98 research outputs found
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches
In the online packet buffering problem (also known as the unweighted FIFO
variant of buffer management), we focus on a single network packet switching
device with several input ports and one output port. This device forwards
unit-size, unit-value packets from input ports to the output port. Buffers
attached to input ports may accumulate incoming packets for later transmission;
if they cannot accommodate all incoming packets, their excess is lost. A packet
buffering algorithm has to choose from which buffers to transmit packets in
order to minimize the number of lost packets and thus maximize the throughput.
We present a tight lower bound of e/(e-1) ~ 1.582 on the competitive ratio of
the throughput maximization, which holds even for fractional or randomized
algorithms. This improves the previously best known lower bound of 1.4659 and
matches the performance of the algorithm Random Schedule. Our result
contradicts the claimed performance of the algorithm Random Permutation; we
point out a flaw in its original analysis
An Improved Algorithm For Online Min-Sum Set Cover
We study a fundamental model of online preference aggregation, where an
algorithm maintains an ordered list of elements. An input is a stream of
preferred sets . Upon seeing and without
knowledge of any future sets, an algorithm has to rerank elements (change the
list ordering), so that at least one element of is found near the list
front. The incurred cost is a sum of the list update costs (the number of swaps
of neighboring list elements) and access costs (position of the first element
of on the list). This scenario occurs naturally in applications such as
ordering items in an online shop using aggregated preferences of shop
customers. The theoretical underpinning of this problem is known as Min-Sum Set
Cover.
Unlike previous work (Fotakis et al., ICALP 2020, NIPS 2020) that mostly
studied the performance of an online algorithm ALG against the static optimal
solution (a single optimal list ordering), in this paper, we study an arguably
harder variant where the benchmark is the provably stronger optimal dynamic
solution OPT (that may also modify the list ordering). In terms of an online
shop, this means that the aggregated preferences of its user base evolve with
time. We construct a computationally efficient randomized algorithm whose
competitive ratio (ALG-to-OPT cost ratio) is and prove the existence
of a deterministic -competitive algorithm. Here, is the maximum
cardinality of sets . This is the first algorithm whose ratio does not
depend on : the previously best algorithm for this problem was -competitive and is a lower bound on the
performance of any deterministic online algorithm.Comment: Presented at AAAI 202
Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration
In this paper, we construct a deterministic 4-competitive algorithm for the online file migration problem, beating the currently best 20-year old, 4.086-competitive MTLM algorithm by Bartal et al. (SODA 1997). Like MTLM, our algorithm also operates in phases, but it adapts their lengths dynamically depending on the geometry of requests seen so far. The improvement was obtained by carefully analyzing a linear model (factor-revealing LP) of a single phase of the algorithm. We also show that if an online algorithm operates in phases of fixed length and the adversary is able to modify the graph between phases, no algorithm can beat the competitive ratio of 4.086
A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location
In the online non-metric variant of the facility location problem, there is a
given graph consisting of a set of facilities (each with a certain opening
cost), a set of potential clients, and weighted connections between them.
The online part of the input is a sequence of clients from , and in response
to any requested client, an online algorithm may open an additional subset of
facilities and must connect the given client to an open facility.
We give an online, polynomial-time deterministic algorithm for this problem,
with a competitive ratio of . The
result is optimal up to loglog factors. Our algorithm improves over the
-competitive
construction that first reduces the facility location instance to a set cover
one and then later solves such instance using the deterministic algorithm by
Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical
scenario where .
We achieve this by a more direct approach: we design an algorithm for a
fractional relaxation of the non-metric facility location problem with
clustered facilities. To handle the constraints of such non-covering LP, we
combine the dual fitting and multiplicative weight updates approach. By
maintaining certain additional monotonicity properties of the created
fractional solution, we can handle the dependencies between facilities and
connections in a rounding routine.
Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the
first deterministic algorithm for the online node-weighted Steiner tree
problem. The resulting competitive ratio is on
graphs of nodes and terminals.Comment: STACS 202
A Subquadratic Bound for Online Bisection
In the online bisection problem one has to maintain a partition of
elements into two clusters of cardinality . During runtime, an online
algorithm is given a sequence of requests, each being a pair of elements: an
inter-cluster request costs one unit while an intra-cluster one is free. The
algorithm may change the partition, paying a unit cost for each element that
changes its cluster.
This natural problem admits a simple deterministic -competitive
algorithm [Avin et al., DISC 2016]. While several significant improvements over
this result have been obtained since the original work, all of them either
limit the generality of the input or assume some form of resource augmentation
(e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the
best known competitive ratio even if randomization is allowed.
In this paper, we present a first randomized online algorithm that breaks
this natural barrier and achieves a competitive ratio of
without resource augmentation and for an arbitrary sequence of requests
Dynamic sharing of a multiple access channel
In this paper we consider the mutual exclusion problem on a multiple access
channel. Mutual exclusion is one of the fundamental problems in distributed
computing. In the classic version of this problem, n processes perform a
concurrent program which occasionally triggers some of them to use shared
resources, such as memory, communication channel, device, etc. The goal is to
design a distributed algorithm to control entries and exits to/from the shared
resource in such a way that in any time there is at most one process accessing
it. We consider both the classic and a slightly weaker version of mutual
exclusion, called ep-mutual-exclusion, where for each period of a process
staying in the critical section the probability that there is some other
process in the critical section is at most ep. We show that there are channel
settings, where the classic mutual exclusion is not feasible even for
randomized algorithms, while ep-mutual-exclusion is. In more relaxed channel
settings, we prove an exponential gap between the makespan complexity of the
classic mutual exclusion problem and its weaker ep-exclusion version. We also
show how to guarantee fairness of mutual exclusion algorithms, i.e., that each
process that wants to enter the critical section will eventually succeed
An Improved Online Algorithm for the Traveling Repairperson Problem on a Line
In the online variant of the traveling repairperson problem (TRP), requests arrive in time at points of a metric space X and must be eventually visited by a server. The server starts at a designated point of X and travels at most at unit speed. Each request has a given weight and once the server visits its position, the request is considered serviced; we call such time completion time of the request. The goal is to minimize the weighted sum of completion times of all requests.
In this paper, we give a 5.429-competitive deterministic algorithm for line metrics improving over 5.829-competitive solution by Krumke et al. (TCS 2003). Our result is obtained by modifying the schedule by serving requests that are close to the origin first. To compute the competitive ratio of our approach, we use a charging scheme, and later evaluate its properties using a factor-revealing linear program which upper-bounds the competitive ratio
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
Dynamic Balanced Graph Partitioning
This paper initiates the study of the classic balanced graph partitioning
problem from an online perspective: Given an arbitrary sequence of pairwise
communication requests between nodes, with patterns that may change over
time, the objective is to service these requests efficiently by partitioning
the nodes into clusters, each of size , such that frequently
communicating nodes are located in the same cluster. The partitioning can be
updated dynamically by migrating nodes between clusters. The goal is to devise
online algorithms which jointly minimize the amount of inter-cluster
communication and migration cost.
The problem features interesting connections to other well-known online
problems. For example, scenarios with generalize online paging, and
scenarios with constitute a novel online variant of maximum matching. We
present several lower bounds and algorithms for settings both with and without
cluster-size augmentation. In particular, we prove that any deterministic
online algorithm has a competitive ratio of at least , even with significant
augmentation. Our main algorithmic contributions are an -competitive deterministic algorithm for the general setting with
constant augmentation, and a constant competitive algorithm for the maximum
matching variant
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