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    "description": "We first present a fixed-parameter algorithm for the NP-hard problem of deciding if there are two matchings M1 and M2 in a given graph G such that |M1|\u2009+\u2009|M2| is no less than a given number k. The algorithm runs in \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$O\\left(m + k\\cdot k!\\cdot \\left(2\\sqrt{2}\\right)^k\\cdot n^2\\log n \\ \\right)$\\end{document} time, where n (respectively, m) is the number of vertices (respectively, edges) in G. We then present a combinatorial approximation algorithm for the NP-hard problem of finding two disjoint matchings in a given edge-weighted graph G so that their total weight is maximized. The algorithm achieves an approximation ratio of roughly 0.76 and runs in \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$O\\left(m + n^3\\alpha(n)\\right)$\\end{document} time, where \u03b1 is the inverse Ackermann function.", 
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