On-line scheduling of jobs with fixed start and end times
Theoretical Computer Science - Special issue on dynamic and on-line algorithms
Note on scheduling intervals on-line
Discrete Applied Mathematics
On the k-coloring of intervals
Discrete Applied Mathematics
Bounding the Power of Preemption in Randomized Scheduling
SIAM Journal on Computing
Online computation and competitive analysis
Online computation and competitive analysis
An Improved Randomized On-Line Algorithm for a Weighted Interval Selection Problem
Journal of Scheduling
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Randomized online interval scheduling
Operations Research Letters
On-line scheduling of equal-length intervals on parallel machines
Information Processing Letters
Space-constrained interval selection
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
SIGACT news online algorithms column 21: APPROX and ALGO
ACM SIGACT News
Online selection of intervals and t-intervals
Information and Computation
| Hi-index | 5.23 |
Online interval selection is a problem in which intervals arrive one by one, sorted by their left endpoints. Each interval has a length and a non-negative weight associated with it. The goal is to select a non-overlapping set of intervals with maximal total weight and run them to completion. The decision regarding a possible selection of an arriving interval must be done immediately upon its arrival. The interval may be preempted later in favor of selecting an arriving overlapping interval, in which case the weight of the preempted interval is lost. We follow Woeginger (1994) [12] and study the same models. The types of instances we consider are C-benevolent instances, where the weight of an interval is a monotonically increasing (convex) function of length, and D-benevolent instances, where the weight of an interval is a monotonically decreasing function of length. Some of our results can be extended to the case of unit length intervals with arbitrary costs. We significantly improve the previously known bounds on the performance of online randomized algorithms for the problem, namely, we introduce a new algorithm for the D-benevolent case and for unit intervals, which uses a parameter @q and has a competitive ratio of at most @q^2ln@q(@q-1)^2. This value is equal to approximately 2.4554 for @q~3.513 being the solution of the equation x-1=2lnx. We further design a lower bound of 1+ln2~1.693 on the competitive ratio of any randomized algorithm. The lower bound is valid for any C-benevolent instance, some D-benevolent functions and for unit intervals. We further show a lower bound of 32 for a wider class of D-benevolent instances. This improves over previously known lower bounds. We also design a barely random online algorithm for the D-benevolent case and the case of unit intervals, which uses a single random bit, and has a competitive ratio of 3.22745.