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 \(\alpha\) -asymptotic approximation algorithm and provides a polynomial-time \((\alpha+\varepsilon)\) -competitive algorithm for online bin packing under the i.i.d. model, where \(\varepsilon > 0\) is a small constant. Using the AFPTAS for offline bin packing, we thus provide a linear time \((1+\varepsilon)\) -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 chooses the instance, but the order of arrival of items in the instance is drawn uniformly at random from the set of all permutations of the items. Kenyon’s seminal result (1996) 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 \(\approx 1.15\) . However, it has been a long-standing open problem to break the barrier of \(3/2\) even for special cases. Recently, Albers et al. (2021) showed an improvement by proving that in the special case when all the item sizes are greater than \(1/3\) , Best-Fit has a competitive ratio of at most \(5/4\) in the random-order model. In this work, we settle this special case by showing that Best-Fit has a competitive ratio of exactly 1, i.e., Best-Fit performs almost optimally in this special case in the random-order model. We also 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 \((1/4,1/2]\) .

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 $α$-asymptotic approximation algorithm and provides a polynomial-time $(α+ \varepsilon)$-competitive algorithm for online bin packing under the i.i.d. model, where $\varepsilon$>0 is a small constant. Using the AFPTAS for offline bin packing, we thus provide a linear time $(1+\varepsilon)$-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].

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 article, 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 $\ln k$.