We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization problem which are not polynomial time in the degree, but are polynomial time for polynomials of fixed degree. For two-variable polynomials we derive an irreducibility criterion which is almost always satisfied, even for fixed degree, and is polynomial time in the degree. We prove there are unique least common multiples of tropical polynomials, but not unique greatest common divisors. We show that if two polynomials in one variable have a common tropical factor, then their eliminant matrix is singular in the tropical sense. We prove the problem of determining tropical rank is NP-hard.
The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.
This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.
We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinator…
Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lov{á}sz and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-wi…
The purpose of this paper is to establish analogues of the classical Lefschetz Section Theorem for smooth tropical varieties. More precisely, we prove tropical analogues of the section theorems of Lefschetz, Andreotti-Frankel, Bott-Milnor-Thom, Hamm-Lê and Kodaira-Spencer, and the vanishing theorems of Andreotti-Frankel and Akizuki-Kodaira-Nakano. We start the paper by resolving a conjecture of Mikhalkin and Ziegler (2008) concerning the homotopy types of certain filtrations of geometric lattices, generalizing several known properties of full geometric lattices. This translates to a crucial in…
We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are enriched with an information of multiplicity, sign, or argument. We obtain existence or uniqueness results, which extend or refine earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss-Seidel type algorithms to solve linear systems in suitably extended tropical sem…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of tropical regularity and invertibility.
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables. Special families of these polynomials have appeared in economics, discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to many open problems of interest in discrete geometry.
The tropical arithmetic operations on $\mathbb{R}$ are defined by $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. Let $A$ be a tropical matrix and $k$ a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices $B\in\mathbb{R}^{m\times k}$ and $C\in\mathbb{R}^{k\times n}$ satisfying $B\otimes C=A$. We show that no algorithm for TMF is likely to work in polynomial time for every fixed $k$, thus resolving a problem proposed by Barvinok in 1993.
Matrix factorization problems over various semirings naturally arise in different contexts of modern pure and applied mathematics. These problems are very hard in general and cause computational difficulties in applications. We give a survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and positive semidefinite factorizations, and we examine the behavior of the corresponding rank functions on matrices of bounded bandwidth. We show that the Boolean, fuzzy, and tropical versions of matrix factorization become polynomial time solvable when restricted to…
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of polynomials, description of varieties, properties of the resultant, and for mathematical analysis and calculus, in particular with respect to multiple roots of polynomials. Explicit examples and comparisons are given for various sorting semirings such as the natural numbers and the positive rational numbers, and we see how this theory relates to some recent devel…
Adapting Lindström's well-known construction, we consider a wide class of functions which are generated by flows in a planar acyclic directed graph whose vertices (or edges) take weights in an arbitrary commutative semiring. We give a combinatorial description for the set of "universal" quadratic relations valid for such functions. Their specializations to particular semirings involve plenty of known quadratic relations for minors of matrices (e.g., Plücker relations) and the tropical counterparts of such relations. Also some applications and related topics are discussed.
It is well known, due to Lindström, that the minors of a (real or complex) matrix can be expressed in terms of weights of flows in a planar directed graph. Another classical fact is that there are plenty of homogeneous quadratic relations involving flag minors, or Plücker coordinates of the corresponding flag manifold. Generalizing and unifying these facts and their tropical counterparts, we consider a wide class of functions on $2^{[n]}$ that are generated by flows in a planar graph and take values in an arbitrary commutative semiring, where $[n]=\{1,2,\ldots,n\}$. We show that the ``universa…
D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, together with a sufficient condition for polynomial complexity that is connected to the saturated Newton polytope property.
This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed.
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.
Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the Minkowski factorization of polytopes. Using our construction we show that for any given polytopal fan there is a polytope factorization basis, i.e. a finite set of polytopes with respect to which any polytope whose normal fan is refined by the original fan can be uniquely written as a signed Minkowski sum. We explicitly study the factorization of polymatroids a…
We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.
Tropical geometry and its applications indicate a "theory of syzygies" over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Gröbner basis that captures the image of an ideal under the Newton polytope map. The image ${\rm New}(I)$ of a graded ideal $I$ under the Newton polytope is a graded sub-semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of ${\rm N…
Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We show that paving tropical ideals of degree d+1 are in bijection with $\mathbb Z^n$-invariant d-partitions of $\mathbb Z^n$. This implies that zer…
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 b…
\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is e…
