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hw3 - Industrial Engineering Integer Programming Fall...

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Industrial Engineering Fall Integer Programming Homework Last updated: October , Due: Friday October , in class Note: only a proper subset of these problems will be graded. Prob em . (Bertsimas and Weismantel, Exercise . ) Prove that a matrix of zero-one elements, in which each column has consecutive ones, is totally unimodular. Prob em . (Bertsimas and Weismantel, Exercise . ) Let b Z m . Show that { x R n Ax b , x } is integral if and only if {( x , s ) R n + m Ax + s = b , x , s , } is integral. Prob em . (Bertsimas and Weismantel, Exercise . ) a. Show that an incidence matrix of a general undirected graph is not totally unimodular. b. Show that an incidence matrix of an undirected graph is totally unimodular if and only if the graph is bipartite. (See Chapter of Korte and Vygen for a formal de nition of the incidence matrix of an undirected graph.) Prob em . (Bertsimas and Weismantel, Exercise . ) a. Recall the integer programming formulation of the minimum cut problem on an undirected graph we discussed in Lecture . Show how to modify this formulation to
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