By Bauer P.
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Bauer, P. (1994): A Polyhedral Approach to the Weighted Girth Problem. PhD thesis, Universität zu Köln 6. Bauer, P. (1997): The circuit polytope: facets. Mathematics of Operations Research 22, 110–145 7. P. (1998): A branch and cut approach to the cardinality constrained circuit problem. Technical Report TLI-98-04, The Logistics Institute, Georgia Institute of Technology, 1998. pdf 8. , Williamson, D. (1993): A note on the prize collecting traveling salesman problem. Mathematical Programming 59, 413–420 9.
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K is odd and ∃g ∈ (C \ f ) ∩ (E \ T ) such that l gT ≥ k. Suppose first that one of the following occurs: – k ≤ l Tf < 3k/2 − 2, – k ≤ l gT < 3k/2 − 2, or – l gT = 3k/2 − 1 + 2i for some i ∈ +. By definitions of the weights wlTe , it can be shown that either wlTf ≤ (3 − k)/2 and wg ≤ (5 − k)/2, or wlTf ≤ (5 − k)/2 and wg ≤ (3 − k)/2. 3), we can now write (3 − k)/2 + (5 − k)/2 ≥ wlTf + wg ≥ wlTf + we x e ≥ 3 − e∈E\T \ f xe, e∈T which yields x e ≥ k − 1. 1, we have a contradiction since the circuit C is too long.