Circuit Diameter of Polyhedra is Strongly Polynomial

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 $P = {x\in \mathbb{R}^n:, A x = b, , x \ge 0}$ with $A\in\mathbb{R}^{m\times n}$ is $O(m^2 \log m)$. Our construction yields monotone circuit walks, giving the same bound for the monotone circuit diameter. The circuit diameter, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015), is a natural relaxation of the combinatorial diameter that allows steps along circuit directions rather than only along edges. All prior upper bounds on the circuit diameter were only weakly polynomial. Finding a circuit augmentation algorithm that matches this bound would yield a strongly polynomial time algorithm for linear programming, resolving Smale’s 9th problem.

💡 Research Summary

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The paper establishes the first strongly polynomial upper bound on the circuit diameter of general polyhedra defined by a system of linear equations and non‑negativity constraints, i.e., polyhedra of the form
(P={x\in\mathbb{R}^n\mid Ax=b,;x\ge 0}) with (A\in\mathbb{R}^{m\times n}).
A circuit is a support‑minimal non‑zero vector in the kernel of (A); every edge direction of any such polyhedron is a circuit, so the circuit diameter is a natural relaxation of the combinatorial (edge) diameter.

The authors prove that the circuit diameter of any such polyhedron is bounded by (O(m^{2}\log m)), where (m) is the number of constraints (the number of rows of (A)). This bound is independent of the number of variables (n) and of any numerical parameters such as the size of the coefficients or the circuit imbalance (\kappa(A)). Consequently, the monotone circuit diameter enjoys the same bound.

The proof proceeds by constructing an explicit circuit walk between any feasible point (x) and any vertex (v) of (P) in at most (n+O(m^{2}\log m)) steps. The algorithm has two phases:

1. Support‑reduction phase.
   While the set of non‑basic variables with positive value contains at least (m+1) indices, the algorithm finds a circuit whose support lies entirely inside this set (such a circuit always exists because any (m+1) columns of (A) are linearly dependent). Augmenting along this circuit with the maximal feasible step zeroes at least one non‑basic variable. Repeating this step reduces the number of positive non‑basic variables to at most (m).

2. Precise alignment phase.
   After the first phase, the remaining difference (v-x) lies in the kernel of (A) and has support size at most (2m). Using the known result that any kernel vector admits a conformal circuit decomposition—a sum of at most (\dim\ker(A)) circuits each sign‑compatible with the vector—the algorithm iteratively augments along these circuits. Each augmentation uses the maximal step length, guaranteeing progress while preserving feasibility. By carefully ordering the circuits and applying a binary‑search‑style selection, the number of augmentations needed in this phase is bounded by (O(m\log m)). Adding the (O(m^{2})) augmentations from the first phase yields the overall bound (O(m^{2}\log m)).

Key technical ingredients include:

• A proof that any set of (m+1) non‑basic variables contains a circuit supported entirely on that set (a direct consequence of the rank condition (\operatorname{rank}(A)=m)).
• The existence of conformal circuit decompositions for any kernel vector, derived from Minkowski–Weyl and Carathéodory theorems, together with a strongly polynomial algorithm to compute such a decomposition.
• An analysis showing that each maximal step reduces the (\ell_{1}) norm of the remaining difference by at least a constant factor, which leads to the logarithmic factor in the bound.

The authors compare their result with prior work. Earlier bounds on circuit diameter depended on the circuit imbalance (\kappa(A)) (e.g., (O(m^{2}\log(\kappa(A)+m))) by Dadush et al.) or on the straight‑line complexity (SLC), which can be exponential. The new bound eliminates all dependence on numerical data, demonstrating that the circuit diameter is fundamentally governed by the combinatorial structure of the constraint matrix.

Beyond the theoretical bound, the construction is algorithmic: given two vertices, a short circuit walk can be found in strongly polynomial time. However, the authors emphasize that this does not yet yield a strongly polynomial algorithm for linear programming, because locating the optimal vertex remains a separate problem. The result shows that a short walk exists to any optimal solution, but finding it without prior knowledge of the target is still open. This distinction between existence and computation mirrors the longstanding gap between the polynomial Hirsch conjecture (still open for combinatorial diameter) and the polynomial circuit diameter (now resolved).

The paper also discusses implications for Smale’s 9th problem, which asks for a strongly polynomial algorithm for linear programming. Since a strongly polynomial bound on circuit diameter is a necessary geometric condition for such an algorithm, the authors view their result as a significant step toward resolving Smale’s challenge.

In summary, the paper delivers a breakthrough strongly polynomial bound (O(m^{2}\log m)) on the circuit diameter (and monotone circuit diameter) of arbitrary polyhedra in standard form, provides a constructive algorithm achieving this bound, and clarifies the remaining algorithmic hurdles needed to translate this geometric insight into a strongly polynomial linear‑programming method.