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We introduce the complexity class ∃ ℝ $\exists \mathbb {R}$ based on the existential theory of the reals. We show that the definition of ∃ ℝ $\exists \mathbb {R}$ is robust in the sense that even the fragment of the theory expressing solvability of systems of strict polynomial inequalities leads to the same complexity class. Several natural and well-known(More)
We present an algorithm for the parameterized feedback ver-tex set problem that runs in time O((2 lg k + 2 lg lg k + 18) k n 2). This improves the previous O(max{12 k , (4 lg k) k }n ω) algorithm by Raman et al. by roughly a 2 k factor (n w ∈ O(n 2.376) is the time needed to multiply two n × n matrices). Our results are obtained by developing new(More)
We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a drawing realizing the string graph in the plane. This upper bound confirms a conjecture by Kratochv\'{\i}l and Matou\v{s}ek~\cite{KM91} and settles the long-standing open problem of the decidability of(More)
An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this(More)
Digital holographic microscopy provides new facilities for contactless and marker-free quantitative phase contrast imaging. In this work, a digital holographic microscopy method for the integral refractive index determination of living single cells in cell culture medium is presented. Further, the obtained refractive index information is applied to full(More)
In the Ramsey theory of graphs F → (G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H. ARROWING, the problem of deciding whether F → (G, H), lies in Π p 2 = coNP NP and it was shown to be coNP-hard by Burr [Bur90]. We prove that ARROWING is Π p 2-complete, simultaneously settling a conjecture of(More)