Approximation algorithms for stochastic clustering

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results

Multi-level stochastic approximation algorithms

This paper studies multi-level stochastic approximation algorithms. Our aim is to extend the scope of the multilevel Monte Carlo method recently introduced by Giles (Giles 2008) to the framework of stochastic optimization by means of stochastic approximation algorithm. We first introduce and study a two-level method, also referred as statistical Romberg stochastic approximation algorithm. Then, its extension to multi-level is proposed. We prove a central limit theorem for both methods and describe the possible optimal choices of step size sequence. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.Comment: 44 pages, 9 figure

Approximation algorithms for stochastic and risk-averse optimization

We present improved approximation algorithms in stochastic optimization. We prove that the multi-stage stochastic versions of covering integer programs (such as set cover and vertex cover) admit essentially the same approximation algorithms as their standard (non-stochastic) counterparts; this improves upon work of Swamy \& Shmoys which shows an approximability that depends multiplicatively on the number of stages. We also present approximation algorithms for facility location and some of its variants in the $2$-stage recourse model, improving on previous approximation guarantees. We give a $2.2975$-approximation algorithm in the standard polynomial-scenario model and an algorithm with an expected per-scenario $2.4957$-approximation guarantee, which is applicable to the more general black-box distribution model.Comment: Extension of a SODA'07 paper. To appear in SIAM J. Discrete Mat