Graphs and Algorithms Exercise 1 (real Matrices)
نویسندگان
چکیده
V := {ri}i=1 ∪ {ci}i=1 and E := {{ri, cj} | mi,j ≥ 1/n}. The main observation is that if G contains a perfect matching then there exists a desired permutation. To see that, let us assume that P = {e1, . . . , en} is a subset of the edges of G which form a perfect matching, with ei = {ri, cji}. Then setting π(i) := ji satisfies the desired properties: since P is a matching we have that π is a bijection and ei ∈ E(G) implies mi,ji ≥ 1/n. We show that G contains a saturating matching for {ri}i=1 using Hall’s theorem. As partitions are of the same size, this implies the existence of a desired perfect matching. Let S ⊆ {ri}i=1 be an arbitrary subset. Then ∑
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