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

Abstract

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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m×n</sup> , 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> ), where Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> ), depending on Stewart's χ̑ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> , 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> . Plugging in the recent algorithm of van den Brand (SODA '20), our method computes an optimal primal and dual solution using O(mn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω+1+0(1)</sup> log(χ̑ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> +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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> to χ̑ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> *, the minimum value of χ̑ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AD</sub> attainable via column rescalings. Our framework is applicable to achieve the poly (n, m,log χ̑ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> *) 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.