A k-hitting set in a hypergraph is a set of at most k vertices that intersects all hyperedges. We study the union of all inclusion-minimal k-hitting sets in hypergraphs of rank r (where the rank is the maximum size of hyperedges). We show that this union is relevant for certain combinatorial inference problems and give worst-case bounds on its size, depending on r and k. For r = 2 our result is tight, and for each r 3 we have an asymptotically optimal bound and make progress regarding the constant factor. The exact worst-case size for r3 remains an open problem. We also propose an algorithm for counting all k-hitting sets in hypergraphs of rank r. Its asymptotic runtime matches the best one known for the much more special problem of finding one k-hitting set. The results are used for efficient counting of k-hitting sets that contain any particular vertex.
Abstract.In an overdetermined and feasible system of linear equations Ax = b, let vector b be corrupted, in the way that at most k entries are off their true values. Assume that we can check in the restricted system given by any minimal dependent set of rows, the correctness of all corresponding values in b. Furthermore, A has only coefficients 0 and 1, with at most two 1s in each row. We wish to recover the correct values in b and x as much as possible. The problem arises in a certain chemical mixture inference application in molecular biology, where every observable reaction product stems from at most two candidate substances. After formalization we prove that the problem is NP-hard but fixed-parameter tractable in k. The FPT result relies on the small girth of certain graphs.
Peptide Mass Fingerprinting (PMF) for long has been a widely used and reliable method for protein identification. However it faced several problems, the most important of which is inability of classical methods to deal with protein mixtures. To cope with this problem, more costly experimental techniques are employed. We investigate, whether it is possible to extract more information from PMF by more thorough data analysis. To do this, we propose a novel method to remove noise from the data and show how the results can be interpreted in a different way. We also provide simulation results suggesting our method can be used for analysis of small mixtures.
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