Online bin packing with resource augmentation

In competitive analysis, we usually do not put any restrictions on the computational complexity of online algorithms, although efficient algorithms are preferred. Thus if such an algorithm were given the entire input in advance, it could give an optimal solution (in exponential time). Instead of giving the algorithm more knowledge about the input, in this paper we consider the effects of giving an online bin packing algorithm larger bins than the offline algorithm it is compared to. We give new algorithms for this problem that combine items in bins in an unusual way and give bounds on their performance which improve upon the best possible bounded space algorithm. We also give general lower bounds for this problem which are nearly matching for bin sizes b ?

Lower bounds for on-line single-machine scheduling

The problem of scheduling jobs that arrive over time on a single machine is well-studied. We study the preemptive model and the model with restarts. We provide lower bounds for deterministic and randomized algorithms for several optimality criteria: weighted and unweighted total completion time, and weighted and unweighted total flow time. By using new techniques, we provide the first lower bounds for several of these problems, and we significantly improve the bounds that were known

This side up!

We consider two- and three-dimensional bin packing problems where 9

Optimal online bounded space multidimensional packing

We solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space. We show that it is optimal among bounded space algorithms for any dimension $d>1$. Its asymptotic performance ratio is $(Pi_{infty})^d$, where $Pi_{infty}approx1.691$ is the asymptotic performance ratio of the one-dimensional algorithm harm. A modified version of this algorithm for the case where all items are hypercubes is also shown to be optimal. Its asymptotic performance ratio is sublinear in $d$. Additionally, for the special case of packing squares in two-dimensional bins, we present a new unbounded space online algorithm with asymptotic performance ratio of at most $2.271$. We also present an approximation algorithm for the offline problem with approximation ratio of $16/11$. This improves upon all earlier approximation algorithms for this problem, including the algorithm from Caprara, Packing 2-dimensional bins in harmony, Proc. 43rd FOCS, 2002

New results on flow time with resource augmentation

We study the problem of scheduling $n$ jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has $l$ machines. We design an algorithm of competitive ratio O(min(Delta^{1/l,n^{1/l)), where $Delta$ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant $l$. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only $Delta$ is known. This gives a trade-off between the resource augmentation and the competitive ratio. We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has $m$ machines and the on-line algorithm has $lm$ machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines

Minimizing the maximum starting time on-line

We study the scheduling problem of minimizing the maximum starting time on-line. The goal is to minimize the last time that a job starts. We show that while the greedy algorithm has a competitive ratio of $Theta(log m)$, we can give a constant competitive algorithm for this problem. We also show that the greedy algorithm is optimal for resource augmentation in the sense that it requires 2m-1 machines to have a competitive ratio of 1, whereas no algorithm can achieve this with 2m-1 machines

Bounds for online bounded space hypercube packing

In hypercube packing, we receive a sequence of hypercubes that need to be packed into unit hypercubes which are called bins. Items arrive online and each item must be placed within its bin without overlapping with other items in that bin. The goal is to minimize the total number of bins used. We present lower and upper bounds for online bounded space hypercube packing in dimensions 2,...,

Resource augmentation in load balancing

We consider load-balancing in the following setting. The on-line algorithm is allowed to use $n$ machines, whereas the optimal off-line algorithm is limited to $m$ machines, for some fixed $m < n$. We show that while the greedy algorithm has a competitive ratio which decays linearly in the inverse of $n/m$, the best on-line algorithm has a ratio which decays exponentially in $n/m$. Specifically, we give an algorithm with competitive ratio of 1+2^{- frac{n{m (1- o (1)), and a lower bound of 1+ e^{ - frac{n{m (1+ o(1)) on the competitive ratio of any randomized algorithm. We also consider the preemptive case. We show an on-line algorithm with a competitive ratio of 1+ e^{ - frac{n{m (1+ o(1)). We show that the algorithm is optimal by proving a matching lower bound. We also consider the non-preemptive model with temporary tasks. We prove that for $n=m+1$, the greedy algorithm is optimal. (It is not optimal for permanent tasks.

Tight bounds on the competitive ratio on accomodating sequences for the seat reservation problem

The unit price seat reservation problem is investigated. The seat reservation problem is the problem of assigning seat numbers on-line to requests for reservations in a train traveling through $k$ stations. We are considering the version where all tickets have the same price and where requests are treated fairly, i.e., a request which can be fulfilled must be granted. For fair deterministic algorithms, we provide an asymptotically matching upper bound to the existing lower bound which states that all fair algorithms for this problem are frac{1{2-competitive on accommodating sequences, when there are at least three seats. Additionally, we give an asymptotic upper bound of frac{7{9 for fair randomized algorithms against oblivious adversaries. We also examine concrete on-line algorithms, First-Fit and Random, for the special case of two seats. Tight analyses of their performance are given

Properties of the H-alpha-emitting Circumstellar Regions of Be Stars

Long-baseline interferometric observations obtained with the Navy Prototype Optical Interferometer of the H-alpha-emitting envelopes of the Be stars eta Tauri and beta Canis Minoris are presented. For compatibility with the previously published interferometric results in the literature of other Be stars, circularly symmetric and elliptical Gaussian models were fitted to the calibrated H-alpha observations. The models are sufficient in characterizing the angular distribution of the H-alpha-emitting circumstellar material associated with these Be stars. To study the correlations between the various model parameters and the stellar properties, the model parameters for eta Tau and beta CMi were combined with data for other Be stars from the literature. After accounting for the different distances to the sources and stellar continuum flux levels, it was possible to study the relationship between the net H-alpha emission and the physical extent of the H-alpha-emitting circumstellar region. A clear dependence of the net H-alpha emission on the linear size of the emitting region is demonstrated and these results are consistent with an optically thick line emission that is directly proportional to the effective area of the emitting disk. Within the small sample of stars considered in this analysis, no clear dependence on the spectral type or stellar rotation is found, although the results do suggest that hotter stars might have more extended H-alpha-emitting regions.Comment: 24 pages, 16 figures, accepted for publication in Ap