Related papers: On Circuit Diameter and Straight Line Complexity
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 =…
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…
Circuit augmentation schemes are a family of combinatorial algorithms for linear programming that generalize the simplex method. To solve the linear program, they construct a so-called monotone circuit walk: They start at an initial vertex…
The combinatorial diameter of a polytope $P$ is the maximum value of a shortest path between two vertices of $P$, where the path uses the edges of $P$ only. In contrast to the combinatorial diameter, the circuit diameter of $P$ is defined…
Circuits play a fundamental role in polyhedral theory and linear programming. For instance, circuits are used as step directions in various augmentation schemes for solving linear programs or to leave degenerate vertices while running the…
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…
The circuits of a polyhedron are a superset of its edge directions. Circuit walks, a sequence of steps along circuits, generalize edge walks and are "short" if they have few steps or small total length. Both interpretations of short are…
Circuits and extended formulations are classical concepts in linear programming theory. The circuits of a polyhedron are the elementary difference vectors between feasible points and include all edge directions. We study the connection…
We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq…
We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a…
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…
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 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…
Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method. We are interested in better understanding the…
Generalizing the simplex method, circuit augmentation schemes for linear programs follow circuit directions through the interior of the underlying polyhedron. Steepest-descent augmentation is especially promising, but an implementation of…
From the point of view of optimization, a critical issue is relating the combinatorial diameter of a polyhedron to its number of facets $f$ and dimension $d$. In the seminal paper of Klee and Walkup [KW67], the Hirsch conjecture of an upper…
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…
Circuits are fundamental objects in linear programming and oriented matroid theory, representing the elementary difference vectors of a polyhedron between points in its affine space. A recent concept introduced by Ekbatani, Natura, and…
Set Disjointness on a Line is a variant of the Set Disjointness problem in a distributed computing scenario with $d+1$ processors arranged on a path of length $d$. It was introduced by Le Gall and Magniez (PODC 2018) for proving lower…
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…