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Extra info for A branch and cut approach to the cardinality constrained circuit problem
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1. k is even and ∃g ∈ (C \ f ) ∩ (E \ T ) such that l gT ≥ k. In this subcase, it can be shown by definition of wlTe that wlTf ≤ 2 − k/2, and wg ≤ 2 − k/2. Also by definition of wlTe , we know that we ≤ 0 ∀e ∈ E \ T . 3), we may write 2 − k/2 + 2 − k/2 ≥ wlTf + wg ≥ wlTf + we x e ≥ 3 − e∈E\T \ f xe, e∈T which yields x e ≥ k − 1. e∈T Since f, g ∈ C ∩ (E \ T ), the circuit C with incidence vector x has |C| ≥ k + 1. But this contradicts our cardinality constraint. 2. k is odd and ∃g ∈ (C \ f ) ∩ (E \ T ) such that l gT ≥ k.