Circuit imbalance measures and linear programming

Ekbatani, Farbod; Natura, Bento; Végh, László A.

doi:10.48550/arxiv.2108.03616

Cited by 3 publications

(8 citation statements)

References 41 publications

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“…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.…”

Section: Proximity Resultssupporting

confidence: 70%

“…The combinatorial diameter bound O(d 3 log(d/δ M )/δ M ) from [DH16] mentioned above translates to a bound O((n − m) 3 mκ A log(κ A + n)) for the system in the form (P), see [ENV21]. For circuit diameters, the Goldberg-Tarjan minimum-mean cycle cancelling algorithm for minimum-cost flows [GT89] naturally extends to a circuit augmentation algorithm for general LP using the steepest-descent rule.…”

Section: Imbalance and Diametermentioning

confidence: 99%

“…For circuit diameters, the Goldberg-Tarjan minimum-mean cycle cancelling algorithm for minimum-cost flows [GT89] naturally extends to a circuit augmentation algorithm for general LP using the steepest-descent rule. This yields a circuit diameter bound O(n 2 mκ A log(κ A + n)) [ENV21], see also [GD21]. However, note that these bounds may be exponential in the bit-complexity of the input.…”

Section: Imbalance and Diametermentioning

confidence: 99%

“…It can be shown that κ W = 1 if and only if W is a unimodular space, i.e., the kernel of a totally unimodular matrix. This parameter and related variants have been used implicitly or explicitly in many areas of linear programming and discrete optimization, see [ENV21] for a recent survey. 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].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Circuit Diameter Bounds via Circuit Imbalances

Dadush¹,

Koh²,

Natura³

et al. 2021

Preprint

Self Cite

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We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA, 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system {x ∈ R, where κ A is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound e.g. if all entries of A have polynomially bounded encoding length in n. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an optimization LP in O(n 3 log(n + κ A )) augmentation steps.

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Section: Proximity Resultssupporting

confidence: 70%

Section: Imbalance and Diametermentioning

confidence: 99%

Section: Imbalance and Diametermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Circuit Diameter Bounds via Circuit Imbalances

Dadush¹,

Koh²,

Natura³

et al. 2021

Preprint

Self Cite

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“…Note if A and A ′ are equivalent rational matrices, then C(A) = C(A ′ ) and so the parameters c 1 (•) and c ∞ (•) are invariant under row operations. Following the notation from [16], we write κA for the least common multiple of the entries of the circuits of A. Observe that there exists a function f : N → N such that κA ≤ f (c ∞ (A)) for every matrix A.…”

Section: Matricesmentioning

confidence: 99%

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Briański¹,

Koutecký²,

Kráľ³

et al. 2022

Preprint

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An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively.We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ 1 -norm of the Graver basis is bounded by a function of the maximum ℓ 1 -norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists.Our results yield parameterized algorithms for integer programming when parameterized by the ℓ 1 -norm of the Graver basis of the constraint matrix, when parameterized by the ℓ 1 -norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

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Short Circuit Walks on Hirsch Counterexamples

E.¹,

Borgwardt²,

Brugger³

2023

Preprint

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Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for lower bounding the combinatorial diameters of polyhedra. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural approach to this question is to see whether counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the remaining counterexamples for monotone walks and polytopes also do not transfer. The key ingredients for our results are sign-compatible circuit walks on spindles, a class of polytopes underlying all known Hirsch counterexamples. To resolve the monotone case, we enumerate all linear programs with feasible region given by Todd's monotone Hirsch counterexample and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining 7107 satisfy the bound.

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Circuit imbalance measures and linear programming

Cited by 3 publications

References 41 publications

On Circuit Diameter Bounds via Circuit Imbalances

On Circuit Diameter Bounds via Circuit Imbalances

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Short Circuit Walks on Hirsch Counterexamples

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