Morten N. Nielsen

Incremental) priority algorithms

Abstractn We study the question of which optimization problems can benoptimally or approximately solved byn“greedy” or “greedy-like” algorithms. For definiteness, we limit the presentndiscussion to some well-studied scheduling problems althoughnthe underlying issues apply in a much more general setting.nOf course, the main benefit of greedy algorithmsnlies in both their conceptual simplicity and their computationalnefficiency. Based on the experience from onlinencompetitive analysis, it seems plausible that we shouldnbe able to derive approximation bounds for “greedy-like”nalgorithms exploiting only the conceptual simplicity of these algorithms.nTo this end, we need (and will provide) a precise definition of what we mean by greedy and greedy-like.nn

The Accommodating Function: A Generalization of the Competitive Ratio

A new measure, the accommodating function, for the quality of on-line algorithms is presented. The accommodating function, which is a generalization of both the competitive ratio and the competitive ratio on accommodating sequences, measures the quality of an on-line algorithm as a function of the resources that would be sufficient for an optimal off-line algorithm to fully grant all requests. More precisely, if we have some amount of resources n, the function value at

Fair versus unrestricted Bin packing

alpha

Extending the accommodating function

is the usual ratio (still on some fixed amount of resources n), except that input sequences are restricted to those where the optimal off-line algorithm will not obtain a better result by having more than the amount

On-Line Edge-Coloring with a Fixed Number of Colors

alpha n

Fair versus Unrestricted Bin Packing

of resources. nThe accommodating functions for three specific on-line problems are investigated: a variant of bin packing in which the goal is to maximize the number of items put in n bins, the seat reservation problem, and the problem of optimizing total flow time when preemption is allowed. nWe also show that when trying to distinguish between two algorithms, the decision as to which one performs better cannot necessarily be made from the competitive ratio or the competitive ratio on accommodating sequences alone. For the variant of bin-packing considered, we show that Worst-Fit has a strictly better competitive ratio than First-Fit, while First-Fit has a strictly better competitive ratio on accommodating sequences than Worst-Fit.

On the Existence and Construction of Non-Extreme (a,b)-Trees

AbstractAbstract. We consider the on-line Dual Bin Packing problem where we have nunit size bins and a sequence of items. The goal is to maximize the number of items that are packed in the bins by an on-line algorithm. We investigate unrestrictedalgorithms that have the power of performing admission control on the items, i.e., rejecting items while there is enough space to pack them, versus fairalgorithms that reject an item only when there is not enough space to pack it. We show that by performing admission control on the items, we get better performance compared with the performance achieved on the fair version of the problem. Our main result shows that with an unfair variant of First-Fit, we can pack approximately two-thirds of the items for sequences for which an optimal off-line algorithm can pack all the items. This is in contrast to standard First-Fit where we show an asymptotically tight hardness result: if the number of bins can be chosen arbitrarily large, the fraction of the items packed by First-Fit comes arbitrarily close to five-eighths.n

Seat reservation allowing seat changes

Abstract.The applicability of the accommodating function, a relatively new measure for the quality of on-line algorithms, is extended.The standard quality measure for on-line algorithms is the competitive ratio, which is, roughly speaking, the worst case ratio of the on-line performance to the optimal off-line performance. However, for many on-line problems, the competitive ratio gives overly pessimistic results and/or fails to distinguish between algorithms that are known to perform very differently in practice. Many researchers have proposed variations on the competitive ratio to obtain more realistic results. These variations are often tailor-made for specific on-line problems.The concept of the accommodating function applies to any on-line problem with some limited resource, such as bins, seats in a train, or pages in a cache. If a limited amount n of some resource is available, the accommodating function

Explicit construction of constrained instantons

mathcal{A}(alpha)

The Accommodating Function - A Generalization of the Competitive Ratio

is the competitive ratio when input sequences are restricted to those for which the amount