42 research outputs found
A 4/3-competitive randomized algorithm for online scheduling of packets with agreeable deadlines
In 2005 Li et al. gave a phi-competitive deterministic online algorithm for
scheduling of packets with agreeable deadlines with a very interesting
analysis. This is known to be optimal due to a lower bound by Hajek. We claim
that the algorithm by Li et al. can be slightly simplified, while retaining its
competitive ratio. Then we introduce randomness to the modified algorithm and
argue that the competitive ratio against oblivious adversary is at most 4/3.
Note that this still leaves a gap between the best known lower bound of 5/4 by
Chin et al. for randomised algorithms against oblivious adversary.Comment: 11 pages, 3-4 figures; new version due to STACS submissio
Mechanism design for aggregating energy consumption and quality of service in speed scaling scheduling
We consider a strategic game, where players submit jobs to a machine that
executes all jobs in a way that minimizes energy while respecting the given
deadlines. The energy consumption is then charged to the players in some way.
Each player wants to minimize the sum of that charge and of their job's
deadline multiplied by a priority weight. Two charging schemes are studied, the
proportional cost share which does not always admit pure Nash equilibria, and
the marginal cost share, which does always admit pure Nash equilibria, at the
price of overcharging by a constant factor
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
The -Server Problem on Bounded Depth Trees
We study the -server problem in the resource augmentation setting i.e.,
when the performance of the online algorithm with servers is compared to
the offline optimal solution with servers. The problem is very
poorly understood beyond uniform metrics. For this special case, the classic
-server algorithms are roughly -competitive when
, for any . Surprisingly however, no
-competitive algorithm is known even for HSTs of depth 2 and even when
is arbitrarily large.
We obtain several new results for the problem. First we show that the known
-server algorithms do not work even on very simple metrics. In particular,
the Double Coverage algorithm has competitive ratio irrespective of
the value of , even for depth-2 HSTs. Similarly the Work Function Algorithm,
that is believed to be optimal for all metric spaces when , has
competitive ratio on depth-3 HSTs even if . Our main result
is a new algorithm that is -competitive for constant depth trees,
whenever for any . Finally, we give a general
lower bound that any deterministic online algorithm has competitive ratio at
least 2.4 even for depth-2 HSTs and when is arbitrarily large. This gives
a surprising qualitative separation between uniform metrics and depth-2 HSTs
for the -server problem, and gives the strongest known lower bound for
the problem on general metrics.Comment: Appeared in SODA 201
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