LP-based Heuristics for the Traveling Salesman Problem (original) (raw)

An Analysis of Several Heuristics for the Traveling Salesman Problem

SIAM Journal on Computing, 1977

Several polynomial time algorithms finding "good," but not necessarily optimal, tours for the traveling salesman problem are considered. We measure the closeness of a tour by the ratio of the obtained tour length to the minimal tour length. For the nearest neighbor method, we show the ratio is bounded above by a logarithmic function of the number of nodes. We also provide a logarithmic lower bound on the worst case. A class of approximation methods we call insertion methods are studied, and these are also shown to have a logarithmic upper bound. For two specific insertion methods, which we call nearest insertion and cheapest insertion, the ratio is shown to have a constant upper bound of 2, and examples are provided that come arbitrarily close to this upper bound. It is also shown that for any n => 8, there are traveling salesman problems with n nodes having tours which cannot be improved by making n/4 edge changes, but for which the ratio is 2(1-l/n).

The Traveling Salesman Problem: A inclusive study

The traveling salesman problem deals with the problem of a salesman and a lay down of cities. The salesman has in the direction of appointment all one of the cities opening from a definite one and returning to the same city. The confront of the problem is so as to the traveling salesman requirements to minimize the total length of the trip. In spite of that we want to solve this problem we have to show here that the traveling salesman problem is NP-complete since it has two properties. Initial is that in NP in relation to meaning that there is a polynomial-time algorithm to prove solutions to the problem. Subsequently for any additional problem in NP, we be capable of convert instances of the difficulty lying on the technique to corresponding instances of traveling salesman in polynomial time. This means that a polynomial-time way out to traveling salesman would mean a polynomial-time solution to all problems in NP, which most of the people don't think is probable. For an overview of NP-completeness I am going to present this paper

A restricted Lagrangean approach to the traveling salesman problem

Mathematical Programming, 1981

We describe an algorithm for the asymmetric traveling salesman problem (TSP) using a new, restricted Lagrangean relaxation based on the assignment problem (AP). The Lagrange multipliers are constrained so as to guarantee the continued optimality of the initial AP solution, thus eliminating the need for repeatedly solving AP in the process of computing multipliers. We give several polynomially bounded procedures for generating valid inequalities and taking them into the Lagrangean function with a positive multiplier without violating the constraints, so as to strengthen the current lower bound. Upper bounds are generated by a fast heuristic whenever possible. When the bound-strengthening techniques are exhausted without matching the upper with the lower bound, we branch by using two different rules, according to the situation: the usual subtour breaking disjunction, and a new disjunction based on conditional bounds. We discuss computational experience on 120 randomly generated asymmetric TSP's with up to 325 cities, the maximum time used for any single problem being 82 seconds. Though the algorithm discussed here is for the asymmetric TSP, the approach can be extended to the symmetric TSP by using the 2-matching problem instead of AP. the equations Z x -1, itS., and (7a) can be obtained from (7b) by the reverse operation. Nevertheless, the presence of inequalities associated with the same set S in both subsets (7a) and (7b) need not be avoided, since it may enrich the set of dual vectors (u,v,w) satisfying (8) and w > 0. -7-Finally, for any k«N, S t c N\{k} and S t » N\S t , the arc sets K = (S t ,S t \Ck}) and Kj -(S t \{k},S t ) are (directed) cutsets in the subgraph <N\(k}) o f G induced by N\[k }. Proposition 3. The inequalities (7C) are satisfied by every tour.

Heuristics Designed For the Traveling Salesman Problem

International Journal of Research in Advent Technology

The voyaging sales rep issue is a fit perceived enhancement issue. Ideal answers for humble cases can be found in sensible time by straight programming. Be that as it may, since the TSP is NP-hard, it will be very tedious to determine bigger occurrences with ensured optimality. Circumstance optimality aside, there's a lot of calculations opening equivalently quick activity time and still delicate close ideal arrangements. Keywords: NP-hard TSP 1 , HK-Hard TSP 2 , Brach bound TSP 3 , Heuristic algorithm STSP 4 .