Walrasian equilibrium is a prominent market equilibrium notion, but rarely exists in markets with indivisible items. We introduce a new market equilibrium notion, called two-price equilibrium (2PE). A 2PE is a relaxation of Walrasian equilibrium, where instead of a single price per item, every item has two prices: one for the item's owner and a (possibly) higher one for all other buyers. Thus, a 2PE is given by a tuple (S,p_high,p_low) of an allocation S and two price vectors p_high,p_low, where every buyer i is maximally happy with her bundle S_i, given prices p_low for items in S_i and prices p_high for all other items. 2PE generalizes previous market equilibrium notions, such as conditional equilibrium, and is related to relaxed equilibrium notions like endowment equilibrium. We define the discrepancy of a 2PE --- a measure of distance from Walrasian equilibrium --- as the sum of differences p_high_j-p_low_j over all items (normalized by social welfare). We show that the social welfare degrades gracefully with the discrepancy; namely, the social welfare of a 2PE with discrepancy d is at least a fraction 1/d+1 of the optimal welfare. We use this to establish welfare guarantees for markets with subadditive valuations over identical items. In particular, we show that every such market admits a 2PE with at least 1/7 of the optimal welfare. This is in contrast to Walrasian equilibrium or conditional equilibrium which may not even exist. Our techniques provide new insights regarding valuation functions over identical items, which we also use to characterize instances that admit a WE.

Allocating the right amount of resources to each service in any of the datacenters in a cloud environment is a very difficult task. This task becomes much harder due to the dynamic nature of the workload and the fact that while long term statistics about the demand may be known, it is impossible to predict the exact demand in each point in time. As a result, service providers either over allocate resources and hurt the service cost efficiency, or run into situation where the allocated local resources are insufficient to support the current demand. In these cases, the service providers deploy overflow mechanisms such as redirecting traffic to a remote datacenter or temporarily leasing additional resources (at a higher price) from the cloud infrastructure owner. The additional cost is in many cases proportional to the amount of overflow demand. In this paper we study this approach and develop a novel mechanism to assign services to datacenters based on the available resources in each datacenter and the distribution of the demand for each service. We use comprehensive analysis to prove that the overall overflow cost is almost optimal for arbitrary demand distributions, as long as there are no dependencies among the services. We further show, using simulation based on real data that the scheme performs very well on realistic service workloads.

The stochastic knapsack problem is a stochastic version of the well known deterministic knapsack problem, in which some of the input values are random variables. There are several variants of the stochastic problem. In this paper we concentrate on the chance-constrained variant, where item values are deterministic and item sizes are stochastic. The goal is to find a maximum value allocation subject to the constraint that the overflow probability is at most a given value. Previous work showed a PTAS for the problem for various distributions (Poisson, Exponential, Bernoulli and Normal). Some strictly respect the constraint and some relax the constraint by a factor of (1+epsilon). All algorithms use Omega(n^{1/epsilon}) time. A very recent work showed a "almost FPTAS" algorithm for Bernoulli distributions with O(poly(n) * quasipoly(1/epsilon)) time. In this paper we present a FPTAS for normal distributions with a solution that satisfies the chance constraint in a relaxed sense. The normal distribution is particularly important, because by the Berry-Esseen theorem, an algorithm solving the normal distribution also solves, under mild conditions, arbitrary independent distributions. To the best of our knowledge, this is the first (relaxed or non-relaxed) FPTAS for the problem. In fact, our algorithm runs in poly(n/epsilon) time. We achieve the FPTAS by a delicate combination of previous techniques plus a new alternative solution to the non-heavy elements that is based on a non-convex program with a simple structure and an O(n^2 log {n/epsilon}) running time. We believe this part is also interesting on its own right.