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- W163819639 abstract "Given linear representations M1 and M2 of matroids over a field F, we consider the problem (denoted by ECLR[F]), of checking if M1 and M2 represent the same matroid over F. We show that when F = Z2, ECLR[Z2] is complete for ⊕L under logspace many-one reductions. When F = Q, given linear representations M1,M2 ∈ Q m×n as input, any set of indexes X ⊆ {1,...,n} such that columns corresponding to these indexes in X are linearly dependent in one linear representation but are linearly independent in the other linear representation is a witness that M1 and M2 represent different matroids over Q. We show that the decision and the search version of this problem are polynomial time equivalent. We consider the CYCLE MATCHING problem of checking if for a pair of undirected graphs G1 = (V1,E1) and G2 = (V2,E2) given as input with n = |V1| = |V2|, we check if the set of vertices having indexes in X ⊆ {1,...,n} form a cycle in G1 if and only if the corresponding set of vertices having indexes in X ⊆ {1,...,n} form a cycle in G2 for all X ⊆ {1,...,n}. We show that CYCLE MATCHING is complete for L. Also the problem of counting the number of X ⊆ {1,...,n} such that vertices with indexes in X form a cycle in one of the input graphs but not in the other is shown to be #P-complete." @default.
- W163819639 created "2016-06-24" @default.
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- W163819639 date "2009-01-01" @default.
- W163819639 modified "2023-09-26" @default.
- W163819639 title "Checking Equality of Matroid Linear Representations and the Cycle Matching Problem" @default.
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