# Algorithms and Effectivity in Tropical Mathematics and Beyond

## Motivation

Tropical mathematics is a uniting name for different research directions which involve the semi-ring of real numbers endowed with the operations {min, +} (called the tropical semi-ring). It has emerged in several areas of computer science and of pure and applied mathematics. This seminar focusses on effective methods, algorithms and complexity bounds in tropical mathematics, and on their relations with open questions in various areas of computer science, including optimization, game theory and circuit complexity. During the seminar, we plan to study the following issues (the list is not exhaustive).

Tropical linear systems and mean payoff games. One of the oldest open algorithmic challenges in tropical mathematics is the complexity of solving systems of tropical linear equalities and inequalities. It is known to be equivalent to solving mean payoff games. The solvability of these problems is among the few known problems which are contained in the intersection NP ∩ co-NP, but not currently known to be in P. This leads to new approaches in linear programming or convex semialgebraic programming over nonarchimedean fields.

Tropical polynomial systems. Beyond tropical linear systems, much less was done towards tropical polynomial systems. Their solvability is NP-complete analogously to the solvability of classical polynomial equations. A tropical dual version of the weak Hilbert's Nullstellensatz was recently proved which reduces solvability of a tropical polynomial system to a tropical linear one.

Tropical bases and tropical symbolic computation. In computer algebra and symbolic computation, Gröbner bases provide an effective means to handle polynomials, with numerous applications such as formal reasoning, robotics, integer linear programming. Their tropical counterparts, tropical bases of tropical varieties, do exist and provide as well a cornerstone of tropical mathematics. They naturally occur when solving a (classical) polynomial system in Newton-Puiseux series, and there exists an algorithm for constructing a tropical basis in the constant-coefficient case. The seminar will discuss the state-of-the-art of tropical bases (complexity bounds, algorithmics).

Combinatorial complexity. The seminar will discuss combinatorial aspects of tropical systems. This includes the ranks of tropical matrices and their computation, tropical polyhedra, tropical separation problems, as well as face numbers and Betti numbers of systems of tropical systems.

Circuit complexity. Recently, new connections between (max, +) matrix multiplication and circuit complexity have come up. While conventional multiplication of two n x n-matrices can be done in time O(2.3729) it is not yet known whether tropical matrix multiplication can be achieved in subcubic time. Moreover, the problem of tropical matrix multiplication belongs to a large collection of problems (including the All-Pairs Shortest Path Problem APSP), which are either all solvable in subcubic time, or none of them ("APSP-hardness"). Using tools from circuit complexity, it has recently been shown that randomized methods allow for faster algorithms.

Tropical differential equations. The complexity issues of this recent topic will be discussed, together with basic problems of tropical differential algebraic geometry.