Polynomial hierarchy

NP-hard Problems and Languages and complements to algorithmic complexity

Published in J.-P. Barthelemy, G. Cohen, A. Lobstein, Catherine Fritsch-Mignotte, Maurice Mignotte, Algorithmic Complexity and communication problems, 2020

J.-P. Barthelemy, G. Cohen, A. Lobstein

The polynomial hierarchy (Meyer & Stockmeyer, 1972; Stockmeyer, 1976; Wrathall, 1976) extends the positions of P, NP, co-NP to sets of problems “beyond” these three classes, and it allows us to classify problems and languages according to their difficulty. Before describing this hierarchy, let us first check our knowledge of the three classes P, NP and co-NP.

A generic optimization framework for resilient systems

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Journal Information

Published in Optimization Methods and Software, 2023

Marc E. Pfetsch, Andreas Schmitt

It is important to note that we concentrate on the case in which the feasibility sets and are generic, i.e. we do not exploit their particular structure. In fact, since we deal with mixed-integer nonlinear and nonconvex problems, a practically feasible reformulation for the lower level based on optimality conditions seems to be out of reach. Moreover, we show that the decision version of (2) is on the third level of the polynomial hierarchy and thus extremely challenging. To provide solutions for practical instances, we essentially build on the (implicit) assumption that the number of failures k is small. Nevertheless, we show that our algorithm outperforms an algorithm based on the enumeration of failures.

Polynomial hierarchy

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NP-hard Problems and Languages and complements to algorithmic complexity

A generic optimization framework for resilient systems