Packing items from a triangular distribution

We consider the problem of packing n items which are drawn according to a probability distribution whose density function is triangular in shape. For triangles which represent density functions whose expectation is 1/p for p = 3, 4, 5, ..., we give a packing strategy for which the ratio of the number of bins used in the packing to the expected total size of the items asymptotically approaches 1

RGFGA: An efficient representation and crossover for grouping genetic algorithms

There is substantial research into genetic algorithms that are used to group large numbers of objects into mutually exclusive subsets based upon some fitness function. However, nearly all methods involve degeneracy to some degree. We introduce a new representation for grouping genetic algorithms, the restricted growth function genetic algorithm, that effectively removes all degeneracy, resulting in a more efficient search. A new crossover operator is also described that exploits a measure of similarity between chromosomes in a population. Using several synthetic datasets, we compare the performance of our representation and crossover with another well known state-of-the-art GA method, a strawman optimisation method and a well-established statistical clustering algorithm, with encouraging results

Online Bin Covering: Expectations vs. Guarantees

Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin. For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all "reasonable" algorithms have a competitive ratio of 1/2. Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restricted or uniformly distributed item sizes. In this way, our study also supplements the ongoing systematic studies of the relative strengths of various performance measures. Two classic algorithms for online bin packing that have natural dual versions are Harmonic and Next-Fit. Even though the algorithms are quite different in nature, the dual versions are not separated by competitive analysis. We make the case that when guarantees are needed, even under restricted input sequences, dual Harmonic is preferable. In addition, we establish quite robust theoretical results showing that if items come from a uniform distribution or even if just the ordering of items is uniformly random, then dual Next-Fit is the right choice.Comment: IMADA-preprint-c

Number partitioning as random energy model

Number partitioning is a classical problem from combinatorial optimisation. In physical terms it corresponds to a long range anti-ferromagnetic Ising spin glass. It has been rigorously proven that the low lying energies of number partitioning behave like uncorrelated random variables. We claim that neighbouring energy levels are uncorrelated almost everywhere on the energy axis, and that energetically adjacent configurations are uncorrelated, too. Apparently there is no relation between geometry (configuration) and energy that could be exploited by an optimization algorithm. This ``local random energy'' picture of number partitioning is corroborated by numerical simulations and heuristic arguments.Comment: 8+2 pages, 9 figures, PDF onl

Scheduling Storms and Streams in the Cloud

Motivated by emerging big streaming data processing paradigms (e.g., Twitter Storm, Streaming MapReduce), we investigate the problem of scheduling graphs over a large cluster of servers. Each graph is a job, where nodes represent compute tasks and edges indicate data-flows between these compute tasks. Jobs (graphs) arrive randomly over time, and upon completion, leave the system. When a job arrives, the scheduler needs to partition the graph and distribute it over the servers to satisfy load balancing and cost considerations. Specifically, neighboring compute tasks in the graph that are mapped to different servers incur load on the network; thus a mapping of the jobs among the servers incurs a cost that is proportional to the number of "broken edges". We propose a low complexity randomized scheduling algorithm that, without service preemptions, stabilizes the system with graph arrivals/departures; more importantly, it allows a smooth trade-off between minimizing average partitioning cost and average queue lengths. Interestingly, to avoid service preemptions, our approach does not rely on a Gibbs sampler; instead, we show that the corresponding limiting invariant measure has an interpretation stemming from a loss system.Comment: 14 page