On the Circuit Diameter Conjecture

Abstract

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

Circuit imbalance measures and linear programming

“…Note that by the requirement on maximal augmentations (i.e., every intermediate step being on the boundary of the polyhedron), augmentation sequences cannot be reversed; the length of the shortest sequence from vertex x to y may be different from the length of the one from y to x. A circuit-variant of the Hirsch conjecture was investigated in [BFH15] and [BSY18], and bounds were shown for some combinatorial polytopes, e.g. [BFH15,KPS19].…”

Circuit walks in integral polyhedra

“…Simple polyhedra are of interest in the study of diameters as it suffices to only consider this class of polyhedra to bound the combinatorial diameter of any n-dimensional polyhedron with a fixed number of facets [20]. While much harder to prove, the same holds for circuit diameters [11].…”

“…These generalize the concept of walking along the edges of a polyhedron to walking along its circuits. Whereas the famous Hirsch Conjecture is false in general [20,25], the analogous Circuit Diameter Conjecture [8], which asks whether the circuit diameter of a polyhedron is bounded by f − n, remains open [11,26].…”

On Circuit Diameter Bounds via Circuit Imbalances

“…Note that, in contrast to walks in the vertex-edge graph, circuit walks are non-reversible and the minimum length from x to y may be different from the one from y to x; this is due to the maximality requirement. The circuitanalogue of the Hirsch conjecture, formulated in [BFH15], asserts that the circuit diameter of a polytope in d dimensions with n facets is at most n − d; this may be true even for unbounded polyhedra, see [BSY18]. For P in the form (P), d = n − m and hence the conjectured bound is m.…”