Related papers: New Perspectives on Semiring Applications to Dynam…
Dynamic programming (DP) is an algorithmic design paradigm for the efficient, exact solution of otherwise intractable, combinatorial problems. However, DP algorithm design is often presented in an ad-hoc manner. It is sometimes difficult to…
This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are…
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…
To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration…
Arithmetic circuit complexity studies the complexity of computing polynomials using only arithmetic operations such as addition, multiplication, subtraction, and division. Polynomials over rings of integers model counting problems.…
The semiring of discrete dynamical systems is a simple algebraic model for modularity in deterministic systems. The objects of the semiring are finite transformations (viewed as directed graphs and regarded up to isomorphism), the sum of…
Tropical algebra, including max-plus, min-plus, and related idempotent semirings, provides a unifying framework in which many optimization problems that are nonlinear in classical algebra become linear. This property makes tropical methods…
We propose a general method for combinatorial online learning problems whose offline optimization problem can be solved efficiently via a dynamic programming algorithm defined by an arbitrary min-sum recurrence. Examples include online…
We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible…
It is well-known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show…
Constraint programming uses enumeration and search tree pruning to solve combinatorial optimization problems. In order to speed up this solution process, we investigate the use of semidefinite relaxations within constraint programming. In…
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,…
Classical shortest-path methods rely on binary tropical semirings $(\min,+)$, whose dyadic structure limits them to pairwise cost interactions. However, many real-world systems, including logistics, supply chains, communication networks,…
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…
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.…
We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis…
We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…
The field of probabilistic logic programming (PLP) focuses on integrating probabilistic models into programming languages based on logic. Over the past 30 years, numerous languages and frameworks have been developed for modeling, inference…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
We consider a broad class of dynamic programming (DP) problems that involve a partially linear structure and some positivity properties in their system equation and cost function. We address deterministic and stochastic problems, possibly…