557 research outputs found
The number of extreme points of tropical polyhedra
The celebrated upper bound theorem of McMullen determines the maximal number
of extreme points of a polyhedron in terms of its dimension and the number of
constraints which define it, showing that the maximum is attained by the polar
of the cyclic polytope. We show that the same bound is valid in the tropical
setting, up to a trivial modification. Then, we study the natural candidates to
be the maximizing polyhedra, which are the polars of a family of cyclic
polytopes equipped with a sign pattern. We construct bijections between the
extreme points of these polars and lattice paths depending on the sign pattern,
from which we deduce explicit bounds for the number of extreme points, showing
in particular that the upper bound is asymptotically tight as the dimension
tends to infinity, keeping the number of constraints fixed. When transposed to
the classical case, the previous constructions yield some lattice path
generalizations of Gale's evenness criterion.Comment: 26 pages, 4 figures, 1 tabl
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
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
Tropical Fourier-Motzkin elimination, with an application to real-time verification
We introduce a generalization of tropical polyhedra able to express both
strict and non-strict inequalities. Such inequalities are handled by means of a
semiring of germs (encoding infinitesimal perturbations). We develop a tropical
analogue of Fourier-Motzkin elimination from which we derive geometrical
properties of these polyhedra. In particular, we show that they coincide with
the tropically convex union of (non-necessarily closed) cells that are convex
both classically and tropically. We also prove that the redundant inequalities
produced when performing successive elimination steps can be dynamically
deleted by reduction to mean payoff game problems. As a complement, we provide
a coarser (polynomial time) deletion procedure which is enough to arrive at a
simply exponential bound for the total execution time. These algorithms are
illustrated by an application to real-time systems (reachability analysis of
timed automata).Comment: 29 pages, 8 figure
Minimal half-spaces and external representation of tropical polyhedra
We give a characterization of the minimal tropical half-spaces containing a
given tropical polyhedron, from which we derive a counter example showing that
the number of such minimal half-spaces can be infinite, contradicting some
statements which appeared in the tropical literature, and disproving a
conjecture of F. Block and J. Yu. We also establish an analogue of the
Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently
represented internally (in terms of extreme points and rays) or externally (in
terms of half-spaces containing it). A canonical external representation of a
polyhedron turns out to be provided by the extreme elements of its tropical
polar. We characterize these extreme elements, showing in particular that they
are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures
improved, references update
Tropical analogues of a Dempe-Franke bilevel optimization problem
We consider the tropical analogues of a particular bilevel optimization
problem studied by Dempe and Franke and suggest some methods of solving these
new tropical bilevel optimization problems. In particular, it is found that the
algorithm developed by Dempe and Franke can be formulated and its validity can
be proved in a more general setting, which includes the tropical bilevel
optimization problems in question. We also show how the feasible set can be
decomposed into a finite number of tropical polyhedra, to which the tropical
linear programming solvers can be applied.Comment: 11 pages, 1 figur
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm
Combinatorial simplex algorithms can solve mean payoff games
A combinatorial simplex algorithm is an instance of the simplex method in
which the pivoting depends on combinatorial data only. We show that any
algorithm of this kind admits a tropical analogue which can be used to solve
mean payoff games. Moreover, any combinatorial simplex algorithm with a
strongly polynomial complexity (the existence of such an algorithm is open)
would provide in this way a strongly polynomial algorithm solving mean payoff
games. Mean payoff games are known to be in NP and co-NP; whether they can be
solved in polynomial time is an open problem. Our algorithm relies on a
tropical implementation of the simplex method over a real closed field of Hahn
series. One of the key ingredients is a new scheme for symbolic perturbation
which allows us to lift an arbitrary mean payoff game instance into a
non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction
expanded, 18 pages, 3 figure
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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