Using Integer Programming to Search for Counterexamples: A Case Study

Lancia, Giuseppe; Pippia, Eleonora; Rinaldi, Franca

doi:10.1007/978-3-030-49988-4_5

Cited by 6 publications

(4 citation statements)

References 10 publications

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“…The separation routine for the TSP-constraints has been easily adjusted to separate the modified constraints. The use of this decomposition strategy allowed us to reduce by more than one order of magnitude the overall computational time required to solve model M(n, 1t, ¬H) with respect to the straightforward solution reported in [16].…”

Section: Decomposition Strategies For the Solutions Spacementioning

confidence: 99%

“…These two facts seem to suggest that to answer the above questions (and similar open questions involving other pairs of connectivity conditions and hamiltonian properties) for 4-regular graphs one might need a different approach than a "standard" mathematical proof. In this paper, extending our preliminary work [16], we address the issue by adopting an Integer Linear Programming (ILP) approach. We have proceeded as follows.…”

Section: Introductionmentioning

confidence: 99%

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Proving hamiltonian properties in connected 4-regular graphs: an ILP-based approach

Lancia¹,

Pippia²,

Rinaldi³

2021

Preprint

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In this paper we study some open questions related to the smallest order f (C, ¬H) of a 4-regular graph which has a connectivity property C but does not have a hamiltonian property H. In particular, C is either connectivity, 2-connectivity or 1-toughness and H is hamiltonicity, homogeneous traceability or traceability. A standard theoretical approach to these questions had already been used in the literature, but in many cases did not succeed in determining the exact value of f (). Here we have chosen to use Integer Linear Programming and to encode the graphs that we are looking for as the binary solutions to a suitable set of linear inequalities. This way, there would exist a graph of order n with certain properties if and only if the corresponding ILP had a feasible solution, which we have determined through a branch-and-cut procedure. By using our approach, we have been able to compute f (C, ¬H) for all the pairs of considered properties with the exception of C =1-toughness, H =traceability. Even in this last case, we have nonetheless significantly reduced the interval [LB, U B] in which f (C, ¬H) was known to lie. Finally, we have shown that for each n ≥ f (C, ¬H) (n ≥ U B in the last case) there exists a 4-regular graph on n vertices which has property C but not property H.

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Section: Decomposition Strategies For the Solutions Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Proving hamiltonian properties in connected 4-regular graphs: an ILP-based approach

Lancia¹,

Pippia²,

Rinaldi³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This holds in particular when the solution of mixed integer programs is used as a tool in mathematics itself. Examples of recent work that employs MIP to investigate open mathematical questions include [11,12,19,29,30,33]. Some of these approaches are forced to rely on floating-point solvers because the availability, the flexibility, and most importantly the computational performance of MIP solvers with numerically rigorous guarantees is currently limited.…”

Section: Introductionmentioning

confidence: 99%

A computational status update for exact rational mixed integer programming

Eifler

Gleixner

2022

Math. Program.

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The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.

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“…This holds in particular when the solution of mixed integer programs is used as a tool in mathematics itself. Examples of recent work that employs MIP to investigate open mathematical questions include [11,12,18,28,29,32]. Some of these approaches are forced to rely on floating-point solvers because the availability, the flexibility, and most importantly the computational performance of MIP…”

Section: Introductionmentioning

confidence: 99%

A Computational Status Update for Exact Rational Mixed Integer Programming

Eifler

Gleixner

2021

Integer Programming and Combinatorial Optimization

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The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.

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Using Integer Programming to Search for Counterexamples: A Case Study

Cited by 6 publications

References 10 publications

Proving hamiltonian properties in connected 4-regular graphs: an ILP-based approach

Proving hamiltonian properties in connected 4-regular graphs: an ILP-based approach

A computational status update for exact rational mixed integer programming

A Computational Status Update for Exact Rational Mixed Integer Programming

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