Stochastic Combinatorial Optimization via Poisson Approximation

We study several stochastic combinatorial problems, including the expected utility maximization problem, the stochastic knapsack problem and the stochastic bin packing problem. A common technical challenge in these problems is to optimize some function of the sum of a set of random variables. The difficulty is mainly due to the fact that the probability distribution of the sum is the convolution of a set of distributions, which is not an easy objective function to work with. To tackle this difficulty, we introduce the Poisson approximation technique. The technique is based on the Poisson approximation theorem discovered by Le Cam, which enables us to approximate the distribution of the sum of a set of random variables using a compound Poisson distribution. We first study the expected utility maximization problem introduced recently [Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we obtain an additive PTAS if there is a multidimensional PTAS for the multi-objective version of the problem, strictly generalizing the previous result. For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and Tardos, STOC97]), we show there is a polynomial time algorithm which uses at most the optimal number of bins, if we relax the size of each bin and the overflow probability by eps. For stochastic knapsack, we show a 1+eps-approximation using eps extra capacity, even when the size and reward of each item may be correlated and cancelations of items are allowed. This generalizes the previous work [Balghat, Goel and Khanna, SODA11] for the case without correlation and cancelation. Our algorithm is also simpler. We also present a factor 2+eps approximation algorithm for stochastic knapsack with cancelations. the current known approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of the 45th ACM Symposium on the Theory of Computing (STOC13

Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields

The one-dimensional Hubbard model with arbitrary boundary magnetic fields is solved exactly via the Bethe ansatz methods. With the coordinate Bethe ansatz in the charge sector, the second eigenvalue problem associated with the spin sector is constructed. It is shown that the second eigenvalue problem can be transformed into that of the inhomogeneous XXX spin chain with arbitrary boundary fields which can be solved via the off-diagonal Bethe ansatz method.Comment: published version, 15 pages, no figur

Multipotent vascular stem cells contribute to neurovascular regeneration of peripheral nerve.

BackgroundNeurovascular unit restoration is crucial for nerve regeneration, especially in critical gaps of injured peripheral nerve. Multipotent vascular stem cells (MVSCs) harvested from an adult blood vessel are involved in vascular remodeling; however, the therapeutic benefit for nerve regeneration is not clear.MethodsMVSCs were isolated from rats expressing green fluorescence protein (GFP), expanded, mixed with Matrigel matrix, and loaded into the nerve conduits. A nerve autograft or a nerve conduit (with acellular matrigel or MVSCs in matrigel) was used to bridge a transected sciatic nerve (10-mm critical gap) in rats. The functional motor recovery and cell fate in the regenerated nerve were investigated to understand the therapeutic benefit.ResultsMVSCs expressed markers such as Sox 17 and Sox10 and could differentiate into neural cells in vitro. One month following MVSC transplantation, the compound muscle action potential (CMAP) significantly increased as compared to the acellular group. MVSCs facilitated the recruitment of Schwann cell to regenerated axons. The transplanted cells, traced by GFP, differentiated into perineurial cells around the bundles of regenerated myelinated axons. In addition, MVSCs enhanced tight junction formation as a part of the blood-nerve barrier (BNB). Furthermore, MVSCs differentiated into perivascular cells and enhanced microvessel formation within regenerated neurovascular bundles.ConclusionsIn rats with peripheral nerve injuries, the transplantation of MVSCs into the nerve conduits improved the recovery of neuromuscular function; MVSCs differentiated into perineural cells and perivascular cells and enhanced the formation of tight junctions in perineural BNB. This study demonstrates the in vivo therapeutic benefit of adult MVSCs for peripheral nerve regeneration and provides insight into the role of MVSCs in BNB regeneration