16 research outputs found
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
A Logical Product Approach to Zonotope Intersection
We define and study a new abstract domain which is a fine-grained combination
of zonotopes with polyhedric domains such as the interval, octagon, linear
templates or polyhedron domain. While abstract transfer functions are still
rather inexpensive and accurate even for interpreting non-linear computations,
we are able to also interpret tests (i.e. intersections) efficiently. This
fixes a known drawback of zonotopic methods, as used for reachability analysis
for hybrid sys- tems as well as for invariant generation in abstract
interpretation: intersection of zonotopes are not always zonotopes, and there
is not even a best zonotopic over-approximation of the intersection. We
describe some examples and an im- plementation of our method in the APRON
library, and discuss some further in- teresting combinations of zonotopes with
non-linear or non-convex domains such as quadratic templates and maxplus
polyhedra
The tropical analogue of polar cones
We study the max-plus or tropical analogue of the notion of polar: the polar
of a cone represents the set of linear inequalities satisfied by its elements.
We establish an analogue of the bipolar theorem, which characterizes all the
inequalities satisfied by the elements of a tropical convex cone. We derive
this characterization from a new separation theorem. We also establish variants
of these results concerning systems of linear equalities.Comment: 21 pages, 3 figures, example added, figures improved, notation
change
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
Reachability analysis for timed automata using max-plus algebra
International audienceWe show that max-plus polyhedra are usable as a data structure in reachability analysis of timed automata. Drawing inspiration from the extensive work that has been done on difference bound matrices, as well as previous work on max-plus polyhedra in other areas, we develop the algorithms needed to perform forward and backward reachability analysis using max-plus polyhedra. To show that the approach works in practice and theory alike, we have created a proof-of-concept implementation on top of the model checker opaal
Duality between invariant spaces for max-plus linear discrete event systems
We extend the notions of conditioned and controlled invariant spaces to
linear dynamical systems over the max-plus or tropical semiring. We establish a
duality theorem relating both notions, which we use to construct dynamic
observers. These are useful in situations in which some of the system
coefficients may vary within certain intervals. The results are illustrated by
an application to a manufacturing system.Comment: 22 pages, 3 figures (6 eps files
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
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update