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- W2951145498 abstract "Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $mathbb F$, we wish to find a linear layout $V_1,V_2,ldots,V_n$ of the subspaces such that $dim(V_1+V_2+cdots+V_i)cap(V_{i+1}+cdots+V_n)le k$ for all $i$; such a linear layout is said to have width at most $k$. When restricted to $1$-dimensional subspaces, this problem is equivalent to computing the path-width of an $mathbb F$-represented matroid in matroid theory and computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $mathbb F$. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most $k$ for an input $mathbb F$-represented matroid of path-width at most $k$, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most $k$ for an input graph of linear rank-width at most $k$. In both corollaries, no such algorithms were known previously. Our approach is based on dynamic programming combined with the idea developed by Bodlaender and Kloks (1996) for their work on path-width and tree-width of graphs. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field $mathbb F$, there are finitely many forbidden $mathbb F$-representable minors for the class of matroids of path-width at most $k$. However, this indirect approach would not produce an actual path-decomposition even if the complete list of forbidden minors were known. Our algorithm is the first one to construct such a path-decomposition." @default.
- W2951145498 created "2019-06-27" @default.
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- W2951145498 date "2015-07-08" @default.
- W2951145498 modified "2023-09-25" @default.
- W2951145498 title "Constructive algorithm for path-width of matroids" @default.
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