Facets of an Assignment Problem with 0–1 Side Constraint

We show that the problem of finding a perfect matching satisfying a single equality constraint with a 0–1 coefficients in an n × n incomplete bipartite graph, polynomially reduces to a special case of the same peoblem called the partitioned case. Finding a solution matching for the partitioned case in the incomlpete bipartite graph, is equivalent to minimizing a partial sum of the variables over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Q_{n_{1,} n_2 }^{n,r_1 } $$ \end{document} = the convex hull of incidence vectors of solution matchings for the partitioned case in the complete bipartite graph. An important strategy to solve this minimization problem is to develop a polyhedral characterization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Q_{n_{1,} n_2 }^{n,r_1 } $$ \end{document}. Towards this effort, we present two large classes of valid inequalities for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Q_{n_{1,} n_2 }^{n,r_1 } $$ \end{document}, which are proved to be facet inducing using a facet lifting scheme.