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 stronger relaxation is the weighted version of the constraint, corresponding to the Minimum Assignment Problem (MAP). It requires @math time @cite_25 to compute a first minimum cost assignment but then @math time @cite_0 to check consistency and filter incrementally. Some interesting evaluations are provided by @cite_10 , but are mainly related to the TSP with time windows constraints.
@cite_0 @cite_10 @cite_25
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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. We develop a mathematical programming approach for the classicalPSPACE-hard restless bandit problem in stochastic optimization. We introduce a hierarchy ofN (whereN is the number of bandits) increasingly stronger linear programming relaxations, the last of which is exact and corresponds to the (exponential size) formulation of the problem as a Markov decision chain, while the other relaxations provide bounds and are efficiently computed. We also propose a priority-index heuristic scheduling policy from the solution to the firstorder relaxation, where the indices are defined in terms of optimal dual variables. In this way we propose a policy and a suboptimality guarantee. We report results of computational experiments that suggest that the proposed heuristic policy is nearly optimal. Moreover, the second-order relaxation is found to provide strong bounds on the optimal value. The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LP-S: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in (2001), (1998), and (2005) for the general case; (ii) QP-RL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCP-MS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by (2006) for a general label set. We show that the SOCP-MS and the QP-RL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints objective function offered by QP and SOCP, the LP-S relaxation strictly dominates (i.e., provides a better approximation than) QP-RL and SOCP-MS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations which is dominated by the LP-S relaxation. Based on these results we propose some novel SOCP relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field. Using some examples we show that the new SOCP relaxations strictly dominate the previous approaches. We consider a problem known as the restricted assignment version of the max-min allocation problem with indivisible goods. There are n items of various nonnegative values and m players. Every player is interested only in some of the items and has zero value for the other items. One has to distribute the items among the players in a way that maximizes a certain notion of fairness, namely, maximizes the minimum of the sum of values of items given to any player. Bansal and Sviridenko [STOC 2006] describe a linear programming relaxation for this problem, and present a rounding technique that recovers an allocation of value at least Ω(log log log m log log m) of the optimum. We show that the value of this LP relaxation in fact approximates the optimum value to within a constant factor. Our proof is not constructive and does not by itself provide an efficient algorithm for finding an allocation that is within constant factors of optimal.