Efficient separation routines for the symmetric traveling salesman problem I: general tools and comb separation (original) (raw)

A branch‐and‐cut algorithm for the undirected selective traveling salesman problem

Networks, 1998

The Selective Traveling Salesman Problem (STSP) is defined on a graph in which profits are associated with vertices and costs are associated with edges. Some vertices are compulsory. The aim is to construct a tour of maximal profit including all compulsory vertices and whose cost does not exceed a preset constant. We developed several classes of valid inequalities for the symmetric STSP and used them in a branch-and-cut algorithm. Depending on problem parameters, the proposed algorithm can solve instances involving up to 300 vertices.

Computing with Domino-Parity Inequalities for the Traveling Salesman Problem (TSP

Informs Journal on Computing, 2007

W e describe methods for implementing separation algorithms for domino-parity inequalities for the symmetric traveling salesman problem. These inequalities were introduced by Letchford , who showed that the separation problem can be solved in polynomial time when the support graph of the LP solution is planar. In our study we deal with the problem of how to use this algorithm in the general (nonplanar) case, continuing the work of . Our implementation includes pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe-shrinking routine, a random-walk heuristic to extract additional violated constraints, and a tightening procedure to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of a 33,810-city instance from the TSPLIB.

Generalized Domino-Parity Inequalities for the Symmetric Traveling Salesman Problem

Mathematics of Operations Research, 2010

We extend the work of Letchford . Separating a superclass of comb inequalities in planar graphs. Math. Oper. by introducing a new class of valid inequalities for the traveling salesman problem, called the generalized domino-parity (GDP) constraints. Just as Letchford's domino-parity constraints generalize comb inequalities, GDP constraints generalize the most well-known multiple-handle constraints, including clique-tree, bipartition, path, and star inequalities. Furthermore, we show that a subset of GDP constraints containing all of the clique-tree inequalities can be separated in polynomial time, provided that the support graph G * is planar, and provided that we bound the number of handles by a fixed constant h.

Implementing Domino-Parity Inequalities for the Traveling Salesman Problem

2005

We describe an implementation of Letchford's domino-parity inequalities for the (symmetric) traveling salesman problem. The implementation includes pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a set of safe-shrinking routines, a random-walk heuristic to extract additional violated constraints, and a tightening routine to allows us to modify existing domino-parity inequalities as the LP solution changes. We report computational results showing that combining the new separation algorithms with the Concorde TSP code allow us to substantially raise the linear programming bounds that are obtained.

A Branch-and-Cut algorithm for the Balanced Traveling Salesman Problem

HAL (Le Centre pour la Communication Scientifique Directe), 2023

The balanced traveling salesman problem (BTSP) is a variant of the traveling salesman problem, in which one seeks a tour that minimizes the difference between the largest and smallest edge costs in the tour. The BTSP, which is obviously NP-hard, was first investigated by Larusic and Punnen in 2011 [9]. They proposed several heuristics based on the double-threshold framework, which converge to good-quality solutions though not always optimal (e.g. 27 provably optimal solutions were found among 65 TSPLIB instances of at most 500 vertices). In this paper, we design a special-purpose branch-and-cut algorithm for solving exactly the BTSP. In contrast with the classical TSP, due to the BTSP's objective function, the efficiency of algorithms for solving the BTSP depends heavily on determining correctly the largest and smallest edge costs in the tour. In the proposed branch-and-cut algorithm, we develop several mechanisms based on local cutting planes, edge elimination, and variable fixing to locate more and more precisely those edge costs. Other important ingredients of our algorithm are heuristics for improving the lower and upper bounds of the branch-and-bound tree. Experiments on the same TSPLIB instances show that our algorithm was able to solve to optimality 63 out of 65 instances.

Computing with Domino-Parity Inequalities for the TSP

Informs Journal on Computing, 2000

W e describe methods for implementing separation algorithms for domino-parity inequalities for the symmetric traveling salesman problem. These inequalities were introduced by Letchford , who showed that the separation problem can be solved in polynomial time when the support graph of the LP solution is planar. In our study we deal with the problem of how to use this algorithm in the general (nonplanar) case, continuing the work of . Our implementation includes pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe-shrinking routine, a random-walk heuristic to extract additional violated constraints, and a tightening procedure to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of a 33,810-city instance from the TSPLIB.

Branch and Bound Methods for the Traveling Salesman Problem

1983

This paper reviews the state of the art in enumerative solution methods for tu. traveling salesman problem (TSP). The introduction (Section 1) discusses the main ingredients of branch and bound methods for the TSP. Sections 2, 3 and 4 discuss classes of methods based on three different relaxations of the TSP: the assignment problem with the TSP cost function, the 1-tree problem with a Lagrangean objective function, and the assignment problem with a Lagrangean objective function. Section 5 briefly reviews some other relaxations of the TSP, while Section 6 discusses the performance of some state of the art computer codes. Besides material from the literature, the paper also includes the results and statistical analysis of some computational experiments designed for the purposes of this review.

A Study of Domino-Parity and k-Parity Constraints for the TSP

2005

introduced the domino-parity inequalities for the symmetric traveling salesman problem and showed that if the support graph of an LP solution is planar, then the separation problem can be solved in polynomial time. We generalize domino-parity inequalities to multi-handled configurations, introducing a superclass of bipartition and star inequalities. Also, we generalize Letchford's algorithm, proving that for a fixed integer k, one can separate a superclass of k-handled clique-tree inequalities satisfying certain connectivity characteristics with respect to the planar support graph. We describe an implementation of Letchford's algorithm including pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe-shrinking routine, a randomwalk heuristic to extract additional violated constraints, and a tightening procedure to allow us to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of the TSPLIB instance pla33810.

Separating a Superclass of Comb Inequalities in Planar Graphs

Mathematics of Operations Research, 2000

Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x * , ÿnds one or more inequalities in the class which are violated by x * , or proves that none exist. Such algorithms are at the core of the highly successful branch-and-cut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2-matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We deÿne a generalization of comb inequalities and show that the associated separation problem can be solved e ciently when the subgraph induced by the edges with x * e ¿0 is planar. The separation algorithm runs in O(n 3) time, where n is the number of vertices in the graph.

The time dependent traveling salesman problem: Polyhedra and branch-cut-and-price algorithm

Experimental Algorithms, 2010

The Time Dependent Traveling Salesman Problem (TDTSP) is a generalization of the classical Traveling Salesman Problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 60 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet defining cuts. In particular, we also show that some facet defining cuts for the usual Asymmetric TSP formulation define low dimensional faces of the TDTSP formulation and give a way to lift them. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving several instances of reasonable size at the root node.