On the k-coloring of intervals

Carlisle, Martin C.; Lloyd, Errol L.

doi:10.1016/0166-218x(95)80003-m

Cited by 96 publications

(64 citation statements)

References 7 publications

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“…1.3. Finally, in Sect. 4, we give a lower bound of m for unit-weight variable-sized jobs, which is tight due to a trivial 1-competitive algorithm for a single machine [4,6] and the following simple observation.…”

Section: Our Resultsmentioning

confidence: 99%

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Online Scheduling of Jobs with Fixed Start Times on Related Machines

Epstein

Jeż

Sgall

et al. 2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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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 Cbenevolent 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.

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Section: Our Resultsmentioning

confidence: 99%

“…Faigle and Nawijn [6] and Carlisle and Lloyd [4] considered the version of jobs with unit weights on m identical machines. They gave a 1-competitive algorithm for this problem.…”

Section: Previous Workmentioning

confidence: 99%

Online Scheduling of Jobs with Fixed Start Times on Related Machines

Epstein

Jeż

Sgall

et al. 2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…In Section 3.1, we show how the cheapest conflict set Gi can be determined efficiently, which gives rise to a total running time of O{n 2 ) of GREEDY". As a byproduct of this subsection we show how knowing that an interval graph is m-colorable gives you an O{n) algorithm for obtaining a maximumweight (m -I)-colorable subgraph as compared to O{mS{n)) in the general case [7], where S (n) denotes the running time for any algorithm for finding a shortest path in a directed graph with O{n) arcs and positive arc weights. (With an efficient implementation of Dijkstra's algorithm, for example, S{n) can be taken as O{nlogn).…”

Section: Algorithm Greedy Amentioning

confidence: 99%

Interval selection: Applications, algorithms, and lower bounds

Erlebach

Spieksma

2003

Journal of Algorithms

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Given a set of jobs, each consisting of a number of weighted intervals on the real line, and a positive integer m, we study the problem of selecting a maximum weight subset of the intervals such that at most one interval is selected from each job and, for any point p on the real line, at most m intervals containing p are selected. This problem has applications in molecular biology, caching, PCB assembly, combinatorial auctions, and scheduling. It generalizes the problem of finding a (weighted) maximum independent set in an interval graph.We give a parameterized algorithm GREEDY", that belongs to the class of "myopic" algorithms, which are deterministic algorithms that process the given intervals in order of non-decreasing right endpoint and can either reject or select each interval (rejections are irrevocable). We show that there are values of the parameter a so that GREEDY", produces a 2-approximation in the case of unit weights, an 8-approximation in the case of arbitrary weights, and a (3 + 2V2)-approximation in the case where the weights of all intervals corresponding to the same job are equal. If all intervals have the same length, we prove for m = 1,2 that GREEDY", achieves ratio 6.638 in the case of arbitrary weights and 5 in the case of equal weights per job.Concerning lower bounds, we show that for instances with intervals of arbitrary lengths, no deterministic myopic algorithm can achieve ratio better than 2 in the case of unit weights, better than ~ 7.103 in the case of arbitrary weights, and better than 3 + 2V2 in the case where the weights of all intervals corresponding to the same job are equal. If all intervals have the same length, we give lower bounds of 3 + 2V2 for the case of arbitrary weights and 5 for the case of equal weights per job. Furthermore, we give a lower bound of .:1 ~ 1.582 on the approximation ratio of randomized myopic algorithms in the case of unit weights .• A preliIninary version of some of the results in this paper has appeared in [10].

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“…In case each interval has unit weight, and there are m machines, Faigle and Nawijn [30], and independently Carlisle and Lloyd [17], observed that a greedy algorithm is in fact an online algorithm that always outputs an optimal solution. This algorithm is extended to deal with the case of time-windows by Faigle et al [29].…”

Section: Online Interval Scheduling Problemsmentioning

confidence: 99%

“…This variant can be solved using a min-cost flow formulation (see Arkin and Silverberg [3] and Bouzina and Emmons [13]). In case each job has unit weight, a greedy algorithm finds a maximum number of jobs (see Faigle and Nawijn [30], and Carlisle and Lloyd [17]). If each job can only be carried out by an arbitrary given subset of the machines, the problem becomes NP-hard ( [3]).…”

Section: Interval Scheduling With Given Machinesmentioning

confidence: 99%

Interval scheduling: A survey

Kolen

Lenstra²,

Papadimitriou

et al. 2007

Naval Research Logistics

192

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Abstract:In interval scheduling, not only the processing times of the jobs but also their starting times are given. This article surveys the area of interval scheduling and presents proofs of results that have been known within the community for some time. We first review the complexity and approximability of different variants of interval scheduling problems. Next, we motivate the relevance of interval scheduling problems by providing an overview of applications that have appeared in literature. Finally, we focus on algorithmic results for two important variants of interval scheduling problems. In one variant we deal with nonidentical machines: instead of each machine being continuously available, there is a given interval for each machine in which it is available. In another variant, the machines are continuously available but they are ordered, and each job has a given "maximal" machine on which it can be processed. We investigate the complexity of these problems and describe algorithms for their solution.

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On the k-coloring of intervals

Cited by 96 publications

References 7 publications

Online Scheduling of Jobs with Fixed Start Times on Related Machines

Online Scheduling of Jobs with Fixed Start Times on Related Machines

Interval selection: Applications, algorithms, and lower bounds

Interval scheduling: A survey

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