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  "path": "/abs/2602.06958v1",
  "publishedAt": "2026-02-09T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Bento Natura"
  ],
  "textContent": "**Authors:** Bento Natura\n\nWe 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.",
  "title": "Circuit Diameter of Polyhedra is Strongly Polynomial"
}