6,260 research outputs found
Obstructions to within a few vertices or edges of acyclic
Finite obstruction sets for lower ideals in the minor order are guaranteed to
exist by the Graph Minor Theorem. It has been known for several years that, in
principle, obstruction sets can be mechanically computed for most natural lower
ideals. In this paper, we describe a general-purpose method for finding
obstructions by using a bounded treewidth (or pathwidth) search. We illustrate
this approach by characterizing certain families of cycle-cover graphs based on
the two well-known problems: -{\sc Feedback Vertex Set} and -{\sc
Feedback Edge Set}. Our search is based on a number of algorithmic strategies
by which large constants can be mitigated, including a randomized strategy for
obtaining proofs of minimality.Comment: 16 page
Conjunctions of Among Constraints
Many existing global constraints can be encoded as a conjunction of among
constraints. An among constraint holds if the number of the variables in its
scope whose value belongs to a prespecified set, which we call its range, is
within some given bounds. It is known that domain filtering algorithms can
benefit from reasoning about the interaction of among constraints so that
values can be filtered out taking into consideration several among constraints
simultaneously. The present pa- per embarks into a systematic investigation on
the circumstances under which it is possible to obtain efficient and complete
domain filtering algorithms for conjunctions of among constraints. We start by
observing that restrictions on both the scope and the range of the among
constraints are necessary to obtain meaningful results. Then, we derive a
domain flow-based filtering algorithm and present several applications. In
particular, it is shown that the algorithm unifies and generalizes several
previous existing results.Comment: 15 pages plus appendi
New Results for the MAP Problem in Bayesian Networks
This paper presents new results for the (partial) maximum a posteriori (MAP)
problem in Bayesian networks, which is the problem of querying the most
probable state configuration of some of the network variables given evidence.
First, it is demonstrated that the problem remains hard even in networks with
very simple topology, such as binary polytrees and simple trees (including the
Naive Bayes structure). Such proofs extend previous complexity results for the
problem. Inapproximability results are also derived in the case of trees if the
number of states per variable is not bounded. Although the problem is shown to
be hard and inapproximable even in very simple scenarios, a new exact algorithm
is described that is empirically fast in networks of bounded treewidth and
bounded number of states per variable. The same algorithm is used as basis of a
Fully Polynomial Time Approximation Scheme for MAP under such assumptions.
Approximation schemes were generally thought to be impossible for this problem,
but we show otherwise for classes of networks that are important in practice.
The algorithms are extensively tested using some well-known networks as well as
random generated cases to show their effectiveness.Comment: A couple of typos were fixed, as well as the notation in part of
section 4, which was misleading. Theoretical and empirical results have not
change
Cutwidth: obstructions and algorithmic aspects
Cutwidth is one of the classic layout parameters for graphs. It measures how
well one can order the vertices of a graph in a linear manner, so that the
maximum number of edges between any prefix and its complement suffix is
minimized. As graphs of cutwidth at most are closed under taking
immersions, the results of Robertson and Seymour imply that there is a finite
list of minimal immersion obstructions for admitting a cut layout of width at
most . We prove that every minimal immersion obstruction for cutwidth at
most has size at most .
As an interesting algorithmic byproduct, we design a new fixed-parameter
algorithm for computing the cutwidth of a graph that runs in time , where is the optimum width and is the number of vertices.
While being slower by a -factor in the exponent than the fastest known
algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear
time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth
II: Algorithms for partial -trees of bounded degree, J. Algorithms,
56(1):25--49, 2005], our algorithm has the advantage of being simpler and
self-contained; arguably, it explains better the combinatorics of optimum-width
layouts
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