10 research outputs found
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
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
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
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
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
The level set method for the two-sided eigenproblem
We consider the max-plus analogue of the eigenproblem for matrix pencils
Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible
values of lambda), which is a finite union of intervals, can be computed in
pseudo-polynomial number of operations, by a (pseudo-polynomial) number of
calls to an oracle that computes the value of a mean payoff game. The proof
relies on the introduction of a spectral function, which we interpret in terms
of the least Chebyshev distance between Ax and lambda Bx. The spectrum is
obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we
explain relation to mean-payoff games and discrete event systems, and show
that the reconstruction of spectrum is pseudopolynomia
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
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum