624 research outputs found
Classification of eight dimensional perfect forms
In this paper, we classify the perfect lattices in dimension 8. There are
10916 of them. Our classification heavily relies on exploiting symmetry in
polyhedral computations. Here we describe algorithms making the classification
possible.Comment: 14 page
Polyhedral Combinatorics of UPGMA Cones
Distance-based methods such as UPGMA (Unweighted Pair Group Method with
Arithmetic Mean) continue to play a significant role in phylogenetic research.
We use polyhedral combinatorics to analyze the natural subdivision of the
positive orthant induced by classifying the input vectors according to tree
topologies returned by the algorithm. The partition lattice informs the study
of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n
taxa, and compute the normalized volumes of the UPGMA cones for small n.
Keywords: phylogenetic trees, polyhedral combinatorics, partition lattic
The tropical double description method
We develop a tropical analogue of the classical double description method
allowing one to compute an internal representation (in terms of vertices) of a
polyhedron defined externally (by inequalities). The heart of the tropical
algorithm is a characterization of the extreme points of a polyhedron in terms
of a system of constraints which define it. We show that checking the
extremality of a point reduces to checking whether there is only one minimal
strongly connected component in an hypergraph. The latter problem can be solved
in almost linear time, which allows us to eliminate quickly redundant
generators. We report extensive tests (including benchmarks from an application
to static analysis) showing that the method outperforms experimentally the
previous ones by orders of magnitude. The present tools also lead to worst case
bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on
Theoretical Aspects of Computer Science, 2010, Nancy, Franc
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
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