Related papers: How to correctly prune tropical trees
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…
Strategic-form min-max game theory examines the existence, multiplicity, selection of equilibria, and the worst-case computational complexity under perfect rationality. However, in many applications, games are drawn from an ensemble, and…
Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state…
Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to…
We prove the first, even super-polynomial, lower bounds on the size of tropical (min,+) and (max,+) circuits approximating given optimization problems. Many classical dynamic programming (DP) algorithms for optimization problems are pure in…
We study a tropical linear regression problem consisting in finding the best approximation of a set of points by a tropical hyperplane. We establish a strong duality theorem, showing that the value of this problem coincides with the maximal…
We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors over an idempotent semifield,…
A tropical (or min-plus) semiring is a set $\mathbb{Z}$ (or $\mathbb{Z \cup \{\infty\}}$) endowed with two operations: $\oplus$, which is just usual minimum, and $\odot$, which is usual addition. In tropical algebra the vector $x$ is a…
The tropical semiring is an algebraic system with addition ``$\max$'' and multiplication ``$+$''. As well as in conventional algebra, linear programming in the tropical semiring has been developed. In this study, we introduce a new type of…
In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…
Optimization problems are considered in the framework of tropical algebra to minimize and maximize a nonlinear objective function defined on vectors over an idempotent semifield, and calculated using multiplicative conjugate transposition.…
The main goal of this paper is to describe a new pruning method for solving decision trees and game trees. The pruning method for decision trees suggests a slight variant of decision trees that we call scenario trees. In scenario trees, we…
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…
We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…
We revisit classic algorithmic search and optimization problems from the perspective of competition. Rather than a single optimizer minimizing expected cost, we consider a zero-sum game in which an optimization problem is presented to two…
Knuth and Moore presented a theoretical lower bound on the number of leaves that any fixed-depth minimax tree-search algorithm traversing a uniform tree must explore, the so-called minimal tree. Since real-life minimax trees are not…
We consider a deterministic game with alternate moves and complete information, of which the issue is always the victory of one of the two opponents. We assume that this game is the realization of a random model enjoying some independence…
The minmax regret problem for combinatorial optimization under uncertainty can be viewed as a zero-sum game played between an optimizing player and an adversary, where the optimizing player selects a solution and the adversary selects costs…
The connection between game theory, convex optimization, and geometry is deep. There are many applications of linear programming methods and polyhedral representation conversion methods in game theory. In this paper, we discuss two more…
Decoding strategies play a pivotal role in text generation for modern language models, yet a puzzling gap divides theory and practice. Surprisingly, strategies that should intuitively be optimal, such as Maximum a Posteriori (MAP), often…