267 research outputs found

    Improving Christofides' Algorithm for the s-t Path TSP

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    We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.Comment: 31 pages, 5 figure

    Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours

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    We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a θ\theta-cyclic point: xe∈{0,θ,1−θ,1}x_e \in \{0,\theta, 1-\theta, 1\}, where the support graph is subcubic and each vertex is incident to at least one edge with xx-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with xx-value 1 from the 32\frac{3}{2} of Christofides algorithm to 32−θ10\frac{3}{2}-\frac{\theta}{10} while keeping the usage of edges with fractional xx-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a 23\frac{2}{3}-uniform point: xe∈{0,23}x_e \in \{0, \frac{2}{3}\}, we give a 1712\frac{17}{12}-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios of 32\frac{3}{2} of Christofides algorithm and 43\frac{4}{3} implied by the famous "four-thirds conjecture"

    A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSP

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    A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP is at most 4/3. Despite significant efforts, the conjecture remains open. We consider the half-integral case, in which the LP has solution values in {0,1/2,1}\{0, 1/2, 1\}. Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture. Karlin, Klein, and Oveis Gharan, in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993. This result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap is at most 1.5−10−361.5- 10^{-36} in the non-half-integral case. For the half-integral case, the current best-known ratio is 1.4983, a result by Gupta et al. With the improvements on the 3/2 bound remaining very incremental even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct. The previous works on the half-integral case perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to "cycle cuts" and the others to "degree cuts". We show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that the known bad cases for the integrality gap have a gap of 4/3 and have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight.Comment: Comments, questions, and suggestions are welcome

    On Polynomial Kernels for Traveling Salesperson Problem and Its Generalizations

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