68 research outputs found
A -Competitive Algorithm for Scheduling Packets with Deadlines
In the online packet scheduling problem with deadlines (PacketScheduling, for
short), the goal is to schedule transmissions of packets that arrive over time
in a network switch and need to be sent across a link. Each packet has a
deadline, representing its urgency, and a non-negative weight, that represents
its priority. Only one packet can be transmitted in any time slot, so, if the
system is overloaded, some packets will inevitably miss their deadlines and be
dropped. In this scenario, the natural objective is to compute a transmission
schedule that maximizes the total weight of packets which are successfully
transmitted. The problem is inherently online, with the scheduling decisions
made without the knowledge of future packet arrivals. The central problem
concerning PacketScheduling, that has been a subject of intensive study since
2001, is to determine the optimal competitive ratio of online algorithms,
namely the worst-case ratio between the optimum total weight of a schedule
(computed by an offline algorithm) and the weight of a schedule computed by a
(deterministic) online algorithm.
We solve this open problem by presenting a -competitive online
algorithm for PacketScheduling (where is the golden ratio),
matching the previously established lower bound.Comment: Major revision of the analysis and some other parts of the paper.
Another revision will follo
Revealed automorphisms
summary:We study automorphisms in the alternative set theory. We prove that fully revealed automorphisms are not closed under composition. We also construct some special automorphisms. We generalize the notion of revealment and Sd-class
Online Bin Stretching with Three Bins
Online Bin Stretching is a semi-online variant of bin packing in which the
algorithm has to use the same number of bins as an optimal packing, but is
allowed to slightly overpack the bins. The goal is to minimize the amount of
overpacking, i.e., the maximum size packed into any bin.
We give an algorithm for Online Bin Stretching with a stretching factor of
for three bins. Additionally, we present a lower bound of for Online Bin Stretching on three bins and a lower bound of
for four and five bins that were discovered using a computer search.Comment: Preprint of a journal version. See version 2 for the conference
paper. Conference paper split into two journal submissions; see
arXiv:1601.0811
The complexity of coloring graphs without long induced paths
We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a P8-free graph is 5-colorable and of deciding whether a P12-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a P5-free graph is 3-colorable
Solution of David Gale's lion and man problem
AbstractA pursue-and-evasion game is analyzed, including almost optimal bounds on the number of moves needed to win
Better Approximation Bounds for the Joint Replenishment Problem
The Joint Replenishment Problem (JRP) deals with optimizing shipments of
goods from a supplier to retailers through a shared warehouse. Each shipment
involves transporting goods from the supplier to the warehouse, at a fixed cost
C, followed by a redistribution of these goods from the warehouse to the
retailers that ordered them, where transporting goods to a retailer has
a fixed cost . In addition, retailers incur waiting costs for each
order. The objective is to minimize the overall cost of satisfying all orders,
namely the sum of all shipping and waiting costs.
JRP has been well studied in Operations Research and, more recently, in the
area of approximation algorithms. For arbitrary waiting cost functions, the
best known approximation ratio is 1.8. This ratio can be reduced to 1.574 for
the JRP-D model, where there is no cost for waiting but orders have deadlines.
As for hardness results, it is known that the problem is APX-hard and that the
natural linear program for JRP has integrality gap at least 1.245. Both results
hold even for JRP-D. In the online scenario, the best lower and upper bounds on
the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64
applies even to the restricted version of JRP, denoted JRP-L, where the waiting
cost function is linear.
We provide several new approximation results for JRP. In the offline case, we
give an algorithm with ratio 1.791, breaking the barrier of 1.8. In the online
case, we show a lower bound of 2.754 on the competitive ratio for JRP-L (and
thus JRP as well), improving the previous bound of 2.64. We also study the
online version of JRP-D, for which we prove that the optimal competitive ratio
is 2
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