We consider the List Update problem where the cost of each swap is assumed to be 1. This is in contrast to the "standard" model, in which an algorithm is allowed to swap the requested item with previous items for free. We construct an online algorithm Full-Or-Partial-Move (FPM), whose competitive ratio is at most 3.3904, improving over the previous best known bound of 4.

Abstract. In the online packet scheduling problem with deadlines ([Formula: see text], 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 nonnegative weight, which 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 that are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning [Formula: see text] 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 [Formula: see text]-competitive online algorithm for [Formula: see text] (where [Formula: see text] is the golden ratio), matching the previously established lower bound.

We study the resource augmented version of the k-server problem, also known as the k-server problem against weak adversaries or the (h,k)-server problem. In this setting, an online algorithm using k servers is compared to an offline algorithm using h servers, where h ≤ k. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any є>0, the competitive ratio drops to a constant if k=(1+є) · h. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics. We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least Ω(loglogh), even as k→∞. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics.

In this paper we consider the k-server problem where events are generated by selfish agents, known as the selfish k-server problem. In this setting, there is a set of k servers located in some metric space. Selfish agents arrive in an online fashion, each has a request located on some point in the metric space, and seeks to serve his request with the server of minimum distance to the request. If agents choose to serve their request with the nearest server, this mimics the greedy algorithm which has an unbounded competitive ratio. We propose an algorithm that associates a surcharge with each server independently of the agent to arrive (and therefore, yields a truthful online mechanism). An agent chooses to serve his request with the server that minimizes the distance to the request plus the associated surcharge to the server. This paper extends [Ilan Reuven Cohen et al., 2015], which gave an optimal k-competitive dynamic pricing scheme for the selfish k-server problem on the line. We give a k-competitive dynamic pricing algorithm for the selfish k-server problem on tree metric spaces, which matches the optimal online (non truthful) algorithm. We show that an alpha-competitive dynamic pricing scheme exists on the tree if and only if there exists alpha-competitive online algorithm on the tree that is lazy and monotone. Given this characterization, the main technical difficulty is coming up with such an online algorithm.

The generalized k-server problem is an extension of the weighted k-server problem, which in turn extends the classic k-server problem. In the generalized k-server problem, each of k servers s_1, ..., s_k remains in its own metric space M_i. A request is a tuple (r_1,...,r_k), where r_i in M_i, and to service it, an algorithm needs to move at least one server s_i to the point r_i. The objective is to minimize the total distance traveled by all servers. In this paper, we focus on the generalized k-server problem for the case where all M_i are uniform metrics. We show an O(k^2 * log k)-competitive randomized algorithm improving over a recent result by Bansal et al. [SODA 2018], who gave an O(k^3 * log k)-competitive algorithm. To this end, we define an abstract online problem, called Hydra game, and we show that a randomized solution of low cost to this game implies a randomized algorithm to the generalized k-server problem with low competitive ratio. We also show that no randomized algorithm can achieve competitive ratio lower than Omega(k), thus improving the lower bound of Omega(k / log^2 k) by Bansal et al.

We study online competitive algorithms for the \emph{line chasing problem} in Euclidean spaces $\reals^d$, where the input consists of an initial point $P_0$ and a sequence of lines $X_1,X_2,...,X_m$, revealed one at a time. At each step $t$, when the line $X_t$ is revealed, the algorithm must determine a point $P_t\in X_t$. An online algorithm is called $c$-competitive if for any input sequence the path $P_0, P_1,...,P_m$ it computes has length at most $c$ times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets $X_t$ are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was $28.1$, even in the plane. We significantly improve this bound, by providing a~$3$-competitive algorithm for any dimension $d$. We also improve the lower bound on the competitive ratio, from $1.412$ to $1.5358$.

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 ϕ-competitive online algorithm for PacketScheduling (where ϕ ≈ 1.618 is the golden ratio), matching the previously established lower bound.

We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. 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 k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

We study the online bounded-delay packet scheduling problem (PacketScheduling), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission. The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature; yet currently the best published upper bound is 1.828 [8], still quite far from the best lower bound of ϕ≈1.618 [11], [2], [6]. In the variant of PacketScheduling with s-bounded instances, each packet can be scheduled in at most s consecutive slots, starting at its release time. The lower bound of ϕ applies even to the special case of 2-bounded instances, and a ϕ-competitive algorithm for 3-bounded instances was given in [5]. Improving that result, and addressing a question posed by Goldwasser [9], we present a ϕ-competitive algorithm for 4-bounded instances. We also study a variant of PacketScheduling where an online algorithm has the additional power of 1-lookahead, knowing at time t which packets will arrive at time t+1. For PacketScheduling with 1-lookahead restricted to 2-bounded instances, we present an online algorithm with competitive ratio 12(13−1)≈1.303 and we prove a nearly tight lower bound of 14(1+17)≈1.281. In fact, our lower bound result is more general: using only 2-bounded instances, for any integer ℓ≥0 we prove a lower bound of 12(ℓ+1)(1+5+8ℓ+4ℓ2) for online algorithms with ℓ-lookahead, i.e., algorithms that at time t can see all packets arriving by time t+ℓ. Finally, for non-restricted instances we show a lower bound of 1.25 for randomized algorithms with ℓ-lookahead, for any ℓ≥0.

We study the Double Coverage (DC) algorithm for the k-server problem in tree metrics in the (h, k)-setting, i.e., when DC with k servers is compared against an offline optimum algorithm with h ≤ k servers. It is well-known that in such metric spaces DC is k-competitive (and thus optimal) for h = k. We prove that even if k > h the competitive ratio of DC does not improve; in fact, it increases slightly as k grows, tending to h + 1. Specifically, we give matching upper and lower bounds of $\frac {k(h+1)}{k+1}$ on the competitive ratio of DC on any tree metric.

We study the k-server problem in the resource augmentation setting i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with H ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k-server algorithms are roughly (1 + 1/∊)-competitive when k = (1 + ∊)h, for any ∊ > 0. Surprisingly however, no o(h)- competitive algorithm is known even for HSTs of depth 2 and even when k/h is arbitrarily large. We obtain several new results for the problem. First we show that the known k-server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio O(h) irrespective of the value of k, even for depth-2 HSTs. Similarly the Work Function Algorithm, that is believed to be optimal for all metric spaces when k = h, has competitive ratio O(h) on depth-3 HSTs even if k = 2h. Our main result is a new algorithm that is O(1)-competitive for constant depth trees, whenever k = (1 + ∊)h for any ∊ > 0. 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 k/h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the (h, k)-server problem, and gives the strongest known lower bound for the problem on general metrics.

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.

Production and distribution are fundamental operational functions in supply chains. The main challenge is to design algorithms that optimize operational performance by jointly scheduling production and delivery of customer orders. In this paper we study a model of scheduling customer orders on multiple identical machines and their distribution to customers afterwards. The goal is to minimize the total time from release to distribution plus total distribution cost to the customers. We design the first poly-logarithmic competitive algorithm for the problem, improving upon previous algorithms with linear competitive ratios. Our model generalizes two fundamental problems: scheduling of jobs on multiple identical machines (where the goal function is to minimize the total flow time) as well as the TCP Acknowledgment problem.

We consider online preemptive scheduling of jobs with fixed starting times revealed at those times on $$m$$ uniformly related machines, with the goal of maximizing the total weight of completed jobs. Every job has a size and a weight associated with it. A newly released job must be either assigned to start running immediately on a machine or otherwise it is dropped. It is also possible to drop an already scheduled job, but only completed jobs contribute their weights to the profit of the algorithm. In the most general setting, no algorithm has bounded competitive ratio, and we consider a number of standard variants. We give a full classification of the variants into cases which admit constant competitive ratio (weighted and unweighted unit jobs, and C-benevolent instances, which is a wide class of instances containing proportional-weight jobs), and cases which admit only a linear competitive ratio (unweighted jobs and D-benevolent instances). In particular, we give a lower bound of $$m$$ on the competitive ratio for scheduling unit weight jobs with varying sizes, which is tight. For unit size and weight we show that a natural greedy algorithm is $$4/3$$ -competitive and optimal on $$m=2$$ machines, while for large $$m$$ , its competitive ratio is between $$1.56$$ and $$2$$ . Furthermore, no algorithm is better than $$1.5$$ -competitive.

We investigate the online variant of the (Multiple) Knapsack Problem: an algorithm is to pack items, of arbitrary sizes and profits, in k knapsacks (bins) without exceeding the capacity of any bin. We study two objective functions: the sum and the maximum of profits over all bins. With either objective, our problem statement captures and generalizes previously studied problems, e.g. Dual Bin Packing [ 1 , 6 ] in case of the sum and Removable Knapsack [ 10 , 11 ] in case of the maximum. Following previous studies, we consider two variants, depending on whether the algorithm is allowed to remove items (forever) from its bins or not, and two special cases where the profit of an item is a function of its size, in addition to the general setting. We study both deterministic and randomized algorithms; for the latter, we consider both the oblivious and the adaptive adversary model. We classify each variant as either admitting O (1)-competitive algorithms or not. We develop simple O (1)-competitive algorithms for some cases of the max-objective variant believed to be intrac because only 1-bin deterministic algorithms were considered before.

We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforcable decisions, prices are posted without knowing the future events or even the current event, and are thus inherently dominant strategy incentive compatible. In particular we show that one can give efficient posted price mechanisms for metrical task systems, some instances of the k-server problem, and metrical matching problems. We give both deterministic and randomized algorithms. Such posted price mechanisms decrease the social cost dramatically over selfish behavior where no decision incurs a charge. One alluring application of this is reducing the social cost of free parking exponentially.

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 ρ has a fixed cost cρ. In addition, we incur waiting costs for each order, possibly an arbitrary non-decreasing function of time, different 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 ℙ -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. We also show that the integrality gap of the linear program for JRP-L is at least 12/11 ≈ 1.09. 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.

Let $\pi'_{w}$ denote the failure function of the Knuth-Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array $A'[1 \mathinner {\ldotp \ldotp }n]$ , is there a word w over an arbitrary alphabet Σ such that $A'[i]=\pi'_{w}[i]$ for all i? Moreover, what is the minimum cardinality of Σ required? We give an elementary and self-contained $\mathcal{O}(n\log n)$ time algorithm for this problem, thus improving the previously known solution (Duval et al. in Conference in honor of Donald E. Knuth, 2007), which had no polynomial time bound. Using both deeper combinatorial insight into the structure of π′ and advanced algorithmic tools, we further improve the running time to $\mathcal{O}(n)$ .

We consider online competitive algorithms for the problem of collecting weighted items from a dynamic queue S . The content of S varies over time. An update to S can occur between any two consecutive time steps, and it consists in deleting any number of items at the front of S and inserting other items into arbitrary locations in S . At each time step we are allowed to collect one item in S . The objective is to maximize the total weight of collected items. This is a generalization of bounded-delay packet scheduling (also known as buffer management). We present several upper and lower bounds on the competitive ratio for the general case and for some restricted variants of this problem.

We give a memoryless scale-invariant randomized algorithm ReMix for Packet Scheduling that is e /( e −1)-competitive against an adaptive adversary. ReMix unifies most of previously known randomized algorithms, and its general analysis yields improved performance guarantees for several restricted variants, including the s -bounded instances. In particular, ReMix attains the optimum competitive ratio of 4/3 on 2-bounded instances. Our results are applicable to a more general problem, called Item Collection , in which only the relative order between packets’ deadlines is known. ReMix is the optimal memoryless randomized algorithm against adaptive adversary for that problem.

This article concerns an online packet scheduling problem that arises as a natural model for buffer management at a network router. Packets arrive at a router at integer time steps, and are buffered upon arrival. Packets have non-negative weights and integer deadlines that are (weakly) increasing in their arrival times. In each integer time step, at most one packet can be sent. The objective is to maximize the sum of the weights of the packets that are sent by their deadlines. The main results include an optimal (ϕ := (1 + √ 5)/2 ≈ 1.618)-competitive deterministic online algorithm, a (4/3 ≈ 1.33)-competitive randomized online algorithm against an oblivious adversary, and a 2-speed 1-competitive deterministic online algorithm. The analysis does not use a potential function explicitly, but instead modifies the adversary's buffer and credits the adversary to account for these modifications.

In 2005 Li~et~al. gave a \(\phi\)-competitive deterministic online algorithm for scheduling of packets with agreeable deadlines~\cite{DBLP:conf/soda/LiSS05} with a very interesting analysis. This is known to be optimal due to a lower bound by Hajek~\cite{Hajek-det-lb}. 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 (\frac{4}{3}\). Note that this still leaves a gap between the best known lower bound of \(\frac{5}{4}\) by Chin~et~al.~\cite{DBLP:journals/algorithmica/ChinF03} for randomized algorithms against oblivious adversary.

We consider the problem of collecting weighted items from a dynamic queue . Before each step, some items at the front of can be deleted and some other items can be added to at any place. An item, once deleted, cannot be re-inserted — in other words, it “expires”. We are allowed to collect one item from per step. Each item can be collected only once. The objective is to maximize the total weight of the collected items. We study the online version of the dynamic queue problem. It is quite easy to see that the greedy algorithm that always collects the maximum-value item is 2-competitive, and that no deterministic online algorithm can be better than 1.618-competitive. We improve both bounds: We give a 1.89-competitive algorithm for general dynamic queues and we show a lower bound of 1.632 on the competitive ratio. We also provide other upper and lower bounds for restricted versions of this problem. The dynamic queue problem is a generalization of the well-studied buffer management problem, and it is an abstraction of the buffer management problem for network links with intermittent access.