19 research outputs found
On Minimal Tree Realizations of Linear Codes
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 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 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.Comment: Submitted to IEEE Transactions on Information Theory; 29 pages, 11
figure
Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs
A graphical realization of a linear code C consists of an assignment of the
coordinates of C to the vertices of a graph, along with a specification of
linear state spaces and linear ``local constraint'' codes to be associated with
the edges and vertices, respectively, of the graph. The \k-complexity of a
graphical realization is defined to be the largest dimension of any of its
local constraint codes. \k-complexity is a reasonable measure of the
computational complexity of a sum-product decoding algorithm specified by a
graphical realization. The main focus of this paper is on the following
problem: given a linear code C and a graph G, how small can the \k-complexity
of a realization of C on G be? As useful tools for attacking this problem, we
introduce the Vertex-Cut Bound, and the notion of ``vc-treewidth'' for a graph,
which is closely related to the well-known graph-theoretic notion of treewidth.
Using these tools, we derive tight lower bounds on the \k-complexity of any
realization of C on G. Our bounds enable us to conclude that good
error-correcting codes can have low-complexity realizations only on graphs with
large vc-treewidth. Along the way, we also prove the interesting result that
the ratio of the \k-complexity of the best conventional trellis realization
of a length-n code C to the \k-complexity of the best cycle-free realization
of C grows at most logarithmically with codelength n. Such a logarithmic growth
rate is, in fact, achievable.Comment: Submitted to IEEE Transactions on Information Theor
An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and
Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B,
96(4):514--528, 2006]. Motivated from recent development on graph modification
problems regarding classes of graphs of bounded treewidth or pathwidth, we
study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex
Deletion). In the LRW1-Vertex Deletion problem, given an -vertex graph
and a positive integer , we want to decide whether there is a set of at most
vertices whose removal turns into a graph of linear rankwidth at most
and find such a vertex set if one exists. While the meta-theorem of
Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved
in time for some function , it is not clear whether this
problem allows a running time with a modest exponential function.
We first establish that LRW1-Vertex Deletion can be solved in time . The major obstacle to this end is how to handle a long
induced cycle as an obstruction. To fix this issue, we define necklace graphs
and investigate their structural properties. Later, we reduce the polynomial
factor by refining the trivial branching step based on a cliquewidth expression
of a graph, and obtain an algorithm that runs in time . We also prove that the running time cannot be improved to under the Exponential Time Hypothesis assumption. Lastly,
we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201
Pathwidth vs cocircumference
The {\em circumference} of a graph with at least one cycle is the length
of a longest cycle in . A classic result of Birmel\'e (2003) states that the
treewidth of is at most its circumference minus . In case is
-connected, this upper bound also holds for the pathwidth of ; in fact,
even the treedepth of is upper bounded by its circumference (Bria\'nski,
Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether
similar bounds hold when replacing the circumference of by its {\em
cocircumference}, defined as the largest size of a {\em bond} in , an
inclusion-wise minimal set of edges such that has more components
than . In matroidal terms, the cocircumference of is the circumference
of the bond matroid of .
Our first result is the following `dual' version of Birmel\'e's theorem: The
treewidth of a graph is at most its cocircumference. Our second and main
result is an upper bound of on the pathwidth of a -connected graph
with cocircumference . Contrary to circumference, no such bound holds
for the treedepth of . Our two upper bounds are best possible up to a
constant factor.Comment: v2: revised following the referees' comment
Branchwidth is (1,g)-self-dual
A graph parameter is self-dual in some class of graphs embeddable in some
surface if its value does not change in the dual graph by more than a constant
factor. We prove that the branchwidth of connected hypergraphs without bridges
and loops that are embeddable in some surface of Euler genus at most g is an
(1,g)-self-dual parameter. This is the first proof that branchwidth is an
additively self-dual width parameter.Comment: 10 page
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
On the Optimality of Pseudo-polynomial Algorithms for Integer Programming
In the classic Integer Programming (IP) problem, the objective is to decide
whether, for a given matrix and an -vector , there is a non-negative integer -vector such that . Solving
(IP) is an important step in numerous algorithms and it is important to obtain
an understanding of the precise complexity of this problem as a function of
natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981]
for instances of (IP) with a constant number of constraints was only recently
improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder
[ArXiv 2018]. We continue this line of work and show that under the Exponential
Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal.
We also show that when the matrix is assumed to be non-negative, a
component of Papadimitriou's original algorithm is already nearly optimal under
ETH.
This motivates us to pick up the line of research initiated by Cunningham and
Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative
matrices in which the number of constraints may be unbounded, but the
branch-width of the column-matroid corresponding to the constraint matrix is a
constant. We prove a lower bound on the complexity of solving (IP) for such
instances and obtain optimal results with respect to a closely related
parameter, path-width. Specifically, we prove matching upper and lower bounds
for (IP) when the path-width of the corresponding column-matroid is a constant.Comment: 29 pages, To appear in ESA 201