On the approximability of an interval scheduling problem (original) (raw)
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Approximating Interval Scheduling Problems with Bounded Profits
Lecture Notes in Computer Science, 2007
We consider the Generalized Scheduling Within Intervals (GSWI) problem: given a set J of jobs and a set I of intervals, where each job j ∈ J has in interval I ∈ I length (processing time) j,I and profit p j,I , find the highest-profit feasible schedule. The best approximation ratio known for GSWI is (1/2 − ε). We give a (1 − 1/e − ε)approximation scheme for GSWI with bounded profits, based on the work by Chuzhoy, Rabani, and Ostrovsky [5], for the {0, 1}-profit case. We also consider the Scheduling Within Intervals (SWI) problem, which is a particular case of GSWI where for every j ∈ J there is a unique interval I = I j ∈ I with p j,I > 0. We prove that SWI is (weakly) NP-hard even if the stretch factor (the maximum ratio of job's interval size to its processing time) is arbitrarily small, and give a polynomial-time algorithm for bounded profits and stretch factor < 2.
Naval Research Logistics, 2007
In interval scheduling, not only the processing times of the jobs but also their starting times are given. This paper 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.
Optimization problems in multiple-interval graphs
ACM Transactions on Algorithms, 2010
Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them.
On the complexity of interval scheduling with a resource constraint
Theoretical Computer Science, 2011
We consider a scheduling problem where jobs have to be carried out by parallel identical machines. The attributes of a job j are: a fixed start time s j , a fixed finish time f j , and a resource requirement r j . Every machine owns R units of a renewable resource necessary to carry out jobs. A machine can process more than one job at a time, provided the resource consumption does not exceed R. The jobs must be processed in a non-preemptive way. Within this setting, the problem is to decide whether a feasible schedule for all jobs exists or not.
An Efficient Algorithm for Finding a Maximum Weight 2Independent Set on Interval Graphs
Information Processing Letters, 1992
In this paper, we introduce an O(n) time algorithm to solve the maximum weight independent set problem on an interval graph with n vertices given its interval representation with sorted endpoints list. Based on this linear algorithm, we design an O(n2) time algorithm using O(n2) space to solve the maximum weight 2-independent set problem on an interval graph with n vertices. With a slight extension and modification of our algorithm, the maximum weight k-independent set problem on an interval graph with n vertices can be solved in O(nk) time using O(nk) space.
Proceedings of the …, 2002
We consider the problem of scheduling jobs that are given as groups of non-intersecting segments on the real line. Each job J j is associated with an interval, I j , which consists of up to t segments, for some t ≥ 1, and a weight (profit), w j ; two jobs are in conflict if their intervals intersect. Such jobs show up in a wide range of applications, including the transmission of continuous-media data, allocation of linear resources (e.g. bandwidth in linear processor arrays), and in computational biology/geometry. The objective is to schedule a subset of non-conflicting jobs of maximum total weight.
Fixed interval scheduling: Models, applications, computational complexity and algorithms
European Journal of Operational Research, 2007
The defining characteristic of fixed interval scheduling problems is that each job has a finite number of fixed processing intervals. A job can be processed only in one of its intervals on one of the available machines, or is not processed at all. A decision has to be made about a subset of the jobs to be processed and their assignment to the processing intervals such that the intervals on the same machine do not intersect. These problems arise naturally in different real-life operations planning situations, including the assignment of transports to loading/unloading terminals, work planning for personnel, computer wiring, bandwidth allocation of communication channels, printed circuit board manufacturing, gene identification and examining computer memory structures. We present a general formulation of the interval scheduling problem, show its relations to cognate problems in graph theory, and survey existing models, results on computational complexity and solution algorithms.
Consecutive interval query and dynamic programming on intervals
Discrete Applied Mathematics, 1998
Given a set of n points (nodes) on a line and a set of m weighted intervals defined on the nodes, we consider a particular dynamic programming (DP) problem on these intervals. If the weight function of the DP has convex or concave property, we can solve this DP problem efficiently by using matrix searching in Monge matrices, together with a new query data structure, which we call the consecutive query structure. We invoke our algorithm to obtain fast algorithms for sequential partition of a graph and for maximum K-clique of an interval graph.
Interval selection: Applications, algorithms, and lower bounds
Journal of Algorithms, 2003
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.
Lower and Upper Bounds for the Linear Arrangement Problem on Interval Graphs
RAIRO - Operations Research
We deal here with the Linear Arrangement Problem (LAP) on interval graphs, any interval graph being given here together with its representation as the intersection graph of some collection of intervals, and so with related precedence and inclusion relations. We first propose a lower bound LB, which happens to be tight in the case of unit interval graphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with the precedence relation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP on interval graphs.