Bento Natura

STOC Conference 2024 Conference Paper

A Strongly Polynomial Algorithm for Linear Programs with At Most Two Nonzero Entries per Row or Column

• Daniel Dadush
• Zhuan Khye Koh
• Bento Natura
• Neil Olver
• László A. Végh

We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Végh ‍(MOR ’17) and Megiddo ‍(SICOMP ’83), respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) by Allamigeon, Dadush, Loho, Natura, and Végh ‍(FOCS ’22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum ‍(ORL ’04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices and extreme rays. Our approach is black-box and can be applied to any log-barrier path-following method.

FOCS Conference 2022 Conference Paper

Interior point methods are not worse than Simplex

• Xavier Allamigeon
• Daniel Dadush
• Georg Loho
• Bento Natura
• László A. Végh

Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound $O(2^{n}n^{15}\log n)$ for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations. The number of iterations of our algorithm is at most $O(n^{15}\log n)$ times the number of segments of any piecewise linear curve in the wide neighborhood of the central path. In particular, it matches the number of iterations of any path following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the max central path’, a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths. From the existence of a line segment in the wide neighborhood we derive strong implications on the structure of the corresponding segment of the central path. Our algorithm is able to detect this structure from the local geometry at the current iterate, and constructs a step direction that descends along this segment. The bound $O(n^{15}\log n)$ that applies for arbitrarily long line segments is derived from a combinatorial progress measure. Our algorithm falls into the family of layered least squares interior point methods introduced by Vavasis and Ye (Math. Prog. 1996). In contrast to previous layered least squares methods that partition the kernel of the constraint matrix into coordinate subspaces, our method creates layers based on a general subspace providing more flexibility. Our result also implies the same bound on the number of iterations of the trust region interior point method by Lan, Monteiro, and Tsuchiya (SIOPT 2009).

STOC Conference 2020 Conference Paper

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

• Daniel Dadush
• Sophie Huiberts
• Bento Natura
• László A. Végh

FOCS Conference 2020 Conference Paper

Revisiting Tardos's Framework for Linear Programming: Faster Exact Solutions using Approximate Solvers

• Daniel Dadush
• Bento Natura
• László A. Végh

In breakthrough work, Tardos (Oper. Res. '86) gave a proximity based framework for solving linear programming (LP) in time depending only on the constraint matrix in the bit complexity model. In Tardos's framework, one reduces solving the LP min(c, x), Ax=b, x ≥ 0, A Z m×n, to solving O(nm) LPs in A having small integer coefficient objectives and right-hand sides using any exact LP algorithm. This gives rise to an LP algorithm in time poly (n, m log Δ A ), where Δ A is the largest subdeterminant of A. A significant extension to the real model of computation was given by Vavasis and Ye (Math. Prog. '96), giving a specialized interior point method that runs in time poly (n, m, log χ̑ A ), depending on Stewart's χ̑ A, a well-studied condition number. In this work, we extend Tardos's original framework to obtain such a running time dependence. In particular, we replace the exact LP solves with approximate ones, enabling us to directly leverage the tremendous recent algorithmic progress for approximate linear programming. More precisely, we show that the fundamental “accuracy” needed to exactly solve any LP in A is inverse polynomial in n and log χ̑ A. Plugging in the recent algorithm of van den Brand (SODA '20), our method computes an optimal primal and dual solution using O(mn ω+1+0(1) log(χ̑ A +n)) arithmetic operations, outperforming the specialized interior point method of Vavasis and Ye and its recent improvement by Dadush et al (STOC '20). By applying the preprocessing algorithm of the latter paper, the dependence can also be reduced from χ̑ A to χ̑ A *, the minimum value of χ̑ AD attainable via column rescalings. Our framework is applicable to achieve the poly (n, m, log χ̑ A *) bound using essentially any weakly polynomial LP algorithm, such as the ellipsoid method. At a technical level, our framework combines together approximate LP solutions to compute exact ones, making use of constructive proximity theorems-which bound the distance between solutions of “nearby” LPs-to keep the required accuracy low.