We consider a problem of dispersing points on disjoint intervals on a line. Given n pairwise disjoint intervals sorted on a line, we want to find a point in each interval such that the minimum pairwise distance of these points is maximized. Based on a greedy strategy, we present a linear time algorithm for the problem. Further, we also solve in linear time the cycle version of the problem where the intervals are given on a cycle.
On the other hand, problems on intervals usually have applications in other areas. For example, some problems on intervals are related to scheduling because the time period between the release time and the deadline of a job or task in scheduling problems can be considered as an interval on the line. From the interval point of view, @cite_17 studied the following problem on intervals: Given @math intervals on a line, determine whether it is possible to find a unit-length sub-interval in each input interval, such that these sub-intervals do not intersect. An @math time algorithm was given in @cite_17 for this problem. The optimization version of the above problem was also studied @cite_18 @cite_10 , where the goal is to find a maximum number of intervals that contain non-intersecting unit-length sub-intervals. @cite_18 gave an @math time algorithm for the problem, and later Vakhania @cite_10 improved the algorithm to @math time. The online version of the problem was also considered @cite_4 . Other optimization problems on intervals have also been considered, e.g., see @cite_17 @cite_1 @cite_5 @cite_8 .
@cite_18 @cite_4 @cite_8 @cite_1 @cite_5 @cite_10 @cite_17
2031347356 1923579438 2963782499 2047204419
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, a positive weight, w j , and two jobs are in conflict if any of their segments 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.In a single machine environment, our problem can be formulated as the problem of finding a maximum weight independent set in a t-interval graph (the special case of t = 1 is an ordinary interval graph). We show that, for t ≥ 2, this problem is APX-hard, even for highly restricted instances. Our main result is a 2t-approximation algorithm for general instances, based on a novel fractional version of the Local Ratio technique. Previously, the problem was considered only for proper union graphs, a restricted subclass of t-interval graphs, and the approximation factor achieved was (2t - 1 + 1 2t). A bi-criteria polynomial time approximation scheme (PTAS) is developed for the subclass of t-union graphs.In the online case, we consider uniform weight jobs that consist of at most two segments. We show that when the resulting 2-interval graph is proper, a simple greedy algorithm is 3-competitive, while any randomized algorithm has competitive ratio at least 2.5. For general instances, we give a randomized O(log2 R)-competitive (or O((log R)2+e)-competitive) algorithm, where R is the known (unknown) ratio between the longest and the shortest segment in the input sequence. The main aim of this note is to show that a polynomial-time algorithm for the scheduling problem 1|r j ; p j = p| ∑ Uj given by Carlier in (1981) is incorrect. In this problem we are given n jobs with release times and deadlines. All jobs have the same processing time p. The objective is to find a non-preemptive schedule that maximizes the number of jobs completed by their deadlines. The feasibility version of this problem, where we ask whether all jobs can meet their deadlines, has been studied thoroughly. Polynomial-time algorithms for this version were first found, independently, by Simons (1978) and Carlier (1981). A faster algorithm, with running time O(n log n), was subsequently given by (1981). The elegant feasibility algorithm of Carlier (1981) is based on dynamic programming and it processes jobs from left to right on the time-axis. For each time t, it constructs a partial schedule with jobs that complete at or before time t. Carlier also attempted to apply the same technique to design a polynomial-time algorithm for the maximization version, 1|r j ; p j = p| ∑ Uj , and claimed an O(n3 log n)-time algorithm. His result is now widely cited in the literature. We show, however, that this algorithm is not correct, by giving an instance on which it produces a sub-optimal schedule. Our counter-example can be, in fact, extended to support a broader claim, namely that even the general approach from Carlier (1981) cannot yield a polynomial-time algorithm. By this general approach we mean a class of algorithms that processes the input from left to right and make decisions based on the deadline ordering, and not their exact values. The question remains as to how efficiently can we solve the scheduling problem 1|r j ; p j = p| ∑ Uj . Baptiste (1999) gave an O(n7)-time algorithm for the more general version of this problem where jobs have weights. We show how to modify his algorithm to obtain a faster, O(n5)-time algorithm for the non-weighted case. These last two results are discussed only briefly in this note. The complete proofs can be found in the full version of this paper, see (2004). We consider the classical scheduling problem on parallel identical machines to minimize the makespan. There is a long history of studies on this problem, focusing on exact and approximation algorithms, and it is thus natural to consider whether these algorithms are best possible in terms of the running time. Under the Exponential Time Hypothesis (ETH), we achieve the following results in this paper: • The scheduling problem on a constant number m of identical machines, which is denoted as Pm||Cmax, is known to admit a fully polynomial time approximation scheme (FPTAS) of running time O(n) + (1 e)O(m) (indeed, the algorithm works for an even more general problem where machines are unrelated). We prove this algorithm is essentially the best possible in the sense that a (1 e)O(m1-δ) + nO(1) time FPTAS for any δ > 0 implies that ETH fails. • The scheduling problem on an arbitrary number of identical machines, which is denoted as P||Cmax, is known to admit a polynomial time approximation scheme (PTAS) of running time 2O(1 e2 log3(1 e))+nO(1). We prove this algorithm is nearly optimal in the sense that a 2O((1 e)1-δ+nO(1) time PTAS for any δ > 0 implies that ETH fails, leaving a small room for improvement. In addition, we also consider exact algorithms for the scheduling problem and prove the following result: • The traditional dynamic programming algorithm for P||Cmax is known to run in 2O(n) time. We prove this is essentially the best possible in the sense that even if we restrict that there are n jobs and the processing time of each job is bounded by O(n), an exact algorithm of running time 22O(n1-δ) for any δ > 0 implies that ETH fails. To obtain these results we will provide two new reductions from 3SAT, one for Pm||Cmax and another for P||Cmax. Indeed, the new reductions explore the structure of scheduling problems and can also lead to other interesting results. For example, using the framework of our reduction for P||Cmax, [5] are able to prove the APX-hardness of the scheduling problem in which the matrix of job processing times P = (pij)mxn is of rank 3, solving the open problem mentioned in [2]. We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single-processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below @math and show several lower bounds on the competitive ratios. First, we give a barely random @math -competitive algorithm that uses only one random bit. We also show a lower bound of @math on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to @math . Second, we give a deterministic @math -competitive algorithm in the model that allows restarts, and we show that in this model the ratio @math is optimal. For randomized algorithms with restarts we show a lower bound of @math .