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Library of Congress Cataloging-in-Publication Data Complexity and real computation / Lenore Blum ... [et al.].p. cm. IncIudes bibliographical references and index.

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We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting.

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Abstract. The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV,à la Beltrán-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ −1 , where σ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f , of the expected running time of LV with input f . In addition to its dependence on N this bound also depends on the condition of f . Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is N O(log log N) . This is nearly a solution to Smale's 17th problem.

Among the many models for flocking systems of interacting particles, the one introduced by Cucker and Smale has attracted attention due to the fact that a convergence to flocking (i.e., to a common velocity) could be established depending on conditions on the initial state of the system. In this note we extend this model by adding to it a repelling force between particles. We show that, for this modified model, convergence to flocking is established along the same lines while, in addition, avoidance of collisions (i.e., the respect of a minimal distance between particles) is ensured.