11 research outputs found
On Codes of Bounded Trellis Complexity
In this paper, we initiate a structure theory of linear codes with bounded trellis complexity. The theory is based on the observation that the family of linear codes over Fq, some permutation of which has trellis state-complexity at most w, is a minor-closed family. It then follows from a deep result of matroid theory that such codes are characterized by finitely many excluded minors. We provide the complete list of excluded minors for w = 1, and give a partial list for w = 2. We also give a polynomial-time algorithm for determining whether or nor a given code has a permutation with state-complexity at most 1
Finding branch-decompositions of matroids, hypergraphs, and more
Given subspaces of a finite-dimensional vector space over a fixed finite
field , we wish to find a "branch-decomposition" of these subspaces
of width at most , that is a subcubic tree with leaves mapped
bijectively to the subspaces such that for every edge of , the sum of
subspaces associated with leaves in one component of and the sum of
subspaces associated with leaves in the other component have the intersection
of dimension at most . This problem includes the problems of computing
branch-width of -represented matroids, rank-width of graphs,
branch-width of hypergraphs, and carving-width of graphs.
We present a fixed-parameter algorithm to construct such a
branch-decomposition of width at most , if it exists, for input subspaces of
a finite-dimensional vector space over . Our algorithm is analogous
to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To
extend their framework to branch-decompositions of vector spaces, we developed
highly generic tools for branch-decompositions on vector spaces. The only known
previous fixed-parameter algorithm for branch-width of -represented
matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time
where is the number of elements of the input -represented
matroid. But their method is highly indirect. Their algorithm uses the
non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is
finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic
second-order formulas on -represented matroids of small
branch-width. Our result does not depend on such a fact and is completely
self-contained, and yet matches their asymptotic running time for each fixed
.Comment: 73 pages, 10 figure
Typical Sequences Revisited ā Computing Width Parameters of Graphs
In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in polynomial time
The Telecommunications and Data Acquisition Report
This quarterly publication provides archival reports on developments in programs managed by JPL's Telecommunications and Mission Operations Directorate (TMOD), which now includes the former Telecommunications and Data Acquisition (TDA) Office. In space communications, radio navigation, radio science, and ground-based radio and radar astronomy, it reports on activities of the Deep Space Network (DSN) in planning, supporting research and technology, implementation, and operations. Also included are standards activity at JPL for space data and information systems and reimbursable DSN work performed for other space agencies through NASA. The preceding work is all performed for NASA's Office of Space Communications (OSC)
The "Art of Trellis Decoding" is NP-Hard
Given a linear code C, the fundamental problem of trellis decoding is to find a coordinate permutation of C that yields a code C ā² whose minimal trellis has the least state-complexity among all codes obtainable by permuting the coordinates of C. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of finding such a coordinate permutation is NP-hard, thus settling a long-standing conjecture