8 research outputs found
A Match in Time Saves Nine: Deterministic Online Matching With Delays
We consider the problem of online Min-cost Perfect Matching with Delays
(MPMD) introduced by Emek et al. (STOC 2016). In this problem, an even number
of requests appear in a metric space at different times and the goal of an
online algorithm is to match them in pairs. In contrast to traditional online
matching problems, in MPMD all requests appear online and an algorithm can
match any pair of requests, but such decision may be delayed (e.g., to find a
better match). The cost is the sum of matching distances and the introduced
delays.
We present the first deterministic online algorithm for this problem. Its
competitive ratio is , where is the
number of requests. This is polynomial in the number of metric space points if
all requests are given at different points. In particular, the bound does not
depend on other parameters of the metric, such as its aspect ratio. Unlike
previous (randomized) solutions for the MPMD problem, our algorithm does not
need to know the metric space in advance
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
Online Algorithms for Multi-Level Aggregation
In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes
of an edge-weighted tree T, and have to be served eventually. A service is
defined as a subtree X of T that contains its root. This subtree X serves all
requests that are pending in the nodes of X, and the cost of this service is
equal to the total weight of X. Each request also incurs waiting cost between
its arrival and service times. The objective is to minimize the total waiting
cost of all requests plus the total cost of all service subtrees. MLAP is a
generalization of some well-studied optimization problems; for example, for
trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while
for trees of depth 2, it is equivalent to the Joint Replenishment Problem.
Aggregation problem for trees of arbitrary depth arise in multicasting, sensor
networks, communication in organization hierarchies, and in supply-chain
management. The instances of MLAP associated with these applications are
naturally online, in the sense that aggregation decisions need to be made
without information about future requests.
Constant-competitive online algorithms are known for MLAP with one or two
levels. However, it has been open whether there exist constant competitive
online algorithms for trees of depth more than 2. Addressing this open problem,
we give the first constant competitive online algorithm for networks of
arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where
D is the depth of T. The algorithm works for arbitrary waiting cost functions,
including the variant with deadlines.
We also show several additional lower and upper bound results for some
special cases of MLAP, including the Single-Phase variant and the case when the
tree is a path
Online Facility Location with Linear Delay
In the problem of online facility location with delay, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a certain penalty: each client incurs a waiting cost (equal to the difference between its arrival and its connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. That is, an algorithm needs to balance three types of costs: cost of opening facilities, costs of connecting clients, and the waiting costs of clients. We study a natural variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay, where clients may connect to a facility only at its opening time.
We present an O(1)-competitive deterministic algorithm for the two-sided delay variant. Our approach is an extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location. To this end, we substantially simplify the part of the original argument in which a bound on the sequence of factor-revealing LPs is derived. We then show how to transform our O(1)-competitive algorithm for the two-sided delay variant to O(log n / log log n)-competitive deterministic algorithm for one-sided delays. This improves the known O(log n) bound by Azar and Touitou [FOCS 2020]. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios
Online Facility Location with Linear Delay
In the problem of online facility location with delay, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a certain penalty: each client incurs a waiting cost (equal to the difference between its arrival and its connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. That is, an algorithm needs to balance three types of costs: cost of opening facilities, costs of connecting clients, and the waiting costs of clients. We study a natural variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay, where clients may connect to a facility only at its opening time.
We present an O(1)-competitive deterministic algorithm for the two-sided delay variant. Our approach is an extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location. To this end, we substantially simplify the part of the original argument in which a bound on the sequence of factor-revealing LPs is derived. We then show how to transform our O(1)-competitive algorithm for the two-sided delay variant to O(log n / log log n)-competitive deterministic algorithm for one-sided delays. This improves the known O(log n) bound by Azar and Touitou [FOCS 2020]. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios
Online Algorithms for Multi-Level Aggregation
In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests.
Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4*2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting
Online algorithms for multi-level aggregation
In the multilevel aggregation problem (MLAP), requests arrive at the nodes of an edge-weighted tree T and have to be served eventually. A service is defined as a subtree X of T that contains the root of T. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the Transmission Control Protocol acknowledgment problem, whereas for trees of depth 2, it is equivalent to the joint replenishment problem. Aggregation problems for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and supply chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant-competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant-competitive online algorithm for trees of arbitrary (fixed) depth. The competitive ratio is O(D42D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines