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 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 ∆ A), where ∆ 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, logχ A), depending on Stewart's χ 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 logχ 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+o(1) log(χ 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 fromχ A toχ * A , the minimum value ofχ AD attainable via column rescalings. Our framework is applicable to achieve the poly(n, m, logχ * 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.

On Circuit Diameter Bounds via Circuit Imbalances

“…We formulate such statements that will be needed for our analyses. These can be derived from more general results in [DNV20]; see also [ENV21]. The references also explain the background and similar results in previous literature, in particular, to proximity bounds via ∆ A in e.g.…”

“…Her algorithm makes O(nm) calls to a weakly polynomial LP solver for instances with small integer capacities and costs, and uses proximity arguments to gradually learn the support of an optimal solution. This approach was extended to the real model of computation for an poly(n, m, log κ A ) bound [DNV20]. This result uses proximity arguments with circuit imbalances κ A , and eliminates all dependence on bit-complexity.…”

“…The proximity tool Theorem 2.7 derives from [DNV20], and our circuit augmentation algorithms are inspired by the feasibility and optimization algorithms in this paper. However, using circuit augmentation oracles instead of an approixmate LP oracles changes the setup.…”

“…However, using circuit augmentation oracles instead of an approixmate LP oracles changes the setup. Our arguments become simpler since we proceed through a sequence of feasible solutions, whereas much effort in [DNV20] is needed to deal with infeasibility of the solutions returned by the approximate solver. On the other hand, we need to be more careful as all steps must be implemented using circuit augmentations in the original system, in contrast to the higher degree of freedom in [DNV20] where we can make approximate solver calls to arbitrary projections and modifications of the input LP.…”

“…It is closely related to the Dikin-Stewart-Todd condition number χW that plays a key role in layered-least-squares interior point methods introduced by Vavasis and Ye [Vav94]. An LP of the form (LP) for A ∈ R m×n can be solved in time poly(n, m, log κ A ), which is strongly polynomial if κ A ≤ 2 poly(n) ; see [DHNV20,DNV20] for recent developments and references.…”