First Fit bin packing: A tight analysis

In the bin packing problem we are given an instance consisting of a sequence of items with sizes between 0 and 1. The objective is to pack these items into the smallest possible number of bins of unit size. FirstFit algorithm packs each item into the first bin where it fits, possibly opening a new bin if the item cannot fit into any currently open bin. In early seventies it was shown that the asymptotic approximation ratio of FirstFit bin packing is equal to 1.7. We prove that also the absolute approximation ratio for FirstFit bin packing is exactly 1.7. This means that if the optimum needs OPT bins, FirstFit always uses at most lfloor 1.7 OPT rfloor bins. Furthermore we show matching lower bounds for a majority of values of OPT, i.e., we give instances on which FirstFit uses exactly lfloor 1.7 OPT rfloor bins. Such matching upper and lower bounds were previously known only for finitely many small values of OPT. The previous published bound on the absolute approximation ratio of FirstFit was 12/7 approx 1.7143. Recently a bound of 101/59 approx 1.7119 was claimed

Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4*2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting

A Cryptographic Moving-Knife Cake-Cutting Protocol

This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple fairness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet. We show that the moving-knife protocol can be executed asynchronously by a discrete protocol using a secure auction protocol. The number of cuts is n-1 where n is the number of players, which is the minimum.Comment: In Proceedings IWIGP 2012, arXiv:1202.422

Scheduling shared continuous resources on many-cores

© 2017 Springer Science+Business Media New York We consider the problem of scheduling a number of jobs on m identical processors sharing a continuously divisible resource. Each job j comes with a resource requirement [InlineEquation not available: see fulltext.]. The job can be processed at full speed if granted its full resource requirement. If receiving only an x-portion of (Formula presented.), it is processed at an x-fraction of the full speed. Our goal is to find a resource assignment that minimizes the makespan (i.e., the latest completion time). Variants of such problems, relating the resource assignment of jobs to their processing speeds, have been studied under the term discrete–continuous scheduling. Known results are either very pessimistic or heuristic in nature. In this article, we suggest and analyze a slightly simplified model. It focuses on the assignment of shared continuous resources to the processors. The job assignment to processors and the ordering of the jobs have already been fixed. It is shown that, even for unit size jobs, finding an optimal solution is NP-hard if the number of processors is part of the input. Positive results for unit size jobs include a polynomial-time algorithm for any constant number of processors. Since the running time is infeasible for practical purposes, we also provide more efficient algorithm variants: an optimal algorithm for two processors and a [InlineEquation not available: see fulltext.] -approximation algorithm for m processors

Solution Of A Covering Problem Related To Labelled Tournaments

Suppose we have a tournament with edges labelled so that the edges incident with any vertex have at most k distinct labels (and no vertex has outdegree 0). Let m be the minimal size of a subset of labels such that for any vertex there exists an outgoing edge labelled by one of the labels in the subset. It was known that m ≤ \Gamma k+1 2 \Delta for any tournament. We show that this bound is almost best possible, by a probabilistic construction of tournaments with m = Ω(k²/log k). We give explicit tournaments with m = k 2\Gammao(1) . If the number of vertices is bounded by N < 2^k, we have a better upper bound of m = O(k log N), which is again almost optimal. We also consider a relaxation of this problem in which instead of the size of a subset of labels we minimize the total weight of a fractional set with analogous properties. In that case the optimal bound is 2k - 1

Online Scheduling

We survey some recent results on scheduling unit jobs. The emphasis of the talk is both on presenting some basic techniques and providing an overview of the current state of the art. The techniques presented cover charging schemes, potential function arguments, and lower bounds based on Yao\u27s principle. The studied problem is equivalent to the following buffer management problem: packets with specified weights and deadlines arrive at a network switch and need to be forwarded so that the total weight of forwarded packets is maximized. A packet not forwarded before its deadline brings no profit. The presented algorithms improve upon 2-competitive greedy algorithm, the competitive ratio is 1.939 for deterministic and 1.582 for randomized algorithms

A Lower Bound for Randomized On-Line Multiprocessor Scheduling

We significantly improve the previous lower bounds on the performance of randomized algorithms for on-line scheduling jobs on m identical machines. We also show that a natural idea for constructing an algorithm with matching performance does not work. Keywords combinatorial problems, on-line algorithms, randomization, scheduling, worst case bounds. 1 Introduction We study the model for scheduling introduced [7] and studied recently in [6, 1, 8]. This model is essentially a modified version of the game of Tetris. We have some fixed number of columns. Rectangles arrive one by one, each of them is one column wide and extends over one or more rows. We have to put each rectangle in one of the columns. The goal is to minimize the total number of rows that are at least partially used by the rectangles. In this scenario the columns represent the machines, rows represent the time steps and the rectangles represent the jobs with a running time corresponding to the height of a rectangle. More p..

Approximation Schemes for Scheduling on Uniformly Related and Identical Parallel Machines

We give a polynomial approximation scheme for the problem of scheduling on uniformly related parallel machines for a large class of objective functions that depend only on the machine completion times, including minimizing the l p norm of the vector of completion times. This generalizes and simplifies many previous results in this area. 1 Introduction Scheduling is one of fundamental areas of combinatorial optimization. Most multiprocessor scheduling problems are known to be hard to solve optimally (NP-hard, see below). Thus the research focuses on giving efficient approximation algorithms that produce a solution close to the optimal one. Ideally, one hopes to obtain a family of polynomial algorithms such that for any given " ? 0 the corresponding algorithm is guaranteed to produce a solution with a cost within a factor of (1 + ") of the optimum cost; such a family is called a polynomial approximation scheme. A polynomial scheme for the basic problem of minimization of the total..