How Many Needles are in a Haystack, or How to Solve #P-Complete Counting Problems Fast

We present two randomized entropy-based algorithms for approximating quite general #P-complete counting problems, like the number of Hamiltonian cycles in a graph, the permanent, the number of self-avoiding walks and the satisfiability problem. In our algorithms we first cast the underlying counting problem into an associate rare-event probability estimation, and then apply dynamic importance sampling (IS) to estimate efficiently the desired counting quantity. We construct the IS distribution by using two different approaches: one based on the cross-entropy (CE) method and the other one on the stochastic version of the well known minimum entropy (MinxEnt) method. We also establish convergence of our algorithms and confidence intervals for some special settings and present supportive numerical results, which strongly suggest that both ones (CE and MinxEnt) have polynomial running time in the size of the problem.     

A theoretical approach to saturation effects in lasers

Exact numerical solution of a one-dimensional three-level lasing system (homogeneously broadened) is used to establish the relevance of spatially independent rate equations. It is shown that these equations can be applied not only for unsaturated lasing media but also for highly saturated media. However the loss term and the correlation term have to be modified.     

Stability and topology of scale-free networks under attack and defense strategies

We study tolerance and topology of random scale-free networks under attack and defense strategies that depend on the degree k of the nodes. This situation occurs, for example, when the robustness of a node depends on its degree or in an intentional attack with insufficient knowledge of the network. We determine, for all strategies, the critical fraction p(c) of nodes that must be removed for disintegrating the network. We find that, for an intentional attack, little knowledge of the well-connected sites is sufficient to strongly reduce p(c). At criticality, the topology of the network depends on the removal strategy, implying that different strategies may lead to different kinds of percolation transitions.     

Quasi-invariant domain constants

Five domain constants are studied in our paper, all related to the hyperbolic geometry in hyperbolic plane regions which are uniformly perfect (in Pommerenke’s terminology). Relations among these domain constants are obtained, from which bounds are derived for the variance ratio of each constant under conformal mappings of the regions, and we also show that each constant may be used to characterize uniformly perfect regions. The authors wish to thank the University of California, San Diego for its hospitality during the 1988–89 academic year when this research was begun. Supported by the Landau Center for Mathematical Research in Analysis. Research partially supported by NSF Grant No. DMS-8801439.     

Vehicle routing via column generation

This paper explores an approximate method for solving a routing problem in a four-level distribution which has “double-ended” demand. Routes are represented as columns in a linear program and column generation is used to improve the solution by generating new routes. The generation of new routes is based on an LP sub-problem. Its solution is rounded down to integer values to insure its feasibility as a route for inclusion in the restricted master problem. Finally, an illustrative problem is solved.     

Aharonov invariants and univalent functions

Several properties of a certain series of differential operators which are invariant under the Möbius group (Aharonov invariants) are proved, and in terms of this series new conditions for univalence and quasiconformal extendability of meromorphic functions are established.     

Optimality conditions and strong duality in abstract and continuous-time linear programming

In this work, optimality conditions for infinite-dimensional linear programs are considered. Strong duality as an optimality condition is investigated. A new approach to duality in the form of positive extendability of linear functionals is proposed. A necessary and sufficient condition for duality in the form of a boundedness test of a related linear program is developed. Elaborating on the continuous time framework, counter cases where duality is not valid are given. In lieu of duality, other generalized duality conditions are proposed for the purpose of testing the optimality of a solution.