The Travelling Salesman Problem is one the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3 2, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only 4 3. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis , and then by Momke and Svensson. In this paper, we provide an improved analysis for the approach introduced by Momke and Svensson yielding a bound of 13 9 on the approximation factor, as well as a bound of 19 12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics.
In the next section we present previous results relevant to the contributions of this paper, in particular we recall key definitions and theorems of M "omke and Svensson @cite_2 . In we present the improved upper bound on the cost of the core part of the circulation, as well as an almost matching lower bound. In we prove that the correction part of the circulation is essentially free. Finally, in we apply the results of the previous sections to obtain improved approximation algorithms for graphic TSP and TSPP.
@cite_2
2051585135 2288595346 2067993964 2000931246
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. This paper develops a branch-and-bound algorithm to solve the 2-norm soft margin semi-supervised support vector machine. First, the original problem is reformulated as a non-convex quadratically constrained quadratic programming problem with a simple structure. Then, we propose a new lower bound estimator which is conceptually simple and easy to be implemented in the branch-and-bound scheme. Since this estimator preserves both a high efficiency and a relatively good quality in the convex relaxation, it leads to a high total efficiency in the whole computational process. The numerical tests on both artificial and real-world data sets demonstrate the better effectiveness and efficiency of this proposed approach, which is compared to other well-known methods on different semi-supervised support vector machine models. In this paper we give a method for solving certain budgeted optimization problems in strongly polynomial time. The method can be applied to several known budgeted problems, and in addition we show two new applications. The first one extends Frederickson’s and Solis-Oba’s result [G. N. Frederickson and R. Solis-Oba, Combinatorica, 18 (1998), pp. 503-518] to (poly)matroid intersections from single matroids. The second one is the budgeted version of the minimum cost circulation problem. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253