Abstract
Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in...
Highlights
We consider the fundamental problem of covering a subspace of a given finite dimensional linear space by a set of vectors of minimum size
The Minimum Distance problem in coding theory asks for a minimum dependent set of columns in a matrix over GF(2)
This problem can be reduced to Space Cover by finding for each column t in matrix M a minimum set of columns in the remaining part of the matrix that cover T = {t}
Summary
We consider the fundamental problem of covering a subspace of a given finite dimensional linear space (vector space) by a set of vectors of minimum size. The dual of Space Cover, i.e., the variant of Space Cover asking whether there is a set F ⊆ E \ T with w(F ) ≤ k such that T ⊆ span(F ) in the dual matroid M ∗, is equivalent to the Restricted Subset Feedback Set problem In this problem the task is for a given matroid M , a weight function w : E → N0, and a nonnegative integer k, to decide whether there is a set F ⊆ E \ T with w(F ) ≤ k such that matroid M obtained from M by deleting the elements of F has no circuit containing an element of T. While Steiner Tree is FPT parameterized by the number of terminal vertices, the hardness results for Multiway Cut with three terminals yields that Space Cover parameterized by the size of the terminal set T is Para-NP-complete even if restricted to cographic matroids. This occurs when the algorithm processes cographic matroids “glued” with other matroids and for that part of the algorithm the transformation of the decomposition is essential
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