The "Art of Trellis Decoding" Is Fixed-Parameter Tractable

Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a linear layout V 1 , V 2 ,..., V n of the subspaces such that dim((V 1 + V 2 + ··· + V i ) ∩ (V i+1 + ··· + V n )) ≤ k for all i; such a linear layout is said to have width at most k. When restricted...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 63; no. 11; pp. 7178 - 7205
Main Authors: Jeong, Jisu, Kim, Eun Jung, Oum, Sang-il
Format: Journal Article
Language:English
Published: New York IEEE 01.11.2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Institute of Electrical and Electronics Engineers
Subjects:
Algorithms
branch-width
Complexity theory
Computation
Computer Science
Decision theory
Decoding
Decomposition
Dynamic programming
Fields (mathematics)
Heuristic algorithms
Layout
Layouts
linear clique-width
linear code
Linear codes
linear rank-width
matroid
Parameterized complexity
Parameters
path-width
Subspaces
trellis state-complexity
trellis-width
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a linear layout V 1 , V 2 ,..., V n of the subspaces such that dim((V 1 + V 2 + ··· + V i ) ∩ (V i+1 + ··· + V n )) ≤ 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 trellis-width (or minimum trellis state-complexity) of a linear code in coding theory and computing the path-width of an F-represented matroid in matroid 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 F. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input 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 F, there are finitely many forbidden F-representable minors for the class of matroids of path-width at most k. An algorithm by Hlinený (2006) can detect a minor in an input F-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2017.2740283