . Zelikovsky himself improved Berman and Ramaiyer's 1.746 to 1.693 in 1996 [30] , which was followed by Prömel and Steger's improvement to 1.667 in 1997 [26] , and by Karpinski and Zelikovsky to 1.644 in 1997 [25] , after which Hougardy and Prömel improved the bound to 1.598 in 1999 [21] . The best and most recent bound, just under 1.55, is due to Robins and Zelikovsky, in 2005 [27] .. We hope for our work to trigger similar improvements for PCST, PCTSP, and PCS. Indeed, this process has already begun. After we informed Goemans of our new result, he combined some of our ideas with others from [15] (namely, a randomized version of the PCTSP algorithm from [7] ) to improve the ratio for PCTSP below 1.915 [16] .. Although the value of that we achieve for the two prizecollecting problems is small (about 0.01 in all three cases), this is a conceptual breakthrough, since the factor 2 was thought to be a barrier. The natural LP relaxation for PCST used in [7] and [18] is known to have an integrality gap of 2. Thus, it cannot by itself provide a strong enough lower bound to prove an approximation factor better than 2.. PCST and PCTSP are two of the classic optimization problems, deserving of study in their own right. Moreover, work on the PCST has had a large impact, both in theory and practice. At AT&T, PCST code has been used in largescale studies in access network design, both as described in Johnson, Minkoff and Phillips [23] , and in unpublished applied work involving the first author of the present paper.. The impact of PCST within approximation algorithms is also far-reaching. As noted by Chudak, Roughgarden and Williamson [10] , PCST is a Lagrangian relaxation of the k-MST problem, which asks for the minimum-cost tree spanning at least k nodes. Moreover, they note that the PCST algorithm of Goemans and Williamson (henceforth called GW ) is not just a 2-approximation algorithm, but also a Lagrangian preserving 2-approximation (we give a formal definition shortly). This implies the useful property that the total edge cost of the tree T returned by GW is no more than twice the edge cost of the cheapest tree that pays at most π(V (T )) penalty. Thus, if one runs PCST with all penalties set to some constant, and it happens to return a tree spanning exactly k nodes, then this tree is a 2-approximate k-MST. This property has been used in a sequence of papers ( [13] , [4] , [3] ) culminating in a 2-approximation algorithm for k-MST by Garg [14] . It has also been used to improve the approximation ratio and running time of algorithms for the MINIMUM LATENCY problem ( [2] , [8] ). The technique of applying Lagrangian relaxation to a problem with hard constraints, then using a Lagrangianpreserving approximation algorithm on the relaxed problem to recover an approximation algorithm for the original problem, has been successfully applied to the k-MEDIAN and UNCAPACITATED FACILITY LOCATION problems as well, in a sequence of papers starting with [22] .. In this paper, we use the Lagrangian-preserving guarantee of the GW algorithm in a different way. Tree T is a Lagrangian-preserving 2-approximate solution if. To be an ordinary 2-approximation, it would suffice to satisfy (1) with the 2 on the left side changed to 1. One way to view this is that GW essentially achieves an approximation ratio of 1 on the penalty term, and 2 on the tree term. We can show that if either GW or OP T pays a large amount of penalty, then GW achieves a (2 − )-approximation. In the case where both GW and OP T pay a small penalty, intuitively GW is finding approximately the correct set of nodes to span, but it might have failed to construct a cheap tree. (This intuition is somewhat misleading; in actuality, GW might choose a very wrong set of nodes, so we will have to correct for this as well.) We take advantage of this by using GW to help us identify a set of nodes to span, then running the Robins-Zelikovsky (RZ) algorithm for the ordinary Steiner tree problem, using those nodes as terminals. We use RZ here as a black box: for the purpose of achieving a (2 − ) factor for PCST, any (2 − )-approximation for ordinary Steiner tree would suffice. We emphasize that in selecting the terminal set to send to RZ, we do not simply use the set of nodes that GW chose to span; in fact, we carefully select a subset. The proper selection of these terminals is the crux of our algorithm. There are two keys to this whole scheme. First, we must ensure that the extra nodes that GW spanned but that we choose not to designate as terminals have a small total penalty. We do this by using LP duality to bound their penalty against the cost of the tree produced by GW (Lemma 3). The second key is Lemma 4, which guarantees that there exists a tree spanning our chosen terminals whose cost is not much larger than OP T . Thus, when we run RZ on this set of nodes to find a 1.55-approximation to this tree, it will give better than a 2-approximation, even after we add in the penalty term (since both the nodes excluded by GW and the extra ones that we excluded from the terminal set have small total penalty). Our algorithm for PCTSP is very similar, except that we use the Christofides 3 2 -approximation algorithm for TSP [9] instead of RZ. For PCS, we use a 3 2 -approximation for the path version of TSP, due to Hoogeveen [20] .. One novel feature of our analysis is that we use the details of the GW moat-growing algorithm to prove in Lemma 4 that there exists an inexpensive set of edges satisfying certain properties, but not to find it. All other applications of GW of which we are aware use it purely in an algorithmic sense to identify edges; this may be the first time it has been utilized solely to provide an existence proof.. We structure the rest of the paper as follows. Section 2 describes the GW PCST algorithm. Section 2.1 discusses performance guarantees for GW, along with a bad instance that motivates our improved algorithm. Section 3 describes our algorithm for PCST, states the key lemmas, and fits them together to prove the approximation ratio of (2 − ). The heart of our analysis appears in Section 3.1, where we prove the key lemmas. Section 4 applies the same techniques to derive (2 − )-approximation algorithms for PCTSP and PCS. Section 5 explains how several variants of GW relate. Our techniques might be applicable to the PCST problem with submodular penalties, introduced in [19] .

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