A stable set of a graph G = (V,E) is defined as a set of nodes...

 

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A stable set of a graph G = (V,E) is defined as a set of nodes S CV with the property that the nodes in S are pairwise non-adjacent (two nodes are adjacent if there is an edge between them). (a) (2 points) Formulate the problem of finding a maximum cardinality stable set as an MILP. (b) (4 points) An odd hole is a cycle with an odd number of nodes and no edges between non-adjacent nodes of the cycle. Let x; be a binary variable that takes a value 1 if a node j S. If H denotes the set of nodes in an odd hole, show that the following inequality is valid: x; (|H| 1)/2. A stable set of a graph G = (V,E) is defined as a set of nodes S CV with the property that the nodes in S are pairwise non-adjacent (two nodes are adjacent if there is an edge between them). (a) (2 points) Formulate the problem of finding a maximum cardinality stable set as an MILP. (b) (4 points) An odd hole is a cycle with an odd number of nodes and no edges between non-adjacent nodes of the cycle. Let x; be a binary variable that takes a value 1 if a node j S. If H denotes the set of nodes in an odd hole, show that the following inequality is valid: x; (|H| 1)/2.

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