41,726 research outputs found
Two-stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms
We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP-hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max-flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max-flow structure of the subproblems
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
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
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Tree-width for first order formulae
We introduce tree-width for first order formulae \phi, fotw(\phi). We show
that computing fotw is fixed-parameter tractable with parameter fotw. Moreover,
we show that on classes of formulae of bounded fotw, model checking is fixed
parameter tractable, with parameter the length of the formula. This is done by
translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable
fragment L^k of first order logic. For fixed k, the question whether a given
first order formula is equivalent to an L^k formula is undecidable. In
contrast, the classes of first order formulae with bounded fotw are fragments
of first order logic for which the equivalence is decidable.
Our notion of tree-width generalises tree-width of conjunctive queries to
arbitrary formulae of first order logic by taking into account the quantifier
interaction in a formula. Moreover, it is more powerful than the notion of
elimination-width of quantified constraint formulae, defined by Chen and Dalmau
(CSL 2005): for quantified constraint formulae, both bounded elimination-width
and bounded fotw allow for model checking in polynomial time. We prove that
fotw of a quantified constraint formula \phi\ is bounded by the
elimination-width of \phi, and we exhibit a class of quantified constraint
formulae with bounded fotw, that has unbounded elimination-width. A similar
comparison holds for strict tree-width of non-recursive stratified datalog as
defined by Flum, Frick, and Grohe (JACM 49, 2002).
Finally, we show that fotw has a characterization in terms of a cops and
robbers game without monotonicity cost
Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs
Drawings of highly connected (dense) graphs can be very difficult to read.
Power Graph Analysis offers an alternate way to draw a graph in which sets of
nodes with common neighbours are shown grouped into modules. An edge connected
to the module then implies a connection to each member of the module. Thus, the
entire graph may be represented with much less clutter and without loss of
detail. A recent experimental study has shown that such lossless compression of
dense graphs makes it easier to follow paths. However, computing optimal power
graphs is difficult. In this paper, we show that computing the optimal
power-graph with only one module is NP-hard and therefore likely NP-hard in the
general case. We give an ILP model for power graph computation and discuss why
ILP and CP techniques are poorly suited to the problem. Instead, we are able to
find optimal solutions much more quickly using a custom search method. We also
show how to restrict this type of search to allow only limited back-tracking to
provide a heuristic that has better speed and better results than previously
known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of
the same nam
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