Short circuit walks in fixed dimension

2025-10-02

AI Summary

This paper investigates the approximation hardness of computing shortest monotone circuit walks on fixed-dimensional polyhedra. For two-dimensional simple polygons with $m$ edges, we establish the strongest known lower bound: for any $varepsilon > 0$, obtaining an $O(m^{1-varepsilon})$-approximation is NP-hard. Our method introduces a novel reduction from high-dimensional degenerate polyhedra to two-dimensional polygons—carefully preserving simplicity and low dimensionality—while integrating circuit augmentation frameworks, complexity-theoretic techniques, and geometric analysis to transfer hardness. This result demonstrates that even in the simplest nontrivial setting (2D polygons), efficient approximation of shortest monotone circuit walks is fundamentally intractable. It significantly advances the understanding of computational complexity for circuit diameters and circuit walks in linear programming, resolving a key open question regarding the limits of approximation in combinatorial and geometric optimization over polyhedra.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Circuit augmentation schemes are a family of combinatorial algorithms for linear programming that generalize the simplex method. To solve the linear program, they construct a so-called monotone circuit walk: They start at an initial vertex of the feasible region and traverse a discrete sequence of points on the boundary, while moving along certain allowed directions (circuits) and improving the objective function at each step until reaching an optimum. Since the existence of short circuit walks has been conjectured (Circuit Diameter Conjecture), several works have investigated how well one can efficiently approximate shortest monotone circuit walks towards an optimum. A first result addressing this question was given by De Loera, Kafer, and Sanità [SIAM J. Opt., 2022], who showed that given as input an LP and the starting vertex, finding a $2$-approximation for this problem is NP-hard. Cardinal and the third author [Math. Prog. 2023] gave a stronger lower bound assuming the exponential time hypothesis, showing that even an approximation factor of $O(frac{log m}{log log m})$ is intractable for LPs defined by $m$ inequalities. Both of these results were based on reductions from highly degenerate polytopes in combinatorial optimization with high dimension. In this paper, we significantly strengthen the aforementioned hardness results by showing that for every fixed $varepsilon>0$ approximating the problem on polygons with $m$ edges to within a factor of $O(m^{1-varepsilon})$ is NP-hard. This result is essentially best-possible, as it cannot be improved beyond $o(m)$. In particular, this implies hardness for simple polytopes and in fixed dimension.

Problem

• Proving NP-hardness of approximating shortest monotone circuit walks in polygons.
• Strengthening hardness results for circuit augmentation schemes in fixed dimension.
• Establishing best-possible inapproximability bounds for linear programming walks.

Innovation

• NP-hard approximation for short circuit walks
• Fixed dimension polygon hardness proof
• Best-possible inapproximability factor established