Branch-and-Bound Solves Random Binary IPs in Polytime

Dey, Santanu S.; Dubey, Yatharth; Molinaro, Marco

doi:10.1137/1.9781611976465.35

Cited by 18 publications

(11 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It has been observed that using such general partitioning rules can be significantly more efficient than simple branch-and-bound for some instances [1,11], but most commercial solver use the la er. Recently, we showed [15] that for certain classes of random integer programs the simple branch-and bound-tree has polynomial size (number of nodes), with good probability. See also [5] for nice extensions of this direction of results.…”

Section: Known Bounds On Sizes Of Branch-and-bound Treesmentioning

confidence: 99%

Lower Bounds on the Size of General Branch-and-Bound Trees

Dey¹,

Dubey²,

Molinaro³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form ⊤ ≤ 0 ∨ ⊤ ≥ 0 + 1, where is an integer vector and 0 is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to the coefficients of the cross polytope, thus showing that polynomial-size "smoothed analysis" upper bound is not possible. e results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. [7], who proved lower bounds for the Chvátal-Gomory rank.

show abstract

Section: Known Bounds On Sizes Of Branch-and-bound Treesmentioning

confidence: 99%

Lower Bounds on the Size of General Branch-and-Bound Trees

Dey¹,

Dubey²,

Molinaro³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a recent breakthrough work, Dey, Dubey and Molinaro [2] proved that, if the entries of A, c are independently distributed in [0, 1] and α > 0 one has…”

Section: Relation To Branch and Boundmentioning

confidence: 99%

“…Then in section 5, we prove Theorem 1, using a discrepancy result that we prove in section 6. In section 7, we explain how to adapt the upper bound on the size of branch-and-bound trees of [2] for random packing IPs to the setting of random Gaussian IPs.…”

Section: Organizationmentioning

confidence: 99%

“…We denote the integrality gap as IPGAP := val LP − val IP . The integrality gap of integer linear programs forms an important measure for the complexity of solving said problem in a number of works on the average-case complexity of integer programming [11,7,3,4,1,2].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the integrality gap in [3] also had a O(1/β m ) dependence, where β := min i∈[m] β i , whereas the integrality gap above does not depend on the "shape" of b (other than requiring b − 2 ≤ n/10). Due a recent breakthrough [2], the integrality gap above also implies that branch and bound applied to the above IPs produces a branching tree of size at most n poly(m) with good probability.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Integrality Gap of Binary Integer Programs with Gaussian Data

Borst¹,

Dadush²,

Huiberts³

et al. 2020

Preprint

View full text Add to dashboard Cite

For a binary integer program (IP) max c T x, Ax ≤ b, x ∈ {0, 1} n , where A ∈ R m×n and c ∈ R n have independent Gaussian entries and the right-hand side b ∈ R m satisfies that its negative coordinates have ℓ2 norm at most n/10, we prove that the gap between the value of the linear programming relaxation and the IP is upper bounded by poly(m)(log n) 2 /n with probability at least 1 − 1/n 7 − 1/2 Ω(m) . Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math. of O.R., 1989) in the case of random packing IPs. In constrast to the packing case, our integrality gap depends only polynomially on m instead of exponentially. By recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), the bound on the integrality gap immediately implies that branch and bound requires n poly(m) time on random Gaussian IPs with good probability, which is polynomial when the number of constraints m is fixed.

show abstract