Search

Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings, which has a balancedness of 0.4762. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. More precisely, we obtain a balancedness of 0.4326, improving on a prior 0.1997-balanced scheme. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints., (© The Author(s) 2020.)

The immunological properties of myosins, especially their muscle-type, class and species specificity, are still controversial. It is the opinion of the authors that the lack of agreement might at least in part be due to the use of contaminated myosins as immunogens and inappropriate methods. We, therefore, purified myosins to a very high degree (approximately 99%) and induced antisera in guinea-pigs. Studies of quantitative microcomplement fixation and enzyme-inhibition by antisera yielded the following results: myosins of cross-striated muscle have an absolute class, a very pronounced muscle-type and a low species specificity. It can be shown that even a very small contamination of myosins with other proteins could seriously hamper the experiments and that the results obtained depend significantly on the immunological methods employed.