In this paper, we derive new families of piecewise linear facet-defining inequalities for the finite group problem and extreme inequalities for the infinite group problem using approximate lifting. The new valid inequalities for the finite group problem are two- and three-slope facet-defining inequalities as well as the first family of four-slope facet-defining inequalities. The new valid inequalities for the infinite group problem are families of two- and three-slope extreme inequalities, including nontrivial inequalities that are not continuous. These new inequalities not only illustrate the diversity of strong inequalities for the finite and infinite group problems, but also provide a large variety of new cutting planes for solving integer and mixed-integer programming problems.
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