Packing a Knapsack of Unknown Capacity

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items and try to pack the items in this order, independent of the observations made while packing. We give efficient algorithms computing these policies. On the other hand, we show that, for any alpha>1, the problem of deciding whether a given universal policy achieves a factor of alpha is coNP-complete. If alpha is part of the input, the same problem is shown to be coNP-complete for items with unit densities. Finally, we show that it is coNP-hard to decide, for given alpha, whether a set of items admits a universal policy with factor alpha, even if all items have unit densities

Overcommitment in Cloud Services -- Bin packing with Chance Constraints

This paper considers a traditional problem of resource allocation, scheduling jobs on machines. One such recent application is cloud computing, where jobs arrive in an online fashion with capacity requirements and need to be immediately scheduled on physical machines in data centers. It is often observed that the requested capacities are not fully utilized, hence offering an opportunity to employ an overcommitment policy, i.e., selling resources beyond capacity. Setting the right overcommitment level can induce a significant cost reduction for the cloud provider, while only inducing a very low risk of violating capacity constraints. We introduce and study a model that quantifies the value of overcommitment by modeling the problem as a bin packing with chance constraints. We then propose an alternative formulation that transforms each chance constraint into a submodular function. We show that our model captures the risk pooling effect and can guide scheduling and overcommitment decisions. We also develop a family of online algorithms that are intuitive, easy to implement and provide a constant factor guarantee from optimal. Finally, we calibrate our model using realistic workload data, and test our approach in a practical setting. Our analysis and experiments illustrate the benefit of overcommitment in cloud services, and suggest a cost reduction of 1.5% to 17% depending on the provider's risk tolerance

Probabilistic analysis of algorithms for dual bin packing problems

In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and reveal the connection between this problem and the classical bin packing problem as well as to renewal theory.

Pressure screening and fluctuations at the bottom of a granular column

We report sets of precise and reproducible measurements on the static pressure at the bottom of a granular column. We make a quantitative analysis of the pressure saturation when the column height is increased. We evidence a great sensitivity of the measurements with the global packing fraction and the eventual presence of shear bands at the boundaries. We also show the limit of the classical Janssen model and discuss these experimental results under the scope of recently proposed theoretical frameworks.Comment: 17 pages, Latex, 8 eps figures, to appear in the European Physical Journal B (1999

Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

Granular Pressure and the Thickness of a Layer Jamming on a Rough Incline

Dense granular media have a compaction between the random loose and random close packings. For these dense media the concept of a granular pressure depending on compaction is not unanimously accepted because they are often in a "frozen" state which prevents them to explore all their possible microstates, a necessary condition for defining a pressure and a compressibility unambiguously. While periodic tapping or cyclic fluidization have already being used for that exploration, we here suggest that a succession of flowing states with velocities slowly decreasing down to zero can also be used for that purpose. And we propose to deduce the pressure in \emph{dense and flowing} granular media from experiments measuring the thickness of the granular layer that remains on a rough incline just after the flow has stopped.Comment: 10 pages, 2 figure

The Price of Information in Combinatorial Optimization

Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of $n$ independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.Comment: SODA 201