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Бабенко Максим Александрович
Курмакаева Эльвира Ростямовна
A perfect 2-matching in an undirected graph G=(V,E) is a function x:E→0,1,2 such that for each node v∈V the sum of values x(e) on all edges e incident to v equals 2. If supp(x)=e∈E∣x(e)≠0 contains no triangles then x is called triangle-free. Polyhedrally speaking, triangle-free 2-matchings are harder than 2-matchings, but easier than usual 1-matchings. Given edge costs c:E→R + , a natural combinatorial problem consists in finding a perfect triangle-free matching of minimum total cost. For this problem, Cornuéjols and Pulleyblank devised a combinatorial strongly-polynomial algorithm, which can be implemented to run in O(VElogV) time. (Here we write V, E to indicate their cardinalities |V|, |E|.) If edge costs are integers in range [0,C] then for both 1- and 2-matchings some faster scaling algorithms are known that find optimal solutions within O(Vα(E,V)logVElog(VC)) and O(VElog(VC)) time, respectively, where α denotes the inverse Ackermann function. So far, no efficient cost-scaling algorithm is known for finding a minimum-cost perfect triangle-free2-matching. The present paper fills this gap by presenting such an algorithm with time complexity of O(VElogVlog(VC)).
The OPERA experiment was designed to study νμ→ντ oscillations in the appearance mode in the CERN to Gran Sasso Neutrino beam (CNGS). In this Letter, we report the final analysis of the full data sample collected between 2008 and 2012, corresponding to 17.97×1019 protons on target. Selection criteria looser than in previous analyses have produced ten ντ candidate events, thus reducing the statistical uncertainty in the measurement of the oscillation parameters and of ντ properties. A multivariate approach for event identification has been applied to the candidate events and the discovery of ντ appearance is confirmed with an improved significance level of 6.1σ. |Δm232| has been measured, in appearance mode, with an accuracy of 20%. The measurement of the ντ charged-current cross section, for the first time with a negligible contamination from ¯ντ, and the first direct evidence for the ντ lepton number are also reported.