Range Assignment for Biconnectivity and k-Edge Connectivity in Wireless Ad Hoc Networks

Depending on whether bidirectional links or unidirectional links are used for communications, the network topology under a given range assignment is either an undirected graph referred to as the bidirectional topology, or a directed graph referred to as the unidirectional topology. The Min-Power Bidirectional (resp., Unidirectional) k-Node Connectivity problem seeks a range assignment of minimum total power subject to the constraint that the produced bidirectional (resp. unidirectional) topology is k-vertex connected. Similarly, the Min-Power Bidirectional (resp., Unidirectional) k-Edge Connectivity problem seeks a range assignment of minimum total power subject to the constraint the produced bidirectional (resp., unidirectional) topology is k-edge connected. The Min-Power Bidirectional Biconnectivity problem and the Min-Power Bidirectional Edge-Biconnectivity problem have been studied by Lloyd et al. [23]. They show that range assignment based the approximation algorithm of Khuller and Raghavachari [18], which we refer to as Algorithm KR, has an approximation ratio of at most 2(2 – 2/n)(2 + 1/n) for Min-Power Bidirectional Biconnectivity, and range assignment based on the approximation algorithm of Khuller and Vishkin [19], which we refer to as Algorithm KV, has an approximation ratio of at most 8(1 – 1/n) for Min-Power Bidirectional Edge-Biconnectivity. In this paper, we first establish the NP-hardness of Min-Power Bidirectional (Edge-) Biconnectivity. Then we show that Algorithm KR has an approximation ratio of at most 4 for both Min-Power Bidirectional Biconnectivity and Min-Power Unidirectional Biconnectivity, and Algorithm KV has an approximation ratio of at most 2k for both Min-Power Bidirectional k-Edge Connectivity and Min-Power Unidirectional k-Edge Connectivity. We also propose a new simple constant-approximation algorithm for both Min-Power Bidirectional Biconnectivity and Min-Power Unidirectional Biconnectivity. This new algorithm applies only to Euclidean instances, but is best suited for distributed implementation. A preliminary version of this work appeared in the proceedings of the 2nd International Conference on AD-HOC Network and Wireless (Adhoc-Now 2003). Research performed in part while visiting the Max-Plank-Institut fur Informatik. Gruia Calinescu is an Assistant Professor of Computer Science at the Illinois Institute of Technology since 2000. He held postdoc or visiting researcher positions at DIMACS, University of Waterloo, and Max-Plank Institut fur Informatik. Gruia has a Diploma from University of Bucharest and a Ph.D. from Georgia Insitute of Technology. His research interests are in the area of algorithms. Peng-Jun Wan has joined the Computer Science Department at Illinois Institute of Technology in 1997 and has been an Associate Professor since 2004. He received his Ph.D. in Computer Science from University of Minnesota in 1997, M.S. in Operations Research and Control Theory from Chinese Academy of Science in 1993, and B.S. in Applied Mathematics from Tsinghua University in 1990. His research interests include optical networks and wireless networks.     

Scale Invariance Properties in the Simulated Annealing Algorithm

The Boltzmann distribution used in the steady-state analysis of the simulated annealing algorithm gives rise to several scale invariant properties. Scale invariance is first presented in the context of parallel independent processors and then extended to an abstract form based on lumping states together to form new aggregate states. These lumped or aggregate states possess all of the mathematical characteristics, forms and relationships of states (solutions) in the original problem in both first and second moments. These scale invariance properties therefore permit new ways of relating objective function values, conditional expectation values, stationary probabilities, rates of change of stationary probabilities and conditional variances. Such properties therefore provide potential applications in analysis, statistical inference and optimization. Directions for future research that take advantage of scale invariance are also discussed.     

Robust handling of non‐linear constraints for GA optimization of aerodynamic shapes

A new approach to the robust handling of non‐linear constraints for GAs (genetic algorithms) optimization is proposed. A specific feature of the approach consists of the change in the conventional search strategy by employing search paths which pass through both feasible and infeasible points (contrary to the traditional approach where only feasible points may be included in a path). The method (driven by full Navier–Stokes computations) was applied to the problem of multiobjective optimization of aerodynamic shapes subject to various geometrical and aerodynamic constraints. The results demonstrated that the method retains high robustness of conventional GAs while keeping CFD computational volume to an acceptable level, which allowed the algorithm to be used in a demanding engineering environment. Copyright © 2004 John Wiley & Sons, Ltd.     

Gray code for derangements

We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no fixed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of fixed points bounded between two constants.     

Local approximations for maximum partial subgraph problem

We deal with MAXH₀-FREE PARTIAL SUBGRAPH. We mainly prove that 3-locally optimum solutions achieve approximation ratio (δ₀+1)/(B+2+ν₀), where B=max_v∈Vd_G(v), δ₀=min_v∈V(H0)d_H0(v) and ν₀=(|V(H₀)|+1)/δ₀. Next, we show that this ratio rises up to 3/(B+1) when H₀=K₃. Finally, we provide hardness results for MAXK₃-FREE PARTIAL SUBGRAPH.     

Equivalence of two linear programming relaxations for broadcast scheduling

A server needs to compute a broadcast schedule for n pages whose request times are known in advance. Outputting a page satisfies all outstanding requests for the page. The goal is to minimize the average waiting time of a client. In this paper, we show the equivalence of two apparently different relaxations that have been considered for this problem.     

Strongly polynomial-time approximation for a class of bicriteria problems

Consider the following problem: given a ground set and two minimization objectives of the same type find a subset from a given subset-class that minimizes the first objective subject to a budget constraint on the second objective. Using Megiddo's parametric method we improve an earlier weakly polynomial time algorithm.     

Decomposition Methods for Differentiable Optimization Problems over Cartesian Product Sets

This paper presents a unified analysis of decomposition algorithms for continuously differentiable optimization problems defined on Cartesian products of convex feasible sets. The decomposition algorithms are analyzed using the framework of cost approx imation algorithms. A convergence analysis is made for three decomposition algorithms: a sequential algorithm which extends the classical Gauss-Seidel scheme, a synchronized parallel algorithm which extends the Jacobi method, and a partially asynchronous parallel algorithm. The analysis validates inexact computations in both the subproblem and line search phases, and includes convergence rate results. The range of feasible step lengths within each algorithm is shown to have a direct correspondence to the increasing degree of parallelism and asynchronism, and the resulting usage of more outdated information in the algorithms.     

Scheduling orders for multiple product types to minimize total weighted completion time

We consider the problem of scheduling orders for multiple different product types in an environment with m dedicated machines in parallel. The objective is to minimize the total weighted completion time. Each product type is produced by one and only one of the m dedicated machines; that is, each machine is dedicated to a specific product type. Each order has a weight and may also have a release date. Each order asks for certain amounts of various different product types. The different products for an order can be produced concurrently. Preemptions are not allowed. Even when all orders are available at time 0, the problem has been shown to be strongly NP-hard for any fixed number (?2) of machines. This paper focuses on the design and analysis of efficient heuristics for the case without release dates. Occasionally, however, we extend our results to the case with release dates. The heuristics considered include some that have already been proposed in the literature as well as several new ones. They include various static and dynamic priority rules as well as two more sophisticated LP-based algorithms. We analyze the performance bounds of the priority rules and of the algorithms and present also an in-depth comparative analysis of the various rules and algorithms. The conclusions from this empirical analysis provide insights into the trade-offs with regard to solution quality, speed, and memory space.     

On the construction of samples for testing approximate optimization methods for estimate calculation algorithms

To validate approximate optimization schemes for estimate calculation algorithms (ECAs), it is necessary to compute the optimal height, which cannot be done in a reasonable amount of time. A variety of samples are built for which the optimal height of the ECAs is known by construction.