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Generalized Two-Server Problem [chapter]

René A. Sitters
2008 Encyclopedia of Algorithms  
We consider the generalized on-line two-server problem in which at each step each server receives a request, which is a point in a metric space. One of the servers has to be moved to its request. Thus, each of the servers is moving in his own metric space. The special case in which both metric spaces are the real line is known as the CNN-problem. It has been a well-known open question in on-line optimization if an algorithm with a constantcompetitive ratio exists for this problem. We answer
more » ... question in the affirmative sense by providing the first constant competitive algorithm for the generalized two-server problem on any metric space. The basic result in this paper is a characterization of competitiveness for metrical service systems that seems much easier to use when looking for a competitive algorithm. The existence of a competitive algorithm for the generalized two-server problem follows rather easily from this result. Part of this research has been funded by the Dutch BSIK/BRICKS project. in the request. Metrical service systems were introduced by Manasse, McGeoch, and Sleator [14] who used the term forcing task systems and independently by Chrobak and Larmore [6] to provide a formalism for investigating a wide variety of on-line optimization problems. A precise definition is given in Section 2. In the same section we derive the basic theorem of this paper. It provides a sufficient condition for the existence of constant competitive algorithms for general metrical service systems. The result on the generalized two-server problem then becomes a matter of verifying this condition. The generalized k-server problem is a natural generalization of the well-known k-server problem for which M 1 = M 2 = · · · = M k and z 1 = z 2 = · · · = z k at each time step. The k-server problem was introduced by Manasse, McGeoch and Sleator [14], who proved a lower bound of k on the competitive ratio of any deterministic algorithm for any metric space with at least k + 1 points and posed the well-known k-server conjecture saying that there exists a k-competitive algorithm for any metric space. The conjecture has been proved for k = 2 [14], for some special metric spaces such as the line, the star, and for all spaces with at most k + 2 points [1, 4, 5] . For k ≥ 3 the current best upper bound of 2k − 1 is given by Koutsoupias and Papadimitriou [12]. The weighted k-server problem turns out to be much harder. In this problem a weight is assigned to each server and the total cost is the sum of the weighted distances. Fiat and Ricklin [9] prove that for any metric space with at least k + 1 points there exists a set of weights such that the competitive ratio of any deterministic algorithm is at least k Ω(k) . For a uniform metric space (in which all internode distances are one) and k = 2 Feuerstein et al. [8] give a 6.275-competitive algorithm, which was improved by Chrobak and Sgall [7] who provided a 5-competitive algorithm and proved that no better competitive ratio is possible. A weighted k-server algorithm is called competitive if the competitive ratio is independent of the weights. For a general metric space no competitive algorithm was known yet even for k = 2. It is easy to see that the generalized k-server problem is a generalization of the weighted k-server problem as well. The generalized two-server problem in which both servers move on the real line has become well-known as the emphasize the importance of the CNN-problem as one of the simplest problems in a rich class of so-called sum-problems [2]. In the sum-problem each of a set of systems gets a request and only one system has to serve this request. Koutsoupias and Taylor [13] prove a lower bound of 6 + √ 17 on the competitive ratio of any deterministic on-line algorithm for the generalized two-server problem, through an instance of the weighted two-server problem on the real line. They also conjecture that the generalized work function algorithm has constant competitive ratio for the generalized two-server problem. For the generalized two-server problem the situation was even worse than for the k-server problem: the question if any algorithm exists with constant competitive ratio remained unanswered. In Section 3 we answer this question affirmatively, by designing an algorithm and prove an upper bound of 879 on its competitive ratio 4 . Our algorithm is a combination of the well-known balance algorithm and the generalized work function algorithm. The result is merely checking the condition of the general theorem for metrical service systems in Section 2, announced above. Optimal off-line solutions of metrical service systems can easily be found by dynamic programming (see [2] ), which yields an O(k 2 n k ) time algorithm for the generalized k-server problem. For the classical k-server problem this running time can be reduced to O(kn 2 ) by formulating it as a
doi:10.1007/978-0-387-30162-4_162 fatcat:7ed4q3tu3jdprjz4ubf4wbowje

Generalized Two-Server Problem [chapter]

René A. Sitters
2014 Encyclopedia of Algorithms  
We consider the generalized on-line two-server problem in which at each step each server receives a request, which is a point in a metric space. One of the servers has to be moved to its request. Thus, each of the servers is moving in his own metric space. The special case in which both metric spaces are the real line is known as the CNN-problem. It has been a well-known open question in on-line optimization if an algorithm with a constantcompetitive ratio exists for this problem. We answer
more » ... question in the affirmative sense by providing the first constant competitive algorithm for the generalized two-server problem on any metric space. The basic result in this paper is a characterization of competitiveness for metrical service systems that seems much easier to use when looking for a competitive algorithm. The existence of a competitive algorithm for the generalized two-server problem follows rather easily from this result. Part of this research has been funded by the Dutch BSIK/BRICKS project. in the request. Metrical service systems were introduced by Manasse, McGeoch, and Sleator [14] who used the term forcing task systems and independently by Chrobak and Larmore [6] to provide a formalism for investigating a wide variety of on-line optimization problems. A precise definition is given in Section 2. In the same section we derive the basic theorem of this paper. It provides a sufficient condition for the existence of constant competitive algorithms for general metrical service systems. The result on the generalized two-server problem then becomes a matter of verifying this condition. The generalized k-server problem is a natural generalization of the well-known k-server problem for which M 1 = M 2 = · · · = M k and z 1 = z 2 = · · · = z k at each time step. The k-server problem was introduced by Manasse, McGeoch and Sleator [14], who proved a lower bound of k on the competitive ratio of any deterministic algorithm for any metric space with at least k + 1 points and posed the well-known k-server conjecture saying that there exists a k-competitive algorithm for any metric space. The conjecture has been proved for k = 2 [14], for some special metric spaces such as the line, the star, and for all spaces with at most k + 2 points [1, 4, 5] . For k ≥ 3 the current best upper bound of 2k − 1 is given by Koutsoupias and Papadimitriou [12]. The weighted k-server problem turns out to be much harder. In this problem a weight is assigned to each server and the total cost is the sum of the weighted distances. Fiat and Ricklin [9] prove that for any metric space with at least k + 1 points there exists a set of weights such that the competitive ratio of any deterministic algorithm is at least k Ω(k) . For a uniform metric space (in which all internode distances are one) and k = 2 Feuerstein et al. [8] give a 6.275-competitive algorithm, which was improved by Chrobak and Sgall [7] who provided a 5-competitive algorithm and proved that no better competitive ratio is possible. A weighted k-server algorithm is called competitive if the competitive ratio is independent of the weights. For a general metric space no competitive algorithm was known yet even for k = 2. It is easy to see that the generalized k-server problem is a generalization of the weighted k-server problem as well. The generalized two-server problem in which both servers move on the real line has become well-known as the emphasize the importance of the CNN-problem as one of the simplest problems in a rich class of so-called sum-problems [2]. In the sum-problem each of a set of systems gets a request and only one system has to serve this request. Koutsoupias and Taylor [13] prove a lower bound of 6 + √ 17 on the competitive ratio of any deterministic on-line algorithm for the generalized two-server problem, through an instance of the weighted two-server problem on the real line. They also conjecture that the generalized work function algorithm has constant competitive ratio for the generalized two-server problem. For the generalized two-server problem the situation was even worse than for the k-server problem: the question if any algorithm exists with constant competitive ratio remained unanswered. In Section 3 we answer this question affirmatively, by designing an algorithm and prove an upper bound of 879 on its competitive ratio 4 . Our algorithm is a combination of the well-known balance algorithm and the generalized work function algorithm. The result is merely checking the condition of the general theorem for metrical service systems in Section 2, announced above. Optimal off-line solutions of metrical service systems can easily be found by dynamic programming (see [2] ), which yields an O(k 2 n k ) time algorithm for the generalized k-server problem. For the classical k-server problem this running time can be reduced to O(kn 2 ) by formulating it as a
doi:10.1007/978-3-642-27848-8_162-2 fatcat:7gpdbdgk6jashbbepn6fjiy67m

The Chinese deliveryman problem

Martijn van Ee, René Sitters
2019 4OR  
Netherlands Defence Academy, Den Helder, The Netherlands Vrije Universiteit Amsterdam, Amsterdam, The Netherlands B René Sitters r.a.sitters@vu.nl Martijn van Ee m.v.ee.01@mindef.nl 1 2 Table  ...  A PTAS for planar graph TRP is given in Sitters (2019) . That PTAS can be simplified to a large extent for the CDP.  ... 
doi:10.1007/s10288-019-00420-2 fatcat:hh25yrqpqvfsxbpqdpyazynn6e

The Sorting Buffer Problem is NP-hard [article]

Ho-Leung Chan, Nicole Megow, Rob van Stee, Rene Sitters
2010 arXiv   pre-print
We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a random-access buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling strategy that decides upon the order in which items from the buffer are sent to the server. Each type change incurs unit cost, and thus, the cost minimizing objective is to minimize the total number
more » ... f type changes for serving the entire sequence. This problem is motivated by various applications in manufacturing processes and computer science, and it has attracted significant attention in the last few years. The main focus has been on online competitive algorithms. Surprisingly little is known on the basic offline problem. In this paper, we show that the sorting buffer problem with uniform cost is NP-hard and, thus, close one of the most fundamental questions for the offline problem. On the positive side, we give an O(1)-approximation algorithm when the scheduler is given a buffer only slightly larger than double the original size. We also give a dynamic programming algorithm for the special case of buffer size two that solves the problem exactly in linear time, improving on the standard DP which runs in cubic time.
arXiv:1009.4355v1 fatcat:x4nvbn7xwfcx7mrter4tvssxgi

The A Priori Traveling Repairman Problem

Martijn van Ee, René Sitters
2017 Algorithmica  
The field of a priori optimization is an interesting subfield of stochastic combinatorial optimization that is well suited for routing problems. In this setting, there is a probability distribution over active sets, vertices that have to be visited. For a fixed tour, the solution on an active set is obtained by restricting the solution on the active set. In the well-studied a priori traveling salesman problem, the goal is to find a tour that minimizes the expected length. In the a priori
more » ... ng repairman problem (TRP), the goal is to find a tour that minimizes the expected sum of latencies. In this paper, we study the uniform model, where a vertex is in the active set with probability p independently of the other vertices, and give the first constant-factor approximation for a priori TRP.
doi:10.1007/s00453-017-0351-z pmid:31007325 pmcid:PMC6445530 fatcat:hajpiuapdvhtlo323fksy2zweq

Optimal pricing of capacitated networks

Alexander Grigoriev, Joyce van Loon, René Sitters, Marc Uetz
2009 Networks  
We address the algorithmic complexity of a profit maximization problem in capacitated, undirected networks. We are asked to price a set of m capacitated network links to serve a set of n potential customers. Each customer is interested in purchasing a network connection that is specified by a simple path in the network and has a maximum budget that we assume to be known to the seller. The goal is to decide which customers to serve, and to determine prices for all network links in order to
more » ... ze the total profit. We address this pricing problem in different network topologies. More specifically, we derive several results on the algorithmic complexity of this profit maximization problem, given that the network is either a path, a cycle, a tree, or a grid. Our results include approximation algorithms as well as inapproximability results. We consider a profit maximization problem that is defined on a capacitated, undirected network. Given is a simple undirected graph G = (V, E) with |E| = m edges, and given are integral edge capacities c e , e ∈ E. Each edge can be thought of as a network link, and the edge capacity determines the maximum number of customers the link can accommodate. We mainly, but not exclusively, discuss problems where edge capacities c e are finite. Given is a set of n potential customers J = {1, . . . , n} each of which is interested in purchasing a network connection between two vertices of the graph. In contrast to many classical network problems, we assume that each customer wants one specific simple path, denoted E j ⊆ E, rather than any path that connects the two vertices. In literature on auctions, this is also known as single-minded customers [15] . Each customer j ∈ J has an integral budget (or valuation) b j , which is the largest amount that a customer is willing to pay for her path E j .
doi:10.1002/net.20260 fatcat:xzhwp3m3qbhnvhgqyvg5geha24

Efficient Algorithms for Average Completion Time Scheduling [chapter]

René Sitters
2010 Lecture Notes in Computer Science  
We analyze the competitive ratio of algorithms for minimizing (weighted) average completion time on identical parallel machines and prove that the well-known shortest remaining processing time algorithm (SRPT) is 5/4-competitive w.r.t. the average completion time objective. For weighted completion times we give a deterministic algorithm with competitive ratio 1.791 + o(m). This ratio holds for preemptive and non-preemptive scheduling.
doi:10.1007/978-3-642-13036-6_31 fatcat:xj4j57favjabzpky5iwicuhosu

The generalized two-server problem

René A. Sitters, Leen Stougie
2006 Journal of the ACM  
We consider the generalized on-line two-server problem in which at each step each server receives a request, which is a point in a metric space. One of the servers has to be moved to its request. Thus, each of the servers is moving in his own metric space. The special case in which both metric spaces are the real line is known as the CNN-problem. It has been a well-known open question in on-line optimization if an algorithm with a constantcompetitive ratio exists for this problem. We answer
more » ... question in the affirmative sense by providing the first constant competitive algorithm for the generalized two-server problem on any metric space. The basic result in this paper is a characterization of competitiveness for metrical service systems that seems much easier to use when looking for a competitive algorithm. The existence of a competitive algorithm for the generalized two-server problem follows rather easily from this result. Part of this research has been funded by the Dutch BSIK/BRICKS project. in the request. Metrical service systems were introduced by Manasse, McGeoch, and Sleator [14] who used the term forcing task systems and independently by Chrobak and Larmore [6] to provide a formalism for investigating a wide variety of on-line optimization problems. A precise definition is given in Section 2. In the same section we derive the basic theorem of this paper. It provides a sufficient condition for the existence of constant competitive algorithms for general metrical service systems. The result on the generalized two-server problem then becomes a matter of verifying this condition. The generalized k-server problem is a natural generalization of the well-known k-server problem for which M 1 = M 2 = · · · = M k and z 1 = z 2 = · · · = z k at each time step. The k-server problem was introduced by Manasse, McGeoch and Sleator [14], who proved a lower bound of k on the competitive ratio of any deterministic algorithm for any metric space with at least k + 1 points and posed the well-known k-server conjecture saying that there exists a k-competitive algorithm for any metric space. The conjecture has been proved for k = 2 [14], for some special metric spaces such as the line, the star, and for all spaces with at most k + 2 points [1, 4, 5] . For k ≥ 3 the current best upper bound of 2k − 1 is given by Koutsoupias and Papadimitriou [12]. The weighted k-server problem turns out to be much harder. In this problem a weight is assigned to each server and the total cost is the sum of the weighted distances. Fiat and Ricklin [9] prove that for any metric space with at least k + 1 points there exists a set of weights such that the competitive ratio of any deterministic algorithm is at least k Ω(k) . For a uniform metric space (in which all internode distances are one) and k = 2 Feuerstein et al. [8] give a 6.275-competitive algorithm, which was improved by Chrobak and Sgall [7] who provided a 5-competitive algorithm and proved that no better competitive ratio is possible. A weighted k-server algorithm is called competitive if the competitive ratio is independent of the weights. For a general metric space no competitive algorithm was known yet even for k = 2. It is easy to see that the generalized k-server problem is a generalization of the weighted k-server problem as well. The generalized two-server problem in which both servers move on the real line has become well-known as the emphasize the importance of the CNN-problem as one of the simplest problems in a rich class of so-called sum-problems [2]. In the sum-problem each of a set of systems gets a request and only one system has to serve this request. Koutsoupias and Taylor [13] prove a lower bound of 6 + √ 17 on the competitive ratio of any deterministic on-line algorithm for the generalized two-server problem, through an instance of the weighted two-server problem on the real line. They also conjecture that the generalized work function algorithm has constant competitive ratio for the generalized two-server problem. For the generalized two-server problem the situation was even worse than for the k-server problem: the question if any algorithm exists with constant competitive ratio remained unanswered. In Section 3 we answer this question affirmatively, by designing an algorithm and prove an upper bound of 879 on its competitive ratio 4 . Our algorithm is a combination of the well-known balance algorithm and the generalized work function algorithm. The result is merely checking the condition of the general theorem for metrical service systems in Section 2, announced above. Optimal off-line solutions of metrical service systems can easily be found by dynamic programming (see [2] ), which yields an O(k 2 n k ) time algorithm for the generalized k-server problem. For the classical k-server problem this running time can be reduced to O(kn 2 ) by formulating it as a
doi:10.1145/1147954.1147960 fatcat:jekxrctpkjbcleqemqxbgls54i

Split Scheduling with Uniform Setup Times [article]

Frans Schalekamp, Rene Sitters, Suzanne van der Ster, Leen Stougie, Victor Verdugo, Anke van Zuylen
2012 arXiv   pre-print
We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in
more » ... ning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times.
arXiv:1212.1754v1 fatcat:wuaxyweqcvfb7aqlande2w4dia

The traveling salesman problem on cubic and subcubic graphs [article]

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2011 arXiv   pre-print
We study the Travelling Salesman Problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values of the TSP and its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The
more » ... f uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on n vertices a tour of length 4n/3-2 exists, which also implies the 4/3 conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented a randomized algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3 conjecture for this class of graph-TSP. We will present a way to derandomize their algorithm which leads to a smaller running time than the obvious derandomization. All of the latter also works for multi-graphs.
arXiv:1107.1052v1 fatcat:sbklbt5rebaytnvbpsd7atmd7q

Approximability of average completion time scheduling on unrelated machines

René Sitters
2016 Mathematical programming  
Introduction In the last two decades extensive research has been done on approximation algorithms for machine scheduling problems with the objective of minimizing the average (or B René Sitters r.a.sitters  ...  @vu.nl; sitters@cwi.nl 1 Vrije Universiteit, Amsterdam, The Netherlands 2 Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands total) job completion time.  ... 
doi:10.1007/s10107-016-1004-8 fatcat:6u5cw6zic5awlfaaomymfwwiwy

Complexity of preemptive minsum scheduling on unrelated parallel machines

René Sitters
2005 Journal of Algorithms  
doi:10.1016/j.jalgor.2004.06.011 fatcat:lk7nwfleqzhtjgqtmixhdbz5xi

Scheduling over Scenarios on Two Machines [article]

Esteban Feuerstein, Alberto Marchetti-Spaccamela, Frans Schalekamp, Rene Sitters, Suzanne van der Ster, Leen Stougie, Anke van Zuylen
2014 arXiv   pre-print
We consider scheduling problems over scenarios where the goal is to find a single assignment of the jobs to the machines which performs well over all possible scenarios. Each scenario is a subset of jobs that must be executed in that scenario and all scenarios are given explicitly. The two objectives that we consider are minimizing the maximum makespan over all scenarios and minimizing the sum of the makespans of all scenarios. For both versions, we give several approximation algorithms and
more » ... r bounds on their approximability. With this research into optimization problems over scenarios, we have opened a new and rich field of interesting problems.
arXiv:1404.4766v1 fatcat:a5ufdkvpvncy3fs2mouafgjudm

The Itinerant List Update Problem [chapter]

Neil Olver, Kirk Pruhs, Kevin Schewior, René Sitters, Leen Stougie
2018 Lecture Notes in Computer Science  
We introduce the itinerant list update problem (ILU), which is a relaxation of the classic list update problem in which the pointer no longer has to return to a home location after each request. The motivation to introduce ILU arises from the fact that it naturally models the problem of track memory management in Domain Wall Memory. Both online and offline versions of ILU arise, depending on specifics of this application. First, we show that ILU is essentially equivalent to a dynamic variation
more » ... f the classical minimum linear arrangement problem (MLA), which we call DMLA. Both ILU and DMLA are very natural, but do not appear to have been studied before. In this work, we focus on the offline ILU and DMLA problems. We then give an O(log 2 n)-approximation algorithm for these problems. While the approach is based on well-known divide-and-conquer approaches for the standard MLA problem, the dynamic nature of these problems introduces substantial new difficulties. We also show an Ω(log n) lower bound on the competitive ratio for any randomized online algorithm for ILU. This shows that online ILU is harder than online LU, for which O(1)-competitive algorithms, like Move-To-Front, are known.
doi:10.1007/978-3-030-04693-4_19 fatcat:hu3p2ikzrra23lc2yuqmmqwmya

On the complexity of the highway problem

Khaled Elbassioni, Rajiv Raman, Saurabh Ray, René Sitters
2012 Theoretical Computer Science  
In the highway problem, we are given a path, and a set of buyers interested in buying sub-paths of this path; each buyer declares a non-negative budget, which is the maximum amount of money she is willing to pay for that sub-path. The problem is to assign nonnegative prices to the edges of the path such that we maximize the profit obtained by selling the edges to the buyers who can afford to buy their sub-paths, where a buyer can afford to buy her sub-path if the sum of prices in the sub-path
more » ... at most her budget. In this paper, we show that the highway problem is strongly NP-hard; this settles the complexity of the problem in view of the existence of a polynomial-time approximation scheme, as was recently shown in Grandoni and Rothvoß (2011) [15]. We also consider the coupon model, where we allow some items to be priced below zero to improve the overall profit. We show that allowing negative prices makes the problem APX-hard. As a corollary, we show that the bipartite vertex pricing problem is APX-hard with budgets in {1, 2, 3}, both in the cases with negative and non-negative prices.
doi:10.1016/j.tcs.2012.07.028 fatcat:4uwmggkqp5habhui2nxnhhc4ny
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