32,484 research outputs found
New Algorithms for Solving Tropical Linear Systems
The problem of solving tropical linear systems, a natural problem of tropical
mathematics, has already proven to be very interesting from the algorithmic
point of view: it is known to be in but no polynomial time
algorithm is known, although counterexamples for existing pseudopolynomial
algorithms are (and have to be) very complex.
In this work, we continue the study of algorithms for solving tropical linear
systems. First, we present a new reformulation of Grigoriev's algorithm that
brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us
formulate a whole family of new algorithms, and we present algorithms from this
family for which no known superpolynomial counterexamples work. Second, we
present a family of algorithms for solving overdetermined tropical systems. We
show that for weakly overdetermined systems, there are polynomial algorithms in
this family. We also present a concrete algorithm from this family that can
solve a tropical linear system defined by an matrix with maximal
element in time , and this time matches the complexity of the best of
previously known algorithms for feasibility testing.Comment: 17 page
Tropical differential equations
Tropical differential equations are introduced and an algorithm is designed
which tests solvability of a system of tropical linear differential equations
within the complexity polynomial in the size of the system and in its
coefficients. Moreover, we show that there exists a minimal solution, and the
algorithm constructs it (in case of solvability). This extends a similar
complexity bound established for tropical linear systems. In case of tropical
linear differential systems in one variable a polynomial complexity algorithm
for testing its solvability is designed.
We prove also that the problem of solvability of a system of tropical
non-linear differential equations in one variable is -hard, and this
problem for arbitrary number of variables belongs to . Similar to tropical
algebraic equations, a tropical differential equation expresses the (necessary)
condition on the dominant term in the issue of solvability of a differential
equation in power series
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
On a tropical dual Nullstellensatz
Since a tropical Nullstellensatz fails even for tropical univariate
polynomials we study a conjecture on a tropical {\it dual} Nullstellensatz for
tropical polynomial systems in terms of solvability of a tropical linear system
with the Cayley matrix associated to the tropical polynomial system. The
conjecture on a tropical effective dual Nullstellensatz is proved for tropical
univariate polynomials
Tropical geometries and dynamics of biochemical networks. Application to hybrid cell cycle models
We use the Litvinov-Maslov correspondence principle to reduce and hybridize
networks of biochemical reactions. We apply this method to a cell cycle
oscillator model. The reduced and hybridized model can be used as a hybrid
model for the cell cycle. We also propose a practical recipe for detecting
quasi-equilibrium QE reactions and quasi-steady state QSS species in
biochemical models with rational rate functions and use this recipe for model
reduction. Interestingly, the QE/QSS invariant manifold of the smooth model and
the reduced dynamics along this manifold can be put into correspondence to the
tropical variety of the hybridization and to sliding modes along this variety,
respectivelyComment: conference SASB 2011, to be published in Electronic Notes in
Theoretical Computer Scienc
On the frontiers of polynomial computations in tropical geometry
We study some basic algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension: deciding whether this intersection
is nonempty, whether it is a tropical variety, and whether it is connected, as
well as counting the number of connected components. We characterize the
borderline between tractable and hard computations by proving
-hardness and #-hardness results under various
strong restrictions of the input data, as well as providing polynomial time
algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio
The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average
We introduce an algorithm which solves mean payoff games in polynomial time
on average, assuming the distribution of the games satisfies a flip invariance
property on the set of actions associated with every state. The algorithm is a
tropical analogue of the shadow-vertex simplex algorithm, which solves mean
payoff games via linear feasibility problems over the tropical semiring
. The key ingredient in our approach is
that the shadow-vertex pivoting rule can be transferred to tropical polyhedra,
and that its computation reduces to optimal assignment problems through
Pl\"ucker relations.Comment: 17 pages, 7 figures, appears in 41st International Colloquium, ICALP
2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part
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