Related papers: The Open Algebraic Path Problem
The Longest Path Problem is a question of finding the maximum length between pairs of vertices of a graph. In the general case, the problem is NP-complete. However, there is a small collection of graph classes for which there exists an…
The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the…
The qubit routing problem, also known as the swap minimization problem, is a (classical) combinatorial optimization problem that arises in the design of compilers of quantum programs. We study the qubit routing problem from the viewpoint of…
Autonomous exploration is a complex task where the robot moves through an unknown environment with the goal of mapping it. The desired output of such a process is a sequence of paths that efficiently and safely minimise the uncertainty of…
This paper reviews the overview of the dynamic shortest path routing problem and the various neural networks to solve it. Different shortest path optimization problems can be solved by using various neural networks algorithms. The routing…
We study a group of new methods to solve an open problem that is the shortest paths problem on a given fix-weighted instance. It is the real significance at a considerable altitude to reach our aim to meet these qualities of generic,…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
In this paper we consider several problems concerning packet routing in distributed systems. Each problem is formulated using terms from Graph Theory and for each problem we present efficient, novel, algorithmic techniques for computing…
Axiomatizing covarieties of coalgebras for an endofunctor is less intuitive than axiomatizing varieties of algebras via equations (Dahlqvist and Schmid, 2022). Existing techniques come from coalgebraic modal logic, pattern avoidance…
Routing and scheduling problems are fundamental problems in combinatorial optimization, and also have many applications. Most variations of these problems are NP-Hard, so we need to use heuristics to solve these problems on large instances,…
In this paper we address several constrained transportation optimization problems (e.g. vehicle routing, shortest Hamiltonian path), for which we present novel algorithmic solutions and extensions, considering several optimization…
The shortest path problem in graphs is a cornerstone of AI theory and applications. Existing algorithms generally ignore edge weight computation time. We present a generalized framework for weighted directed graphs, where edge weight can be…
In this paper, we introduce a graph matching method that can account for constraints of arbitrary order, with arbitrary potential functions. Unlike previous decomposition approaches that rely on the graph structures, we introduce a…
We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the…
Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in…
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
Path calculus, or graphical linear algebra, is a string diagram calculus for the category of matrices over a base ring. It is the usual string diagram calculus for a symmetric monoidal category, where the monoidal product is the direct sum…
We propose a new approach to querying graph databases. Our approach balances competing goals of expressive power, language clarity and computational complexity. A distinctive feature of our approach is the ability to express properties of…
We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…