Optimal Traveling Salesman Tour Research Articles

Aggregating courier deliveries

AbstractWe consider the problem of efficiently scheduling deliveries by an uncapacitated courier from a central location under online arrivals. We consider both adversary‐controlled and Poisson arrival processes. In the adversarial setting we provide a randomized (3βΔ/2δ − 1)‐competitive algorithm, where β is the approximation ratio of the traveling salesman problem, δ is the minimum distance between the central location and any customer, and Δ is the length of the optimal traveling salesman tour overall customer locations and the central location. We provide instances showing that this analysis is tight. We also prove a 1 + 0.271Δ/δ lower‐bound on the competitive ratio of any algorithm in this setting. In the Poisson setting, we relax our assumption of deterministic travel times by assuming that travel times are distributed with a mean equal to the excursion length. We prove that optimal policies in this setting follow a threshold structure and describe this structure. For the half‐line metric space we bound the performance of the randomized algorithm in the Poisson setting, and show through numerical experiments that the performance of the algorithm is often much better than this bound.

Multi-depot traveling salesmen location problems on networks with special structure

Suppose that on any day, calls for service are generated by nodes of a transportation network independently with known probabilities. The calls are allocated to p service units that visit the allocated customers on shortest closed tours. That is, each service unit performs an optimal traveling salesman tour on its set of allocated calls (customers). We need to find optimal locations for the service units to minimize the expected travel distance. For this problem on a path, we present an $$O(n^4)$$ algorithm, where n is the number of nodes of the network. We also present an $$O(n^5)$$ algorithm for the problem on a tree with $$p=2$$.

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

We give an algorithm requiring $O(c^{1/\epsilon^2}n)$ time to find an $\epsilon$-optimal traveling salesman tour in the shortest-path metric defined by an undirected planar graph with nonnegative edge-lengths. For the case of all lengths equal to 1, the time required is $O(c^{1/\epsilon} n)$.

The Probabilistic Relationship Between the Assignment and Asymmetric Traveling Salesman Problems

We consider the gap between the cost of an optimal assignment in a complete bipartite graph with random edge weights, and the cost of an optimal traveling salesman tour in a complete directed graph with the same edge weights. Using an improved “patching” heuristic, we show that with high probability the gap is $O((\ln n)^2/n)$, and that its expectation is $\Omega(1/n)$. One of the underpinnings of this result is that the largest edge weight in an optimal assignment has expectation $\Theta(\ln n / n)$. A consequence of the small assignment–TSP gap is an $e^{\tilde{O}(\sqrt{n})}$‐time algorithm which, with high probability, exactly solves a random asymmetric traveling salesman instance. In addition to the assignment–TSP gap, we also consider the expected gap between the optimal and second‐best assignments; it is at least $\Omega(1/n^2)$ and at most $O(\ln n/n^2)$.

Hadwiger's conjecture for circular colorings of edge-weighted graphs

Let G w = ( V , E , w ) be a weighted graph, where G = ( V , E ) is its underlying graph and w : E → [ 1 , ∞ ) is the edge weight function. A (circular) p-coloring of G w is a mapping c of its vertices into a circle of perimeter p so that every edge e = uv satisfies dist ( c ( u ) , c ( v ) ) ⩾ w ( uv ) . The smallest p allowing a p-coloring of G w is its circular chromatic number, χ c ( G w ) . A p-basic graph is a weighted complete graph, whose edge weights satisfy triangular inequalities, and whose optimal traveling salesman tour has length p. Weighted Hadwiger's conjecture (WHC) at p ⩾ 1 states that if p is the largest real number so that G w contains some p-basic graph as a weighted minor, then χ c ( G w ) ⩽ p . We prove that WHC is true for p < 4 and false for p ⩾ 4 , and also that WHC is true for series–parallel graphs.

A new asymmetric pyramidally solvable class of the traveling salesman problem

We present a new method of identifying a class of asymmetric matrices for which an optimal traveling salesman tour exists that is pyramidal. The new class generalizes two previously known classes of matrices and includes some new matrices as well.

On approximately fair cost allocation in Euclidean TSP games

We consider the problem of allocating the cost of an optimal traveling salesman tour in a fair way among the nodes visited; in particular, we focus on the case where the distance matrix of the underlying TSP problem satisfies the triangle inequality. We thereby use the model of TSP games in the sense of cooperative game theory. We give examples showing that the core of such games may be empty, even for the case of Euclidean distances. On the positive, we develop an LP-based allocation rule guaranteeing that no coalition pays more thanα times its own cost, whereα is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation, which is also known as the solution over the “subtour polytope”. A well-known conjecture states thatα≤4/3. We also exhibit examples showing that this ratio cannot be improved below 4/3.

Estimating the length of the optimal TSP tour: An empirical study using regression and neural networks

Over the last 35 years or so, researchers have proposed a variety of ways to estimate the length of an optimal traveling salesman tour. Some of the estimators are asymptotic in nature, while others are equations that relate tour length to various independent variables such as the number of points in a problem, size of a problem's service area, and density of points. In this paper, we develop simple, empirically based estimators of the optimal tour length using regression and neural network models and show that these models can produce reasonably good estimates easily.

Equidistribution in all Dimensions of Worst-case Point Sets for the Traveling Salesman Problem

Given a set $S$ of $n$ points in the unit square $[0,1]^d$, an optimal traveling salesman tour of $S$ is a tour of $S$ that is of minimum length. A worst-case point set for the traveling salesman problem in the unit square is a point set $S^{(n)}$ whose optimal traveling salesman tour achieves the maximum possible length among all point sets $S\subset [0,1]^d$, where $|S|=n$. An open problem is to determine the structure of $S^{(n)}$. We show that for any rectangular parallelepiped $R$ contained in $[0,1]^d$, the number of points in $S^{(n)}\cap R$ is asymptotic to $n$ times the volume of $R$. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like $S^{(n)}$.

Technical Note—Heuristics for Delivery Problems with Constant Error Guarantees

In this paper Q Iterated Optimal Tour Partitioning, and Best Optimal Tour Partitioning algorithms are studied and analyzed for their worst case error. Both algorithms are based on partitioning an optimal traveling salesman tour in order to generate a feasible solution to the unit weight delivery problem. They have a worst case error bound of 2 − 1/Q where Q is the maximal number of customers a vehicle could visit and N is the total number of customers. Similar worst case error bounds are shown when the algorithms are applied to an α-optimal traveling salesman tour.

Testing the necklace condition for shortest tours and optimal factors in the plane

A tour T of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in T . It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour. In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O( n 2 log n) time. If a tour is given, then we can test in O( n 2) time whether or not this tour is a necklace-tour. Both algorithms can be generalized to ƒ-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.

Worst-Case Growth Rates of Some Classical Problems of Combinatorial Optimization

A method is presented for determining the asymptotic worst-case behavior of quantities like the length of the minimal spanning tree or the length of an optimal traveling salesman tour of n points in the unit d-cube. In each of these classical problems, the worst-case lengths are proved to have the exact asymptotic growth rate of $\beta _n^{{{(d - 1)} / d}} $, where $\beta $ is a positive constant depending on the problem and the dimension. These results complement known results on the growth rates for the analogous quantities under probabilistic assumptions on the points, but the results given here are free of any probabilistic hypotheses.

On the relationship between the biconnectivity augmentation and travelling salesman problems

A strategy for solving the traveling salesman problem is adapted to the problem of finding a biconnected subgraph of a weighted graph whose cost function satisfies the triangle inequality. An approximation algorithm similar to Christofides' algorithm [5] for the traveling salesman problem is shown to possess the same worst-case bound of 3 2 when applied to the biconnectivity augmentation problem. A tight inequality is derived relating the cost of an optimal traveling salesman tour to the cost of an optimal biconnection.