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

Abstract

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.

A note on the approximability of deepest-descent circuit steps

“…Secondly, we make use of [5] for our positive results on OCNP and the n-approximability of dd-SP. Most importantly, the set of circuit directions appear as a subset of the extreme rays of a polyhedral cone constructed from the original input [5, Theorem 3].…”

“…Apart from the directly related papers mentioned in the previous subsection, there is vast literature revolving around pivoting rules for circuit augmentation algorithms, and circuits of linear programs in general. Without any pretense of being comprehensive, let us point to a couple of seminal works (below) and refer to [5] with respect to circuits, and to [8] for circuit augmentation and the references therein for a more extensive treatment.…”

“…Recall that the circuit directions of a polyhedron P appear as a subset S of the extreme rays of a polyhedral cone C A,B , as in Proposition 1 [5]. We construct d S := (d, y + , y − ), where y ± i = max{±(Bd) i , 0} as in the definition of S. The construction of d S is efficient: d is copied over and y ± is derived from a matrix-vector product on the original input and component-wise comparisons.…”

An implementation of steepest-descent augmentation for linear programs

“…It is shown in [9] that such a circuit can be computed by finding a vertex solution to the following LP over a polyhedral model of the set of circuits of P :…”

Circuit walks in integral polyhedra

“…The challenge here lies in finding a greedy circuit direction -it is open whether this can be done in polynomial time. However, it is possible to efficiently compute circuits for a steepest-descent augmentation scheme [13], which terminates in at most |C(A, B)| steps and runs in strongly polynomial time for polyhedra defined by totally unimodular matrices [12].…”