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Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area, most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios. In this work, we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing a submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic 1/2-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.
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polynomial-time'' means
efficient''. That algorithm is sequential and deterministic. We have also known since the 1980s that the matching problem has efficient parallel algorithms if the use of randomness is allowed. Formally, it is in the class RNC, i.e., it has randomized algorithms that use polynomially many processors and run in polylogarithmic time. However, we do not know if randomness is necessary - that is, whether the matching problem is in the class NC.
In this thesis we show that the matching problem is in quasi-NC. That is, we give a deterministic parallel algorithm that runs in O(log^3 n) time on n^{O(log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani to obtain an RNC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf, who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.Amin Karbasi, Morteza Zadimoghaddam