Traveling Salesman Games Research Articles

Cooperation of customers in traveling salesman problems with profits

In the profitable tour problem, the carrier decides whether to visit a particular customer with respect to the prize the customer offers for being visited and traveling cost associated with the visit, all in the context of other customers. Our focus is on the prizes customers need to offer to ensure being visited by the carrier. This can be formulated as a cooperative game where customers may form coalitions and make decisions on the prize values cooperatively. We define the profitable tour game describing this situation and analyze the cost associated with each coalition of customers and prizes that help to achieve it. We derive properties of the optimal prizes to be offered when the grand coalition is formed. These properties describe relationship between the prizes and the underlying traveling salesman game to provide connection with extensive literature on core allocations in traveling salesman games. The most important result states that the set of optimal prizes coincides with the core of the underlying traveling salesman game if this core is nonempty.

On the submodularity of multi-depot traveling salesman games

The Steiner traveling salesman problem is the problem of finding a minimum cost tour for a salesman that must visit a set of locations while traveling along costly streets before returning to his starting point at the depot. A solution to the problem is a minimum cost Steiner tour that both starts and ends at the depot and visits all the required locations. If different players are associated with the locations to be visited, the problem induces a cooperative traveling salesman (TS) game that poses the question of how to allocate the total cost of the tour between the different players involved. This cost allocation problem can be tackled using tools and solutions from cooperative game theory.The purpose of this paper is to generalize the notion of a traveling salesman game to allow for multiple depots in the underlying traveling salesman problem (TSP) and to analyze the submodularity of such multi-depot TS games. A multi-depot TSP can be represented by a connected (di)graph in which a fixed set of vertices are denoted depots, and a non-negative weight function is defined on the edges of the graph. The submodularity of multi-depot TS games is analyzed by characterizing graphs and digraphs that induce submodular multi-depot TS games for any choice of depots and for at least one choice of depots, respectively.

On the core of traveling salesman games

We define excess rate to study the core of traveling salesman games from a perspective of optimization, propose a new variant of the traveling salesman problem, and build a link between the two problems. An exact formula for the lowest achievable excess rate is found, which explains the existence of core emptiness. We then develop an implementable method to check whether empty core exists in general case. The results apply for both symmetric and asymmetric traveling salesman games.

Technical Note—On Traveling Salesman Games with Asymmetric Costs

We consider cooperative traveling salesman games with nonnegative asymmetric costs satisfying the triangle inequality. We construct a stable cost allocation with budget balance guarantee equal to the Held-Karp integrality gap for the asymmetric traveling salesman problem, using the parsimonious property and a previously unknown connection to linear production games. We also show that our techniques extend to larger classes of network design games. We then provide a simple example showing that our cost allocation does not necessarily achieve the best possible budget balance guarantee, even among cost allocations stable for the game defined by the Held-Karp relaxation, and discuss its implications on future work on traveling salesman games.

Characterizing the Shapley value in fixed-route traveling salesman problems with appointments

Starting from her home, a service provider visits several customers, following a predetermined route, and returns home after all customers are visited. The problem is to find a fair allocation of the total cost of this tour among the customers served. A transferable-utility cooperative game can be associated with this cost allocation problem. We introduce a new class of games, which we refer as the fixed-route traveling salesman games with appointments. We characterize the Shapley value in this class using a property which requires that sponsors do not benefit from mergers, or splitting into a set of sponsors.

Approximately Fair Cost Allocation in Metric Traveling Salesman Games

A traveling salesman game is a cooperative game ${\mathcal{G}}=(N,c_{D})$. Here N, the set of players, is the set of cities (or the vertices of the complete graph) and c D is the characteristic function where D is the underlying cost matrix. For all S⊆N, define c D(S) to be the cost of a minimum cost Hamiltonian tour through the vertices of S∪{0} where $0\not \in N$ is called as the home city. Define Core$({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)=c_{D}(N) \mbox{and} \forall S\subseteq N,x(S)\le c_{D}(S)\}$ as the core of a traveling salesman game ${\mathcal{G}}$. Okamoto (Discrete Appl. Math. 138:349–369, [2004]) conjectured that for the traveling salesman game ${\mathcal{G}}=(N,c_{D})$ with D satisfying triangle inequality, the problem of testing whether Core $({\mathcal{G}})$ is empty or not is $\mathsf{NP}$-hard. We prove that this conjecture is true. This result directly implies the $\mathsf{NP}$-hardness for the general case when D is asymmetric. We also study approximately fair cost allocations for these games. For this, we introduce the cycle cover games and show that the core of a cycle cover game is non-empty by finding a fair cost allocation vector in polynomial time. For a traveling salesman game, let $\epsilon\mbox{-Core}({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)\ge c_{D}(N)$ and ∀S⊆N, x(S)≤e⋅cD(S)} be an e-approximate core, for a given e>1. By viewing an approximate fair cost allocation vector for this game as a sum of exact fair cost allocation vectors of several related cycle cover games, we provide a polynomial time algorithm demonstrating the non-emptiness of the log 2(|N|−1)-approximate core by exhibiting a vector in this approximate core for the asymmetric traveling salesman game. We improve it further by finding a $(\frac{4}{3}\log_{3}(|N|)+c)$-approximate core in polynomial time for some constant c. We also show that there exists an e0>1 such that it is $\mathsf{NP}$-hard to decide whether e0-Core $({\mathcal{G}})$ is empty or not.

On the core of multiple longest traveling salesman games

In this paper we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except to his home location, in such an order that maximizes the total reward. First it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions. Secondly it is shown that MLTS games with five or less players have a nonempty core. However, a six player MLTS game may have an empty core. For the special instance in which the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.

Traveling salesman games with the Monge property

Several works have indicated the relationships between polynomially solvable combinatorial optimization problems and the core non-emptiness of cooperative games associated with them, and between intractable combinatorial optimization problems and the hardness of the problem to decide the core non-emptiness of the associated games. In this paper, we study the core of a traveling salesman game, which is associated with the traveling salesman problem. First, we show that in general the problem to test the core non-emptiness of a given traveling salesman game is NP-hard. This corresponds to the NP-hardness of the traveling salesman problem. Second, we show that the core of a traveling salesman game is non-empty if the distance matrix is a symmetric Monge matrix, and also that a traveling salesman game is submodular (or concave) if the distance matrix is a Kalmanson matrix. These correspond to the fact that the Monge property and the Kalmanson property are polynomially solvable special cases of the traveling salesman problem.

On the Core of Multiple Longest Traveling Salesman Games

In this paper, we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except its home location, in an order that maximizes the total reward. First, it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions. Secondly, it is shown that MLTS games with five or less players have a nonempty core. However, a six player MLTS game may have an empty core. For the special instance where the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.

Traveling salesman games

In this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited. We introduce two types of cooperative games—fixed-route traveling salesman games and traveling salesman games—as a tool to attack this problem. Under very mild conditions we prove that fixed-route traveling salesman games have non-empty cores if the fixed route is a solution of the classical traveling salesman problem. Core elements provide us with fair cost allocations. A traveling salesman game may have an empty core, even if the cost matrix satisfies the triangle inequality. In this paper we introduce a class of matrices defining TS-games with non-empty cores.