Related papers: On the Circuit Diameter of some Combinatorial Poly…
The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of…
The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical"…
The circuit diameter of a polyhedron is the maximum length (number of steps) of a shortest circuit walk between any two vertices of the polyhedron. Introduced by Borgwardt, Finhold and Hemmecke (SIDMA 2015), it is a relaxation of the…
The Circuit diameter of polytopes was introduced by Borgwardt, Finhold and Hemmecke as a fundamental tool for the study of circuit augmentation schemes for linear programming and for estimating combinatorial diameters. Determining the…
We prove a strongly polynomial bound on the circuit diameter of polyhedra, resolving the circuit analogue of the polynomial Hirsch conjecture. Specifically, we show that the circuit diameter of a polyhedron $P = \{x\in \mathbb{R}^n:\, A x =…
The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of…
In this paper we introduce the circuit diameter of polyhedra, which is always bounded from above by the combinatorial diameter. We consider dual transportation polyhedra defined on general bipartite graphs. For complete $M{\times}N$…
It has been shown that the edge structure of the characteristic imset polytope is closely connected to the question of causal discovery. The diameter of a polytope is an indicator of how connected the polytope is and moreover gives us a…
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…
Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective…
Both the combinatorial and the circuit diameters of polyhedra are of interest to the theory of linear programming for their intimate connection to a best-case performance of linear programming algorithms. We study the diameters of dual…
A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a…
Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called circuits, that is a superset of the edges of a polytope. We show that in the…
Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to…
The polygon $P$ is small if its diameter equals one. When $n=2^s$, it is still an open problem to find the maximum perimeter or the maximum width of a small $n$-gon. Motivated by Bingane's series of works, we improve the lower bounds for…
The diameter of a polytope is a fundamental geometric parameter that plays a crucial role in understanding the efficiency of the simplex method. Despite its central nature, the computational complexity of computing the diameter of a given…
The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the…
Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space…
The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its…
The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch…