Edges versus circuits: a hierarchy of diameters in polyhedra

Discrete Applied Mathematics

Viss

2022

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 simplex method. However, there are significant challenges when implementing these approaches: The set of circuits of a polyhedron may be of exponential size and is highly sensitive to the representation of the polyhedron. In this paper, we provide a universal framework for enumerating the set of circuits and optimizing over sets of circuits of a polyhedron in any representation-we propose a polyhedral model in which the circuits of the original polyhedron are encoded as extreme rays or vertices. Many methods in the literature and software assume that a polyhedron is in standard form; our framework is a direct generalization. We demonstrate its value by showing that the conversion of a general representation to standard form may introduce exponentially many new circuits. We then discuss the main advantages of the generalized polyhedral model. It enables the direct enumeration of useful subsets of circuits such as strictly feasible circuits or sign-compatible circuits, as well as optimization over these sets. In particular, this leads to the efficient computation of a steepest-descent circuit, which can be used in an augmentation scheme for solving linear programs or the construction of sign-compatible circuit walks with additional properties.

Section: Sign-compatible Circuitsmentioning

confidence: 99%

Section: Constructing Sign-compatible Sums and Circuit Walksmentioning

confidence: 99%

Section: Constructing Sign-compatible Sums and Circuit Walksmentioning

confidence: 99%

See 1 more Smart Citation

A polyhedral model for enumeration and optimization over the set of circuits

Discrete Applied Mathematics

Viss

2022

“…For the maximum combinatorial diameter in this class of polyhedra, we use ∆ E (f, d). Additionally, we use the terms ∆ F (P ), and ∆ F (f, d) for the corresponding notions for the so-called feasible circuit walks [BLF16], where the walk does not have to take steps of maximal length, but is only required to stay feasible. To distinguish these walks from the original circuit walks, the original ones are sometimes called maximal circuit walks.…”

Section: Circuit Walksmentioning

confidence: 99%

On the Circuit Diameter Conjecture

Stephen

Yusun

2018

Discrete Comput Geom

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 bound of f − d was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron U4 with 8 facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos [San12]. We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the polyhedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and d-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equivalences in [KW67], the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra -in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the 4-step conjecture. These results offer some hope that the circuit version of the Hirsch conjecture may hold, even for unbounded polyhedra. A challenge in the circuit setting is that different realizations of polyhedra with the same combinatorial structure may have different diameters. We adapt the notion of simplicity to work with circuits in the form of C-simple and wedge-simple polyhedra. We show that it suffices to consider such polyhedra for studying circuit analogues of the Hirsch conjecture. MSC[2012]: 52B05, 90C05

“…In an attempt to understand the behavior of the graph diameter we introduced a hierarchy of distances and diameters for polyhedra that extend the usual edge walk [5,6]: Instead of only going along actual edges of the polyhedron, we walk along circuits, which are all potential edge directions of the polyhedron. This means in particular that we can possibly enter the interior of the polyhedron.…”

Section: Introductionmentioning

confidence: 99%

The hierarchy of circuit diameters and transportation polytopes

Discrete Applied Mathematics

Loera

Finhold

et al. 2018

Self Cite

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 conjecture is true for general m×n-transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. The key reason was that this hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarchy of diameters for the m×n-transportation polytopes. We show that the Hirsch conjecture bound of m + n − 1 is actually valid in most of these diameter notions. Second, we prove that for 3×n-transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for 2×n-transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter.