1,262 research outputs found
Characterizations of Decomposable Dependency Models
Decomposable dependency models possess a number of interesting and useful
properties. This paper presents new characterizations of decomposable models in
terms of independence relationships, which are obtained by adding a single
axiom to the well-known set characterizing dependency models that are
isomorphic to undirected graphs. We also briefly discuss a potential
application of our results to the problem of learning graphical models from
data.Comment: See http://www.jair.org/ for any accompanying file
Potential Maximal Clique Algorithms for Perfect Phylogeny Problems
Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard
combinatorial optimization problems related to minimal triangulations, are
broken into subproblems by block subgraphs defined by minimal separators. These
ideas were expanded on by Bouchitt\'e and Todinca, who used potential maximal
cliques to solve these problems using a dynamic programming approach in time
polynomial in the number of minimal separators of a graph. It is known that
solutions to the perfect phylogeny problem, maximum compatibility problem, and
unique perfect phylogeny problem are characterized by minimal triangulations of
the partition intersection graph. In this paper, we show that techniques
similar to those proposed by Bouchitt\'e and Todinca can be used to solve the
perfect phylogeny problem with missing data, the two- state maximum
compatibility problem with missing data, and the unique perfect phylogeny
problem with missing data in time polynomial in the number of minimal
separators of the partition intersection graph
Relating threshold tolerance graphs to other graph classes
A graph G=(V, E) is a threshold tolerance if it is possible to associate weights and tolerances with each node of G so that two nodes are adjacent exactly when the sum of their weights exceeds either one of their tolerances. Threshold tolerance graphs are a special case of the well-known class of tolerance graphs and generalize the class of threshold graphs which are also extensively studied in literature. In this note we relate the threshold tolerance graphs with other important graph classes. In particular we show that threshold tolerance graphs are a proper subclass of co-strongly chordal graphs and strictly include the class of co-interval graphs. To this purpose, we exploit the relation with another graph class, min leaf power graphs (mLPGs)
On strongly chordal graphs that are not leaf powers
A common task in phylogenetics is to find an evolutionary tree representing
proximity relationships between species. This motivates the notion of leaf
powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V
and a threshold k such that uv is an edge if and only if the distance between u
and v in T is at most k. Characterizing leaf powers is a challenging open
problem, along with determining the complexity of their recognition. This is in
part due to the fact that few graphs are known to not be leaf powers, as such
graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf
powers could be characterized by strong chordality and a finite set of
forbidden subgraphs.
In this paper, we provide a negative answer to this question, by exhibiting
an infinite family \G of (minimal) strongly chordal graphs that are not leaf
powers. During the process, we establish a connection between leaf powers,
alternating cycles and quartet compatibility. We also show that deciding if a
chordal graph is \G-free is NP-complete, which may provide insight on the
complexity of the leaf power recognition problem
On Sharing, Memoization, and Polynomial Time (Long Version)
We study how the adoption of an evaluation mechanism with sharing and
memoization impacts the class of functions which can be computed in polynomial
time. We first show how a natural cost model in which lookup for an already
computed value has no cost is indeed invariant. As a corollary, we then prove
that the most general notion of ramified recurrence is sound for polynomial
time, this way settling an open problem in implicit computational complexity
On Patchworks and Hierarchies
Motivated by questions in biological classification, we discuss some
elementary combinatorial and computational properties of certain set systems
that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample
patchworks' and 'saturated patchworks' and also outline how these concepts
relate to an apparently new 'duality theory' for cluster systems that is based
on the fundamental concept of 'compatibility' of clusters.Comment: 17 pages, 2 figure
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