Approximation Algorithms for the Joint Replenishment Problem with Deadlines

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness

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 $\rho$ has a fixed cost $c_\rho$. 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

Frameworks for Nonclairvoyant Network Design with Deadlines or Delay

Clairvoyant network design with deadlines or delay has been studied extensively, culminating in an O(log n)-competitive general framework, where n is the number of possible request types (Azar and Touitou, FOCS 2020). In the nonclairvoyant setting, the problem becomes much harder, as ?(?n) lower bounds are known for certain problems (Azar et al., STOC 2017). However, no frameworks are known for the nonclairvoyant setting, and previous work focuses only on specific problems, e.g., multilevel aggregation (Le et al., SODA 2023). In this paper, we present the first nonclairvoyant frameworks for network design with deadlines or delay. These frameworks are nearly optimal: their competitive ratio is O?(?n), which matches known lower bounds up to logarithmic factors

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