Exercise: Let P &#x2208;<!-- ∈ --> <mi mathvariant="double-struck">R V </msup>

Wayne Steele 2022-05-28 Answered
Exercise: Let P R V be defined by the inequalities
x u 1  for every  u V , ( 1 ) x u + x v 1  for every edge  u v E , ( 2 )
node set V = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }
Starting from the system ( 1 ) ( 2 ), give the cutting-plane proof of the inequality
x 1 + x 2 + x 6 2
What I've tried: I know that I need to show that there exists a nonnegative combination from the inequalities ( 1 ) and ( 2 ) such that x 1 + x 2 + x 3 3 holds. I don't know how unfortunately.
Question: How do I solve this exercise?
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Answers (1)

Krish Finley
Answered 2022-05-29 Author has 14 answers
If we put x 1 = x 2 = x 6 = 1 / 2 and all other x i ’s equal 1 then all the defining inequalities will be satisfied but x 1 + x 2 + x 6 = 3 / 2 < 2.
But the claim holds provided we require all xi are integer. Indeed, if x 1 + x 2 + x 6 < 2 then al least two of these x i (say, x u and x v ) are zero. Then u v E, but x u + x v = 0, a contradiction to (2).
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