This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex-type algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the so-called self-dual method, is analyzed. The algorithm is not started at the traditional point (1,..., but points of the form (1, e, e2,...)T, with t sufficiently small, are used. The result is better, in two respects, than those of the previous analyses. First, it is shown that the expected number of steps is bounded between two quadratic functions cl(min(m, n))' and cz(min(m, n)) ' of the smaller dimension of the problem. This should be compared with the previous two major results in the field. Borgwardt proves an upper bound of 0(n4m1'(n-1') under a model that implies that the zero vector satisfies all the constraints, and also the algorithm under his consideration solves only problems from that particular subclass. Smale analyzes the self-dual algorithm starting at (1,..., He shows that for any fixed m there is a constant c(m) such the expected number of steps is less than ~(m)(lnn)"'("+~); Megiddo has shown that, under Smale's model, an upper bound C(m) exists. Thus, for the first time, a polynomial upper bound with no restrictions (except for nondegeneracy) on the problem is proved, and, for the first time, a nontrivial lower bound of precisely the same order of magnitude is established. Both Borgwardt and Smale require the input vectors to be drawn from

In this paper we analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem of n variables with m constraints. Smale established that for every number of constraints m, there is a constant c(m) such that the number of pivot steps of the self-dual algorithm, p(m, n), is less than c(m)(ln n)"""'+". We improve upon this estimate by showing that p(m, n) is bounded by a function of m only. The symmetry of the function in m and n implies that p(m, n) is in fact bounded by a function of the smaller of m and n.

In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of the distribution tails, we improve the existing moment estimates, and we prove various limit theorems for the cases where either n or m and n tend to in nity. Our results are of complexity theoretic interest, because interior-point methods and relaxation methods for the solution of Ax 0, x 6= 0 have running times that are bounded in terms of log C (A) and C (A) respectively. AMS Classi cation: primary 90C31,15A52; secondary 90C05,90C60,62H10. Key Words: condition number, random matrices, linear programming, probabilistic analysis, complexity theory.

A problem that has recently attracted the attention of the research community is the autonomous formation of robot teams to perform complex multirobot tasks. The corresponding problem for software agents is also known in the multi-agent community as the coalition formation problem. Numerous algorithms for software agent coalition formation have been provided that allow for efficient cooperation in both competitive and cooperative environments. However, despite the plethora of relevant literature on the software agent coalition formation problem, and the existence of similar problems in theoretical computer science, the multirobot coalition formation problem has not been sufficiently grounded for different tasks and task environments. In this paper, comparisons are drawn to highlight the differences between software agents and robotics, and parallel problems from theoretical computer science are identified. This paper further explores robot coalition formation in different practical robotic environments. A heuristic-based coalition formation algorithm from our previous work was extended to operate in precedence ordered cooperative environments. In order to explore coalition formation in competitive environments, the paper also studies the RACHNA system, a market based coalition formation system. Finally, the paper investigates the notion of task preemption for complex multi-robot tasks in random allocation environments.

Summary: This lecture introduces the Ellipsoid Method, the first polynomialtime algorithm to solve LP. We start by discussing the historical significance

There has been increasing attention recently on average case algorithmic performance measures since worst case measures can be qualitatively quite different. An important characteristic of a linear program, relating to Simplex Method performance, is the number of vertices of the feasible region. We show 2 ~ to be an upper bound on the mean number of extreme points of a randomly generated feasible region with arbitrary probability distributions on the constraint matrix and right hand side vector. The only assumption made is that inequality directions are chosen independently in accordance with a series of independent fair coin tosses.

Abstract not found