Consider the following variant of the set cover problem. We are given a universe @math and a collection of subsets @math where @math . For every element @math we need to find a set @math such that @math . Once we construct and fix the mapping @math a subset @math of the universe is revealed, and we need to cover all elements from @math with exactly @math . The goal is to find a mapping such that the cover @math is as cheap as possible. This is an example of a universal problem where the solution has to be created before the actual instance to deal with is revealed. Such problems appear naturally in some settings when we need to optimize under uncertainty and it may be actually too expensive to begin finding a good solution once the input starts being revealed. A rich body of work was devoted to investigate the approximability of such problems under the regime of worst case analysis or when the input instance is drawn randomly from some probability distribution. Here one typically compares the quality of the produced solution with the optimal offline solution. In this paper we consider a different viewpoint: What if we would compare our approximate universal solution against an optimal universal solution that obeys the same rules as we do? We show that under this viewpoint it is possible to achieve improved approximation algorithms for the stochastic version of universal set cover. Our result is based on rounding a proper configuration IP that captures the optimal universal solution, and using tools from submodular optimization. The same basic approach leads to improved approximation algorithms also for other related problems.
One can of course consider the (offline) stochastic version of optimization problems. For example, @math -stage stochastic set cover is studied in @cite_32 @cite_41 , with an improved approximation factor (independent from @math ) later given in @cite_21 .
@cite_41 @cite_21 @cite_32
2019875157 2112058494 2117399775 2008298964
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 that shows an approximability which 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 study two-stage, finite-scenario stochastic versions of several combinatorial optimization problems, and provide nearly tight approximation algorithms for them. Our problems range from the graph-theoretic (shortest path, vertex cover, facility location) to set-theoretic (set cover, bin packing), and contain representatives with different approximation ratios.The approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart. Furthermore, we show that common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be adapted to obtain these results. Stochastic optimization problems attempt to model uncertainty in the data by assuming that (part of) the input is specified in terms of a probability distribution. We consider the well-studied paradigm of 2-stage models with recourse: first, given only distributional information about (some of) the data one commits on initial actions, and then once the actual data is realized (according to the distribution), further (recourse) actions can be taken. We give the first approximation algorithms for 2-stage discrete stochastic optimization problems with recourse for which the underlying random data is given by a "black box" and no restrictions are placed on the costs in the two stages, based on an FPRAS for the LP relaxation of the stochastic problem (which has exponentially many variables and constraints). Among the range of applications we consider are stochastic versions of the set cover, vertex cover, facility location, multicut (on trees), and multicommodity flow problems. We consider two- and multistage versions of stochastic combinatorial optimization problems with recourse: in this framework, the instance for the combinatorial optimization problem is drawn from a known probability distribution @math and is only revealed to the algorithm over two (or multiple) stages. At each stage, on receiving some more information about the instance, the algorithm is allowed to build some partial solution. Since the costs of elements increase with each passing stage, there is a natural tension between waiting for later stages, to gain more information about the instance, and purchasing elements in earlier stages, to take advantages of lower costs. We provide approximation algorithms for stochastic combinatorial optimization problems (such as the Steiner tree problem, the Steiner network problem, and the vertex cover problem) by means of a simple sampling-based algorithm. In every stage, our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample. We show that if one can construct cost-sharing functions associated with the algorithms used to construct these partial solutions, then this strategy results in provable approximation guarantees for the overall stochastic optimization problem. We also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem, a problem that does not fit into the simpler framework of our main model.