Related papers: Circuits in Extended Formulations
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…
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…
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…
Let $P$ be a polytope. The hitting number of $P$ is the smallest size of a hitting set of the facets of $P$, i.e., a subset of vertices of $P$ such that every facet of $P$ has a vertex in the subset. An extended formulation of $P$ is the…
Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on 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…
This paper presents a general framework for linear circuit analysis based on elementary aspects of projective geometry. We use a flexible approach in which no a priori assignment of an electrical nature to the circuit branches is necessary.…
Probabilistic circuits are a unifying representation of functions as computation graphs of weighted sums and products. Their primary application is in probabilistic modeling, where circuits with non-negative weights (monotone circuits) can…
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…
Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for finding lower bounds on combinatorial diameters. The main open problem in this area is 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 =…
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…
We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of…
An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear…
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…
This paper studies extended formulations for radial cones at vertices of polyhedra, where the radial cone of a polyhedron $ P $ at a vertex $ v \in P $ is the polyhedron defined by the constraints of $ P $ that are active at $ v $. Given an…
We study properties and applications of various circuit imbalance measures associated with linear spaces. These measures describe possible ratios between nonzero entries of support-minimal nonzero vectors of the space. The fractional…
An extended formulation of a polyhedron $P$ is a linear description of a polyhedron $Q$ together with a linear map $\pi$ such that $\pi(Q)=P$. These objects are of fundamental importance in polyhedral combinatorics and optimization theory,…
Probabilistic circuits compute multilinear polynomials that represent multivariate probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in…
A binarization of a bounded variable $x$ is a linear formulation with variables $x$ and additional binary variables $y_1,\dots, y_k$, so that integrality of $x$ is implied by the integrality of $y_1,\dots, y_k$. A binary extended…