2,213 research outputs found
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
On the complexity of strongly connected components in directed hypergraphs
We study the complexity of some algorithmic problems on directed hypergraphs
and their strongly connected components (SCCs). The main contribution is an
almost linear time algorithm computing the terminal strongly connected
components (i.e. SCCs which do not reach any components but themselves).
"Almost linear" here means that the complexity of the algorithm is linear in
the size of the hypergraph up to a factor alpha(n), where alpha is the inverse
of Ackermann function, and n is the number of vertices. Our motivation to study
this problem arises from a recent application of directed hypergraphs to
computational tropical geometry.
We also discuss the problem of computing all SCCs. We establish a superlinear
lower bound on the size of the transitive reduction of the reachability
relation in directed hypergraphs, showing that it is combinatorially more
complex than in directed graphs. Besides, we prove a linear time reduction from
the well-studied problem of finding all minimal sets among a given family to
the problem of computing the SCCs. Only subquadratic time algorithms are known
for the former problem. These results strongly suggest that the problem of
computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure
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
Minimal external representations of tropical polyhedra
Tropical polyhedra are known to be representable externally, as intersections
of finitely many tropical half-spaces. However, unlike in the classical case,
the extreme rays of their polar cones provide external representations
containing in general superfluous half-spaces. In this paper, we prove that any
tropical polyhedral cone in R^n (also known as "tropical polytope" in the
literature) admits an essentially unique minimal external representation. The
result is obtained by establishing a (partial) anti-exchange property of
half-spaces. Moreover, we show that the apices of the half-spaces appearing in
such non-redundant external representations are vertices of the cell complex
associated with the polyhedral cone. We also establish a necessary condition
for a vertex of this cell complex to be the apex of a non-redundant half-space.
It is shown that this condition is sufficient for a dense class of polyhedral
cones having "generic extremities".Comment: v1: 32 pages, 10 figures; v2: minor revision, 34 pages, 10 figure
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