867328
Approximating the Radii of Point Sets
We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers $n, d$, and $k$, where $k \le d$, and a set $P$ of $n$ points in $\Re^d$. The goal is to compute the outer $k$-radius of $P$, denoted by ${\cal R}_k(P)$, which is the minimum over all $(d-k)$-dimensional flats $F$ of $\max_{p \in P} d(p,F)$, where $d(p,F)$ is the Euclidean distance between the point $p$ and flat $F$. Computing the radii of point sets is a fundamental problem in computational convexity with many significant applications. The problem admits a polynomial time algorithm when the dimension $d$ is constant [U. Faigle, W. Kern, and M. Streng, Math. Program., 73 (1996), pp. 1-5]. Here we are interested in the general case in which the dimension $d$ is not fixed and can be as large as $n$, where the problem becomes NP-hard even for $k=1$. It is known that $R_k(P)$ can be approximated in polynomial time by a factor of $(1 + \varepsilon)$ for any $\varepsilon > 0$ when $d - k$ is a fixed constant [M. Babdoiu, S. Har-Peled, and P. Indyk, in Proceedings of the ACM Symposium on the Theory of Computing, 2002; S. Har-Peled and K. Varadarajan, in Proceedings of the ACM Symposium on Computing Geometry, 2002]. A polynomial time algorithm that guarantees a factor of $O(\sqrt{\log n})$ approximation for $R_1(P)$, the width of the point set $P$, is implied by the results of Nemirovski, Roos, and Terlaky [Math. Program., 86 (1999), pp. 463-473] and Nesterov [Handbook of Semidefinite Programming Theory, Algorithms, Kluwer Academic Publishers, Norwell, MA, 2000]. In this paper, we show that $R_k(P)$ can be approximated by a ratio of $O(\sqrt{\log n})$ for any $1 \leq k \leq d$, thus matching the previously best known ratio for approximating the special case $R_1 (P)$, the width of point set $P$. Our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure. We also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of $k$. We show that there exists a constant $\delta > 0$ such that the following holds for any $0 < \eps < 1$: there is no polynomial time algorithm that approximates $R_k(P)$ within $(\log n)^{\delta}$ for all $k$ such that $k \leq d - d^{\varepsilon}$ unless NP $\subseteq$ DTIME $[2^{(\log m)^{O(1)}}]$. Our inapproximability result for $R_k(P)$ extends a previously known hardness result of Brieden [Discrete Comput. Geom., 28 (2002), pp. 201-209] and is proved by modifying Brieden’s construction using basic ideas from probabilistically checkable proofs (PCP) theory.
6954893
How Partial Knowledge Helps to Solve Linear Programs
We present results on how partial knowledge helps to solve linear programs. In particular, if a linear system,Ax=bandx?0, has an interior feasible point, then we show that finding a feasible point to this system can be done inO(n2.5c(A)) iterations by the layered interior-point method, and each iteration solves a least-squares problem, wherenis the dimension of vectorxandc(A) is the condition number of matrixAdefined by Vavasis and Ye. This complexity bound is reduced by a factornfrom that when this property does not exists. We also present a result for solving the problem using a little strong knowledge.
206798229
On affine scaling algorithms for nonconvex quadratic programming
We investigate the use of interior algorithms, especially the affine-scaling algorithm, to solve nonconvex — indefinite or negative definite — quadratic programming (QP) problems. Although the nonconvex QP with a polytope constraint is a “hard” problem, we show that the problem with an ellipsoidal constraint is “easy”. When the “hard” QP is solved by successively solving the “easy” QP, the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions.
10134807
A semidefinite programming approach to tensegrity theory and realizability of graphs
Recently, Connelly and Sloughter [14] have introduced the notion of d-realizability of graphs and have, among other things, given a complete characterization of the class of 3-realizable graphs. However, their work has left open the question of finding an algorithm for realizing those graphs. In this paper, we resolve that question by showing that the semidefinite programming (SDP) approach of [11, 32] can be used for realizing 3-realizable graphs. Specifically, we use SDP duality theory to show that given a graph G and a set of lengths on its edges, the optimal dual multipliers of a certain SDP give rise to a proper equilibrium stress for some realization of G. Using this result and the techniques in [14, 31], we then obtain a polynomial time algorithm for (approximately) realizing 3-realizable graphs. Our results also establish a little-explored connection between SDP and tensegrity theories and allow us to derive some interesting properties of tensegrity frameworks.
12762926
Theory of semidefinite programming for Sensor Network Localization
AbstractWe analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior-point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we show, for the first time, that these networks can be localized in polynomial time. We also give a simple and efficient criterion for checking whether a given instance of the localization problem has a unique realization in
$$\mathcal{R}^2$$ using graph rigidity theory. Finally, we introduce a notion called strong localizability and show that the SDP model will identify all strongly localizable sub-networks in the input network.
9510976
An interior point potential reduction algorithm for the linear complementarity problem
AbstractThe linear complementarity problem (LCP) can be viewed as the problem of minimizingxTy subject toy=Mx+q andx, y⩾0. We are interested in finding a point withxTy  0. The algorithm proceeds by iteratively reducing the potential function
$$f(x,y) = \rho \ln x^T y - \Sigma \ln x_j y_j ,$$
 where, for example,ρ=2n. The direction of movement in the original space can be viewed as follows. First, apply alinear scaling transformation to make the coordinates of the current point all equal to 1. Take a gradient step in the transformed space using the gradient of the transformed potential function, where the step size is either predetermined by the algorithm or decided by line search to minimize the value of the potential. Finally, map the point back to the original space.A bound on the worst-case performance of the algorithm depends on the parameterλ*=λ*(M, ε), which is defined as the minimum of the smallest eigenvalue of a matrix of the form
$$(I + Y^{ - 1} MX)(I + M^T Y^{ - 2} MX)^{ - 1} (I + XM^T Y^{ - 1} )$$
 whereX andY vary over the nonnegative diagonal matrices such thateTXYe ⩾ε andXjjYjj⩽n2. IfM is a P-matrix,λ* is positive and the algorithm solves the problem in polynomial time in terms of the input size, |log ε|, and 1/λ*. It is also shown that whenM is positive semi-definite, the choice ofρ = 2n+
$$\sqrt {2n} $$
 yields a polynomial-time algorithm. This covers the convex quadratic minimization problem.
6298363
Supporting reuse by delivering task-relevant and personalized information
Technical, cognitive, and social factors inhibit the widespread success of systematic software reuse. Our research is primarily concerned with the cognitive and social challenges faced by software developers: how to motivate them to reuse and how to reduce the difficulty of locating components from a large reuse repository. Our research has explored a new interaction style between software developers and reuse repository systems enabled by information delivery mechanisms. Instead of passively waiting for software developers to explore the reuse repository with explicit queries, information delivery autonomously locates and presents components by using the developers' partially written programs as implicit queries.We have designed, implemented, and evaluated a system called CodeBroker, which illustrates different techniques to address the essential challenges in information delivery: to make the delivered information relevant to the task-at-hand and personalized to the background knowledge of an individual developer. Empirical evaluations of CodeBroker show that information delivery is effective in promoting reuse.
1476378
Toward an understanding of the motivation of open source software developers
An Open Source Software (OSS) project is unlikely to be successful unless there is an accompanied community that provides the platform for developers and users to collaborate. Members of such communities are volunteers whose motivation to participate and contribute is of essential importance to the success of OSS projects. In this paper, we aim to create an understanding of what motivates people to participate in OSS communities. We theorize that learning is one of the motivational forces. Our theory is grounded in the learning theory of Legitimate Peripheral Participation, and is supported by analyzing the social structure of OSS communities and the co-evolution between OSS systems and communities. We also discuss practical implications of our theory for creating and maintaining sustainable OSS communities as well as for software engineering research and education.
1677204
Personalizing Delivered Information in a Software Reuse Environment
Browsing- and querying-oriented schemes have long served as the principal techniques for software developers to locate software components from a component repository for reuse. Unfortunately, the problem remains that software developers simply will not actively search for components when they are unaware that they need components or that relevant components even exist. Thus, to assist software developers in making full use of large component repositories, information access need to be complemented by information delivery. Effective delivery of components calls for the personalization of the components to the task being performed and the knowledge of the user performing it.We have designed, implemented, and evaluated the CodeBroker system to support personalized component delivery to increase the usefulness of a Java software reuse environment.
18302519
Dynamic community: a new conceptual framework for supporting knowledge collaboration in software development
Software development is a knowledge intensive activity and software developers are knowledge workers. Knowledge needed for software development is often distributed among different developers. Supporting efficient knowledge collaboration and transfer is thus essential for software development organizations to remain competitive. This paper proposes a new theory /sub y/namic community to support knowledge collaboration and discusses its application in software development. Dynamic community integrates the traditional knowledge management approach in which knowledge is formalized and accumulated in knowledge repositories, and the community-based knowledge collaboration approach in which knowledge is transferred through informal community participation and human contacts. The dynamic community theory takes into full consideration individual knowledge workers' ever-changing needs for new knowledge as well as the role of social relationship of knowledge workers in effective knowledge collaboration. This paper introduces the dynamic community theory, a general system architecture of sociotechnical environments in support of dynamic community, and its application in software development.
12123304
Integrating active information delivery and reuse repository systems
Although software reuse can improve both the quality and productivity of software development, it will not do so until software developers stop believing that it is not worth their effort to find a component matching their current problem. In addition, if the developers do not anticipate the existence of a given component, they will not even make an effort to find it in the first place.
Even the most sophisticated and powerful reuse repositories will not be effective if developers don't anticipate a certain component exists, or don't deem it worthwhile to seek for it. We argue that this crucial barrier to reuse is overcome by integrating active information delivery, which presents information without explicit queries from the user, and reuse repository systems. A prototype system, CodeBroker, illustrates this integration and raises several issues related to software reuse.