4 research outputs found
The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential
The weighted k-server problem is a natural generalization of the k-server problem in which the cost incurred in moving a server is the distance traveled times the weight of the server. Even after almost three decades since the seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem remains poorly understood even on the simplest class of metric spaces - the uniform metric spaces. In particular, in the case of randomized algorithms against the oblivious adversary, neither a better upper bound that the doubly exponential deterministic upper bound, nor a better lower bound than the logarithmic lower bound of unweighted k-server, is known. In this paper, we make significant progress towards understanding the randomized competitive ratio of weighted k-server on uniform metrics. We cut down the triply exponential gap between the upper and lower bound to a singly exponential gap by proving that the competitive ratio is at least exponential in k, substantially improving on the previously known lower bound of about ln k
On Minimizing Generalized Makespan on Unrelated Machines
We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given n jobs and m machines and each job j has arbitrary processing time p_{ij} on machine i. Additionally, there is a general symmetric monotone norm ?_i for each machine i, that determines the load on machine i as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load.
Recently, Deng, Li, and Rabani [Deng et al., 2023] gave a 3 approximation for GMP when the ?_i are top-k norms, and they ask the question whether an O(1) approximation exists for general norms ?? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant ? > 0, such that GMP is ?(log^? n) hard to approximate. We also give an ?(log^{1/2} n) integrality gap for the natural configuration LP
Near-optimal Algorithms for Stochastic Online Bin Packing
We study the online bin packing problem under two stochastic settings. In the
bin packing problem, we are given n items with sizes in (0,1] and the goal is
to pack them into the minimum number of unit-sized bins. First, we study bin
packing under the i.i.d. model, where item sizes are sampled independently and
identically from a distribution in (0,1]. Both the distribution and the total
number of items are unknown. The items arrive one by one and their sizes are
revealed upon their arrival and they must be packed immediately and irrevocably
in bins of size 1. We provide a simple meta-algorithm that takes an offline
-asymptotic approximation algorithm and provides a polynomial-time
-competitive algorithm for online bin packing under the
i.i.d. model, where >0 is a small constant. Using the AFPTAS for
offline bin packing, we thus provide a linear time
-competitive algorithm for online bin packing under i.i.d.
model, thus settling the problem.
We then study the random-order model, where an adversary specifies the items,
but the order of arrival of items is drawn uniformly at random from the set of
all permutations of the items. Kenyon's seminal result [SODA'96] showed that
the Best-Fit algorithm has a competitive ratio of at most 3/2 in the
random-order model, and conjectured the ratio to be around 1.15. However, it
has been a long-standing open problem to break the barrier of 3/2 even for
special cases. Recently, Albers et al. [Algorithmica'21] showed an improvement
to 5/4 competitive ratio in the special case when all the item sizes are
greater than 1/3. For this special case, we settle the analysis by showing that
Best-Fit has a competitive ratio of 1. We make further progress by breaking the
barrier of 3/2 for the 3-Partition problem, a notoriously hard special case of
bin packing, where all item sizes lie in (1/4,1/2]