Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Introduction to Coding Theory
Dynamic Programming on Graphs with Bounded Treewidth
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
A Parametrized Algorithm for Matroid Branch-Width
SIAM Journal on Computing
European Journal of Combinatorics - Special issue on Eurocomb'03 - graphs and combinatorial structures
The branchwidth of graphs and their cycle matroids
Journal of Combinatorial Theory Series B
Matroid Pathwidth and Code Trellis Complexity
SIAM Journal on Discrete Mathematics
Finding Branch-Decompositions and Rank-Decompositions
SIAM Journal on Computing
Addendum to matroid tree-width
European Journal of Combinatorics
Constraint complexity of realizations of linear codes on arbitrary graphs
IEEE Transactions on Information Theory
Convolutional codes over groups
IEEE Transactions on Information Theory - Part 1
The generalized distributive law
IEEE Transactions on Information Theory
Factor graphs and the sum-product algorithm
IEEE Transactions on Information Theory
Codes on graphs: normal realizations
IEEE Transactions on Information Theory
Codes on graphs: constraint complexity of cycle-free realizations of linear codes
IEEE Transactions on Information Theory
The structure of tail-biting trellises: minimality and basic principles
IEEE Transactions on Information Theory
A Decomposition Theory for Binary Linear Codes
IEEE Transactions on Information Theory
The Extraction and Complexity Limits of Graphical Models for Linear Codes
IEEE Transactions on Information Theory
Width Parameters Beyond Tree-width and their Applications
The Computer Journal
Constraint complexity of realizations of linear codes on arbitrary graphs
IEEE Transactions on Information Theory
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A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood (ML) decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations. It is known that among all tree realizations of a linear code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state-space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.