On Circuit Diameter Bounds via Circuit Imbalances

Dadush, Daniel; Koh, Zhuan Khye; Natura, Bento; Végh, László A.

doi:10.1007/978-3-031-06901-7_11

Cited by 2 publications

(1 citation statement)

References 29 publications

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“…Despite results on circuit diameters for some polyhedra in combinatorial optimization (see, e.g., [4,22]), and general upper bounds involving the so-called circuit imbalance [10,16] or the input bit-size [13], not much is known about the potential validity of the circuit diameter conjecture. It was shown in [5] that ∆(8, 4) = 4.…”

Section: Circuit Diameters and The Circuit Diameter Conjecturementioning

confidence: 99%

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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Section: Circuit Diameters and The Circuit Diameter Conjecturementioning

confidence: 99%

Short Circuit Walks on Hirsch Counterexamples

E.¹,

Borgwardt²,

Brugger³

2023

Preprint

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

et al. 2023

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Following the breakthrough work of Tardos (Oper Res 34:250–256, 1986) in the bit-complexity model, Vavasis and Ye (Math Program 74(1):79–120, 1996) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) $$\max \, c^\top x,\, Ax = b,\, x \ge 0,\, A \in \mathbb {R}^{m \times n}$$ max c ⊤ x , A x = b , x ≥ 0 , A ∈ R m × n , Vavasis and Ye developed a primal-dual interior point method using a ‘layered least squares’ (LLS) step, and showed that $$O(n^{3.5} \log (\bar{\chi }_A+n))$$ O ( n 3.5 log ( χ ¯ A + n ) ) iterations suffice to solve (LP) exactly, where $$\bar{\chi }_A$$ χ ¯ A is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J Optim 13(4):1054–1079, 2003), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure $$\bar{\chi }^*_A$$ χ ¯ A ∗ , defined as the minimum $$\bar{\chi }_{AD}$$ χ ¯ AD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an $$O(m^2 n^2 + n^3)$$ O ( m 2 n 2 + n 3 ) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying $$\bar{\chi }_{AD} \le n(\bar{\chi }_A^*)^3$$ χ ¯ AD ≤ n ( χ ¯ A ∗ ) 3 . This algorithm also allows us to approximate the value of $$\bar{\chi }_A$$ χ ¯ A up to a factor $$n (\bar{\chi }_A^*)^2$$ n ( χ ¯ A ∗ ) 2 . This result is in surprising contrast to that of Tunçel (Math Program 86(1):219–223, 1999), who showed NP-hardness for approximating $$\bar{\chi }_A$$ χ ¯ A to within $$2^{\textrm{poly}(\textrm{rank}(A))}$$ 2 poly ( rank ( A ) ) . The key insight for our algorithm is to work with ratios $$g_i/g_j$$ g i / g j of circuits of A—i.e., minimal linear dependencies $$Ag=0$$ A g = 0 —which allow us to approximate the value of $$\bar{\chi }_A^*$$ χ ¯ A ∗ by a maximum geometric mean cycle computation in what we call the ‘circuit ratio digraph’ of A. While this resolves Monteiro and Tsuchiya’s question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved $$O(n^{2.5} \log (n)\log (\bar{\chi }^*_A+n))$$ O ( n 2.5 log ( n ) log ( χ ¯ A ∗ + n ) ) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor $$n/\log n$$ n / log n improvement on the iteration complexity bound of the original Vavasis–Ye algorithm.

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On Circuit Diameter Bounds via Circuit Imbalances

Cited by 2 publications

References 29 publications

Short Circuit Walks on Hirsch Counterexamples

Short Circuit Walks on Hirsch Counterexamples

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

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