Derandomization of Online Assignment Algorithms for Dynamic Graphs

This paper analyzes different online algorithms for the problem of assigning weights to edges in a fully-connected bipartite graph that minimizes the overall cost while satisfying constraints. Edges in this graph may disappear and reappear over time. Performance of these algorithms is measured using simulations. This paper also attempts to derandomize the randomized online algorithm for this problem

Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model

We consider online scheduling on multiple machines for jobs arriving one-by-one with the objective of minimizing the makespan. For any number of identical parallel or uniformly related machines, we provide a competitive-ratio approximation scheme that computes an online algorithm whose competitive ratio is arbitrarily close to the best possible competitive ratio. We also determine this value up to any desired accuracy. This is the first application of competitive-ratio approximation schemes in the online-list model. The result proves the applicability of the concept in different online models. We expect that it fosters further research on other online problems

Online Assignment Algorithms for Dynamic Bipartite Graphs

This paper analyzes the problem of assigning weights to edges incrementally in a dynamic complete bipartite graph consisting of producer and consumer nodes. The objective is to minimize the overall cost while satisfying certain constraints. The cost and constraints are functions of attributes of the edges, nodes and online service requests. Novelty of this work is that it models real-time distributed resource allocation using an approach to solve this theoretical problem. This paper studies variants of this assignment problem where the edges, producers and consumers can disappear and reappear or their attributes can change over time. Primal-Dual algorithms are used for solving these problems and their competitive ratios are evaluated

Online Bin Stretching with Three Bins

Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of $11/8 = 1.375$ for three bins. Additionally, we present a lower bound of $45/33 = 1.\overline{36}$ for Online Bin Stretching on three bins and a lower bound of $19/14$ for four and five bins that were discovered using a computer search.Comment: Preprint of a journal version. See version 2 for the conference paper. Conference paper split into two journal submissions; see arXiv:1601.0811

Deterministic Monotone Algorithms for Scheduling on Related Machines

We consider the problem of designing monotone deterministic algorithms for scheduling tasks on related machines in order to minimize the makespan. Several recent papers showed that monotonicity is a fundamental property to design truthful mechanisms for this scheduling problem. We give both theoretical and experimental results. First of all we consider the case of two machines when speeds of the machines are restricted to be powers of a given constant c>0. We prove that algorithm Largest Processing Time (LPT) is monotone for any c≥2 while it is not monotone for c≤1.78; algorithm List Scheduling (LS), instead, is monotone only for c>2. In the case of m>2 machines we restrict our attention to the class of “greedy-like” monotone algorithms defined in [Vincenzo Auletta, Roberto De Prisco, Paolo Penna, Giuseppe Persiano, Deterministic truthful approximation mechanisms for scheduling related machines, in: Proceedings of 21st Annual Symposium on Theoretical Aspects of Computer Science. STACS ’04, in: Lecture Notes in Computer Science, vol. 2996, Springer, 2004, pp. 608–619]. It has been shown that greedy-like monotone algorithms can be used to design a family of 2+ε-approximate truthful mechanisms. In particular, in [Vincenzo Auletta, Roberto De Prisco, Paolo Penna, Giuseppe Persiano, Deterministic truthful approximation mechanisms for scheduling related machines, in: Proceedings of 21st Annual Symposium on Theoretical Aspects of Computer Science. STACS ’04, in: Lecture Notes in Computer Science, vol. 2996, Springer, 2004, pp. 608–619], the greedy-like algorithm Uniform is proposed and it is proved that it is monotone when machine speeds are powers of a given integer constant c>0. In this paper we propose a new algorithm, called Uniform_RR, that is still monotone when speeds are powers of a given integer constant c>0 and we prove that its approximation factor is not worse than that of Uniform. We also experimentally compare the performance of Uniform, Uniform_RR, LPT, and several other monotone and greedy-like heuristics

Online Bipartite Matching with Decomposable Weights

We study a weighted online bipartite matching problem: G(V1, V2, E) is a weighted bipartite graph where V1 is known beforehand and the vertices of V2 arrive online. The goal is to match vertices of V2 as they arrive to vertices in V1, so as to maximize the sum of weights of edges in the matching. If assignments to V1 cannot be changed, no bounded competitive ratio is achievable. We study the weighted online matching problem with free disposal, where vertices in V1 can be assigned multiple times, but only get credit for the maximum weight edge assigned to them over the course of the algorithm. For this problem, the greedy algorithm is 0.5-competitive and determining whether a better competitive ratio is achievable is a well known open problem. We identify an interesting special case where the edge weights are decomposable as the product of two factors, one corresponding to each end point of the edge. This is analogous to the well studied related machines model in the scheduling literature, although the objective functions are different. For this case of decomposable edge weights, we design a 0.5664 competitive randomized algorithm in complete bipartite graphs. We show that such instances with decomposable weights are non-trivial by establishing upper bounds of 0.618 for deterministic and 0.8 for randomized algorithms. A tight competitive ratio of 1 − 1/e ≈ 0.632 was known previously for both the 0-1 case as well as the case where edge weights depend on the offline vertices only, but for these cases, reassignments cannot change the quality of the solution. Beating 0.5 for weighted matching where reassignments are necessary has been a significant challenge. We thus give the first online algorithm with competitive ratio strictly better than 0.5 for a non-trivial case of weighted matching with free disposal.

Rejecting Jobs to Minimize Load and Maximum Flow-time

Online algorithms are usually analyzed using the notion of competitive ratio which compares the solution obtained by the algorithm to that obtained by an online adversary for the worst possible input sequence. Often this measure turns out to be too pessimistic, and one popular approach especially for scheduling problems has been that of "resource augmentation" which was first proposed by Kalyanasundaram and Pruhs. Although resource augmentation has been very successful in dealing with a variety of objective functions, there are problems for which even a (arbitrary) constant speedup cannot lead to a constant competitive algorithm. In this paper we propose a "rejection model" which requires no resource augmentation but which permits the online algorithm to not serve an epsilon-fraction of the requests. The problems considered in this paper are in the restricted assignment setting where each job can be assigned only to a subset of machines. For the load balancing problem where the objective is to minimize the maximum load on any machine, we give O(\log^2 1/\eps)-competitive algorithm which rejects at most an \eps-fraction of the jobs. For the problem of minimizing the maximum weighted flow-time, we give an O(1/\eps^4)-competitive algorithm which can reject at most an \eps-fraction of the jobs by weight. We also extend this result to a more general setting where the weights of a job for measuring its weighted flow-time and its contribution towards total allowed rejection weight are different. This is useful, for instance, when we consider the objective of minimizing the maximum stretch. We obtain an O(1/\eps^6)-competitive algorithm in this case. Our algorithms are immediate dispatch, though they may not be immediate reject. All these problems have very strong lower bounds in the speed augmentation model

Online makespan scheduling with job migration on uniform machines

In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and ≈1.4659. They show that k=O(m) is sufficient to achieve this bound and no k=o(n) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a δ=Θ(1) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than 1.4659+δ with k=o(n). We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and ≈1.7992 with k=O(m). We also show that k=Ω(m) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines