Recent works on cost based relaxations have improved Constraint Programming (CP) models for the Traveling Salesman Problem (TSP). We provide a short survey over solving asymmetric TSP with CP. Then, we suggest new implied propagators based on general graph properties. We experimentally show that such implied propagators bring robustness to pathological instances and highlight the fact that graph structure can significantly improve search heuristics behavior. Finally, we show that our approach outperforms current state of the art results.
A more accurate relaxation is the Minimum Spanning Arborescence (MSA) relaxation, since it does not relax the orientation of the graph. This relaxation has been studied by @cite_29 @cite_28 who provide a @math time filtering algorithm based on primal dual linear programs. The best algorithm for computing a MSA has been provided by @cite_4 . Their algorithm runs in @math worst case time, but it does not provide reduced costs that are used for pruning. Thus, it could be used to create a constraint with a @math time consistency checking but the complete filtering algorithm remains in @math time. The Lagrangian MSA relaxation, with a MSA computation based on Edmonds' algorithm, has been suggested in @cite_17 . This method was very accurate but unfortunately unstable. Also, @cite_3 report that the MSA based Held and Karp scheme lead to disappointing results.
@cite_4 @cite_28 @cite_29 @cite_3 @cite_17
1521197246 2051585135 2126448893 1979269423
We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order- @math tensor @math of the form @math , where @math is a signal-to-noise ratio, @math is a unit vector, and @math is a random noise tensor, the goal is to recover the planted vector @math . For the case that @math has iid standard Gaussian entries, we give an efficient algorithm to recover @math whenever @math , and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of @math . The previous best algorithms with provable guarantees required @math . In the regime @math , natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree- @math sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree- @math sum-of-squares relaxations break down for @math , which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings. We present a deterministic (1+√5 2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on @math 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 had 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+√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. Linear programming relaxations have been used extensively in designing approximation algorithms for optimization problems. For vertex cover, linear programming and a thresholding technique gives a 2-approximate solution, rivaling the best known performance ratio. For a generalization of vertex cover we call vc t, in which we seek to cover t edges, this technique may not yield a feasible solution at all. We introduce a new method for massaging a linear programming solution to get a good, feasible solution for vc t. Our technique manipulates the values of the linear programming solution to prepare them for thresholding. We prove that this method achieves a performance ratio of 2 for vc t with unit weights. A second algorithm extends this result, giving a 2-approximation for vc t with arbitrary weights. We show that this is tight in the sense that any α-approximation algorithm for vc t with α < 2 implies a breakthrough α-approximation algorithm for vertex cover. The bottleneck of the state-of-the-art algorithms for geometric Steiner problems is usually the concatenation phase, where the prevailing approach treats the generated full Steiner trees as edges of a hypergraph and uses an LP-relaxation of the minimum spanning tree in hypergraph (MSTH) problem. We study this original and some new equivalent relaxations of this problem and clarify their relations to all classical relaxations of the Steiner problem. In an experimental study, an algorithm of ours which is designed for general graphs turns out to be an efficient alternative to the MSTH approach.