Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Boyd, Sylvia C.; Carr, Robert D.

doi:10.1016/j.disopt.2011.05.002

Cited by 25 publications

(44 citation statements)

References 12 publications

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“…thus lim k→∞ OPT (c (k) )LP(c (k) ) ≥4 3 . Along with Theorem 8, this gives the following: The integrality gap for square points lies between4 Graph G x for a k-donut x, k=4.…”

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confidence: 73%

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The salesman’s improved tours for fundamental classes

Boyd

Sebő

2019

Math. Program.

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Finding the exact integrality gap α for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that 4/3 ≤ α ≤ 3/2, and a famous conjecture states α = 4/3. It has also been conjectured that there exist halfinteger basic solutions of the linear program for which the highest integrality gap is reached. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most ρ for all instances of the metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the 1/2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3/2 to 10/7 for a superclass of these points for which a lower bound of 4/3 is proved.A key role in the proof of this result is played by finding Hamiltonian cycles whose existence is equivalent to Kotzig's result on "compatible Eulerian tours", and which lead us to delta-matroids for developing the related algorithms. Our arguments also involve other innovative tools from combinatorial optimization with the potential of a broader use.

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“…thus lim k→∞ OPT (c (k) )LP(c (k) ) ≥4 3 . Along with Theorem 8, this gives the following: The integrality gap for square points lies between4 Graph G x for a k-donut x, k=4.…”

mentioning

confidence: 73%

“…Two main classes of such vertices have been introduced, one by Carr and Vempala [8], the other by Boyd and Carr [4]. In this paper we will focus on the latter one, that is, we define a Boyd-Carr point [4] to be a point x ∈ S n that satisfies the following conditions:…”

Section: Introductionmentioning

confidence: 99%

The salesman’s improved tours for fundamental classes

Boyd

Sebő

2019

Math. Program.

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“…But we do not know whether there is an optimal solution that obeys the degree constraints if the comb inequalities are added. 3 The contribution of this paper is to improve our state of knowledge for the subtour LP by proving Conjecture 1.…”

Section: Conjecture 1 (Boyd and Carrmentioning

confidence: 99%

“…We will work extensively with fractional 2-matchings; that is, optimal solutions x to the LP (SUBT ) with only constraints (1) and (3). For convenience we will abbreviate "fractional 2-matching" by F2M and "2-matching" by 2M.…”

Section: Preliminariesmentioning

confidence: 99%

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A Proof of the Boyd-Carr Conjecture

Schalekamp

Williamson

Zuylen

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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Determining the precise integrality gap for the subtour LP relaxation of the traveling salesman problem is a significant open question, with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality. Boyd and Carr [3] observe that we do not even know the worst-case upper bound on the ratio of the optimal 2-matching to the subtour LP; they conjecture the ratio is at most 10/9.In this paper, we prove the Boyd-Carr conjecture. In the case that a fractional 2-matching has no cut edge, we can further prove that an optimal 2-matching is at most 10/9 times the cost of the fractional 2-matching.

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References

2019

Fundamentals of Supply Chain Theory

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Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Cited by 25 publications

References 12 publications

The salesman’s improved tours for fundamental classes

The salesman’s improved tours for fundamental classes

A Proof of the Boyd-Carr Conjecture

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