Algorithms for Secretary Problems on Graphs and Hypergraphs

We examine several online matching problems, with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G, with partite sets L, R. We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l \in L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched. Dimitrov and Plaxton show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l \in L, the weights on all edges incident to l are identical.) We use a similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Perhaps of more interest is the fact that our analysis is easily extended to obtain competitive algorithms for similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups. Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.Comment: 15 pages, 2 figure

Some recent results in the analysis of greedy algorithms for assignment problems

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems

Fast Parallel Fixed-Parameter Algorithms via Color Coding

Fixed-parameter algorithms have been successfully applied to solve numerous difficult problems within acceptable time bounds on large inputs. However, most fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no use of the parallel hardware present in modern computers. We show that parallel fixed-parameter algorithms do not only exist for numerous parameterized problems from the literature -- including vertex cover, packing problems, cluster editing, cutting vertices, finding embeddings, or finding matchings -- but that there are parallel algorithms working in \emph{constant} time or at least in time \emph{depending only on the parameter} (and not on the size of the input) for these problems. Phrased in terms of complexity classes, we place numerous natural parameterized problems in parameterized versions of AC$^0$. On a more technical level, we show how the \emph{color coding} method can be implemented in constant time and apply it to embedding problems for graphs of bounded tree-width or tree-depth and to model checking first-order formulas in graphs of bounded degree

Distributed Interior-point Method for Loosely Coupled Problems

In this paper, we put forth distributed algorithms for solving loosely coupled unconstrained and constrained optimization problems. Such problems are usually solved using algorithms that are based on a combination of decomposition and first order methods. These algorithms are commonly very slow and require many iterations to converge. In order to alleviate this issue, we propose algorithms that combine the Newton and interior-point methods with proximal splitting methods for solving such problems. Particularly, the algorithm for solving unconstrained loosely coupled problems, is based on Newton's method and utilizes proximal splitting to distribute the computations for calculating the Newton step at each iteration. A combination of this algorithm and the interior-point method is then used to introduce a distributed algorithm for solving constrained loosely coupled problems. We also provide guidelines on how to implement the proposed methods efficiently and briefly discuss the properties of the resulting solutions.Comment: Submitted to the 19th IFAC World Congress 201

Explicit memory schemes for evolutionary algorithms in dynamic environments

Copyright @ 2007 Springer-VerlagProblem optimization in dynamic environments has atrracted a growing interest from the evolutionary computation community in reccent years due to its importance in real world optimization problems. Several approaches have been developed to enhance the performance of evolutionary algorithms for dynamic optimization problems, of which the memory scheme is a major one. This chapter investigates the application of explicit memory schemes for evolutionary algorithms in dynamic environments. Two kinds of explicit memory schemes: direct memory and associative memory, are studied within two classes of evolutionary algorithms: genetic algorithms and univariate marginal distribution algorithms for dynamic optimization problems. Based on a series of systematically constructed dynamic test environments, experiments are carried out to investigate these explicit memory schemes and the performance of direct and associative memory schemes are campared and analysed. The experimental results show the efficiency of the memory schemes for evolutionary algorithms in dynamic environments, especially when the environment changes cyclically. The experimental results also indicate that the effect of the memory schemes depends not only on the dynamic problems and dynamic environments but also on the evolutionary algorithm used