György Dósa

The tight bound of first fit decreasing bin-packing algorithm is FFD(I) ≤ 11/9 OPT(I) + 6/9

First Fit Decreasing is a classical bin packing algorithm: the items are ordered into their nonincreasing order, and then in this order the next item is always packed into the first bin where it fits. For an instance I let FFD(I) and OPT(I) denote the number of the used bins by algorithm FFD, and an optimal algorithm, respectively. We show in this paper that FFD(I) ≤ 11/9OPT(I) + 6/9, (1) and that this bound is tight. The tight bound of the additive constant was an open question for many years.

Scheduling with machine cost and rejection

In this paper we consider the scheduling problem with machine cost and rejection penalties. For this problem, we are given a sequence of independent jobs, each being characterized by its processing time (size) and its penalty. No machine is initially provided, and when a job is revealed the algorithm has the option to purchase new machines. Right when a new job arrives, we have the following choices: (i) reject it, in which case we pay its penalty; (ii) non-preemptively process it on an existing machine, which contributes to the machine load; (iii) purchase a new machine, and assign it to this machine. The objective is to minimize the sum of the makespan, the cost for purchasing machines, and the total penalty of all rejected jobs. For the small job case, (where all jobs have sizes no greater than the cost for purchasing one machine, and which is the generalization of the Ski-Rental Problem) we present an optimal online algorithm with a competitive ratio of 2.

Tight absolute bound for First Fit Decreasing bin-packing: FFD(L)≤11/9OPT(L)+6/9

First Fit Decreasing is a classical bin-packing algorithm: the items are ordered by non-increasing size, and then in this order the next item is always packed into the first bin where it fits. For an instance L let FFD(L) and OPT(L) denote the number of bins used by algorithm FFD and by an optimal algorithm, respectively. In this paper we give the first complete proof of the inequalityFFD(L)=<11/9@?OPT(L)+6/9. This result is best possible, as was shown earlier by Dosa (2007) [3]. The asymptotic coefficient 11/9 was proved already in 1973 by Johnson, but the tight bound of the additive constant was an open question for four decades.

Bin packing problems with rejection penalties and their dual problems

In this paper we consider the following problems: we are given a set of n items {u1,....,un} and a number of unit-capacity bins. Each item ui has a size wi ∈ (0, 1] and a penalty pi ≥ 0. An item can be either rejected, in which case we pay its penalty, or put into one bin under the constraint that the total size of the items in the bin is no greater than 1. No item can be spread into more than one bin. The objective is to minimize the total number of used bins plus the total penalty paid for the rejected items. We call the problem bin packing with rejection penalties, and denote it as BPR. For the on-line BPR problem, we present an algorithm with an absolute competitive ratio of 2.618 while the lower bound is 2.343, and an algorithm with an asymptotic competitive ratio arbitrarily close to 1.75 while the lower bound is 1.540. For the off-line BPR problem, we present an algorithm with an absolute worst-case ratio of 2 while the lower bound is 1.5, and an algorithm with an asymptotic worst-case ratio of 1.5. We also study a closely related bin covering version of the problem. In this case pi means some amount of profit. If an item is rejected, we get its profit, or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1. The objective is to maximize the number of covered bins plus the total profit of all rejected items. We call this problem bin covering with rejection (BCR). For the on-line BCR problem, we show that no algorithm can have absolute competitive ratio greater than 0, and present an algorithm with asymptotic competitive ratio 1/2, which is the best possible. For the off-line BCR problem, we also present an algorithm with an absolute worst-case ratio of 1/2 which matches the lower bound.

A note on a selfish bin packing problem

In this paper, we consider a selfish bin packing problem, where each item is a selfish player and wants to minimize its cost. In our new model, if there are k items packed in the same bin, then each item pays a cost 1/k, where k ≥ 1. First we find a Nash Equilibrium (NE) in time O(n log n) within a social cost at most 1.69103OPT + 3, where OPT is the social cost of an optimal packing; where n is the number of items or players; then we give tight bounds for the worst NE on the social cost; finally we show that any feasible packing can be converged to a Nash Equilibrium in O(n2) steps without increasing the social cost.

Optimal Analysis of Best Fit Bin Packing

In early seventies it was shown that the asymptotic approximation ratio of BestFit bin packing is equal to 1.7. We prove that also the absolute approximation ratio for BestFit bin packing is exactly 1.7, improving the previous bound of 1.75. This means that if the optimum needs Opt bins, BestFit always uses at most \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins. Furthermore we show matching lower bounds for all values of Opt, i.e., we give instances on which BestFit uses exactly \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins. Thus we completely settle the worst-case complexity of BestFit bin packing after more than 40 years of its study.

Online scheduling with a buffer on related machines

AbstractOnline scheduling with a buffer is a semi-online problem which is strongly related to the basic online scheduling problem. Jobs arrive one by one and are to be assigned to parallel machines. A buffer of a fixed capacity K is available for storing at most K input jobs. An arriving job must be either assigned to a machine immediately upon arrival, or it can be stored in the buffer for unlimited time. A stored job which is removed from the buffer (possibly, in order to allocate a space in the buffer for a new job) must be assigned immediately as well. We study the case of two uniformly related machines of speed ratio s≥1, with the goal of makespan minimization.Two natural questions can be asked. The first question is whether this model is different from standard online scheduling, that is, is any size of buffer K>0 already helpful to the algorithm, compared to the case K=0. The second question is whether there exists a constant K, so that a larger buffer is no longer beneficial to an algorithm, that is, increasing the size of the buffer above this threshold would not change the best competitive ratio further. Previous work (Kellerer et al., Oper. Res. Lett. 21, 235–242, 1997; Zhang, Inf. Process. Lett. 61, 145–148, 1997; Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008) shows that in the case s=1, already K=1 allows to design a

Preemptive scheduling on a small number of hierarchical machines

\frac{4}{3}

Semi-online scheduling jobs with tightly-grouped processing times on three identical machines

-competitive algorithm, which is best possible for any K≥1, whereas the best possible ratio for K=0 is

On the absolute approximation ratio for First Fit and related results

\frac{3}{2}