Smoothed Complexity Theory

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

Nurturing Breakthroughs: Lessons from Complexity Theory

A general theory of innovation and progress in human society is outlined, based on the combat between two opposite forces (conservatism/inertia and speculative herding "bubble" behavior). We contend that human affairs are characterized by ubiquitous ``bubbles'', which involve huge risks which would not otherwise be taken using standard cost/benefit analysis. Bubbles result from self-reinforcing positive feedbacks. This leads to explore uncharted territories and niches whose rare successes lead to extraordinary discoveries and provide the base for the observed accelerating development of technology and of the economy. But the returns are very heterogeneous, very risky and may not occur. In other words, bubbles, which are characteristic definitions of human activity, allow huge risks to get huge returns over large scales. We outline some underlying mathematical structure and a few results involving positive feedbacks, emergence, heavy-tailed power laws, outliers/kings/black swans, the problem of predictability and the illusion of control, as well as some policy implications.Comment: 14 pages, Invited talk at the workshop Trans-disciplinary Research Agenda for Societal Dynamics (http://www.uni-lj.si/trasd in Ljubljana), organized by J. Rogers Hollingsworth, Karl H. Mueller, Ivan Svetlik, 24 - 25 May 2007, Ljubljana, Sloveni

Complexity Theory, Adaptation, and Administrative Law

Recently, commentators have applied insights from complexity theory to legal analysis generally and to administrative law in particular. This Article focuses on one of the central problems that complexity. theory addresses, the importance and mechanisms of adaptation within complex systems. In Part I, the Article uses three features of complex adaptive systems-emergence from self-assembly, nonlinearity, and sensitivity to initial conditions-and explores the extent to which they may add value as a matter of positive analysis to the understanding of change within legal systems. In Part H, the Article focuses on three normative claims in public law scholarship that depend explicitly or implicitly on notions of adaptation: that states offer advantages over the federal government because experimentation can make them more adaptive, that federal agencies should themselves become more experimentalist using the tool of adaptive management, and that administrative agencies shou Id adopt collaborative mechanisms in policymaking. Using two analytic tools found in the complexity literature, the genetic algorithm and evolutionary game theory, the Article tests the extent to which these three normative claims are borne out

An Introduction to Quantum Complexity Theory

We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight the interplay between the three scenarios and currently-known quantum algorithms.Comment: 28 pages, LaTeX, 11 figures within the text, to appear in "Collected Papers on Quantum Computation and Quantum Information Theory", edited by C. Macchiavello, G.M. Palma, and A. Zeilinger (World Scientific

Reconciling Complexity Theory in Organizations and Christian Spirituality

The lenses of complexity theory have been trained on a variety of subjects in organizations, ranging from assembly lines to strategic planning. While this work has been going on, another group of researchers has been actively pursuing the study of workplace spirituality. No work has been published that endeavors to interrelate complexity theory with workplace spirituality, however. In fact, a striking feature of the complexity theory literature is an absence of consideration of spiritual dimensions or wisdom traditions. Complexity theory books and articles that reference the deep implications of the work subtly or overtly further the scientific tradition of negating religious traditions by alleging that these scientific findings provide additional evidence that certain religious traditions are not valid. Looking at the same evidence, however, I will suggest that many complexity theory philosophies and evidence strengthen, rather than weaken, the case for the existence of a supreme being and the religious traditions associated with such a belief

No occurrence obstructions in geometric complexity theory

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.Comment: Substantial revision. This version contains an overview of the proof of the main result. Added material on the model of power sums. Theorem 4.14 in the old version, which had a complicated proof, became the easy Theorem 5.4. To appear in the Journal of the AM