相关论文: On the frontiers of polynomial computations in tro…
A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values.…
Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the…
There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
This paper is in the continuity of a work program. Its goal is to develop an approach of the paralic confinement usable from the modeling slant, before implementing it in numerical tools. More specifically, we here deal with the multiscale…
We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface…
This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations,…
We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…
In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three…
In the PATH COVER problem, one asks to cover the vertices of a graph using the smallest possible number of (not necessarily disjoint) paths. While the variant where the paths need to be pairwise vertex-disjoint, which we call PATH…
This article is devoted to propose some lower and upper bounds for the coupled-tasks scheduling problem in presence of compatibility constraints according to classical complexity hypothesis ($\mathcal{P} \neq \mathcal{NP}$,…
We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show…
Counting the number of answers to conjunctive queries is a fundamental problem in databases that, under standard assumptions, does not have an efficient solution. The issue is inherently #P-hard, extending even to classes of acyclic…
In this paper we study the convergence of the max-consensus protocol. Tropical algebra is used to formulate the problem. Necessary and sufficient conditions for convergence of the max-consensus protocol over fixed as well as switching…
Although NP-Complete problems are the most difficult decisional problems, it is possible to discover in them polynomial (or easy) observables. We study the Graph Partitioning Problem showing that it is possible to recognize in it two…
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for…