Minimum Cost Spanning Tree Problems Research Articles

Demand operators and the Dutta–Kar rule for minimum cost spanning tree problems

Demand operators and the Dutta–Kar rule for minimum cost spanning tree problems

Algorithms for computing the Folk rule in minimum cost spanning tree problems with groups

This paper focuses on the computation of a rule in minimum cost spanning tree problems with groups, which coincides with the Owen value of the cooperative game for the irreducible cost matrix. To this aim, two polynomial-time algorithms are provided. The first of them is based on the steps of Kruskal algorithm whereas the other is based on a painting procedure.

An egalitarian solution to minimum cost spanning tree problems

An egalitarian solution to minimum cost spanning tree problems

Minimum cost spanning tree problems as value sharing problems

Minimum cost spanning tree (mcst) problems study situations in which agents must connect to a source to obtain a good, with the cost of building an edge being independent of the number of users. We reinterpret mcst problems as value sharing problems, and show that the folk and cycle-complete solutions, two of the most studied cost-sharing solutions for mcst problems, do not share values in a consistent way. More precisely, two mcst problems yielding the same value sharing problem might lead to value being shared in different ways. However, they satisfy a weaker version of the property that applies only to elementary problems, in which the cost on an edge can only be 0 or 1. The folk solution satisfies the version related to the public approach, while the cycle-complete solution satisfies the one related to the private approach, which differ depending if we allow a group to use the nodes of other agents or only their own nodes. We then build axiomatizations built on these properties. While the two solutions are usually seen as competitors in the private approach, the results point towards a different interpretation: the two solutions are based on different interpretations of the mcst problem, but are otherwise conceptually very close.

A non-cooperative approach to the folk rule in minimum cost spanning tree problems

This paper deals with the problem of finding a way to distribute the cost of a minimum cost spanning tree problem between the players. A rule that assigns a payoff to each player provides this distribution. An optimistic point of view is considered to devise a cooperative game. Following this optimistic approach, a sequential game provides this construction to define the action sets of the players. The main result states the existence of a unique cost allocation in subgame perfect equilibria. This cost allocation matches the one suggested by the folk rule.

The Folk Rule for Minimum Cost Spanning Tree Problems with Multiple Sources

In this paper, we introduce minimum cost spanning tree problems with multiple sources. This new setting is an extension of the classical model where there is a single source. We extend several definitions of the folk rule, the most prominent rule in the classical model, to this new context: first as the Shapley value of the irreducible game; second as an obligation rule; third as a partition rule and finally through a cone-wise decomposition. We prove that all the definitions provide the same cost allocation and present two axiomatic characterizations.

A claims problem approach to the cost allocation of a minimum cost spanning tree

We propose to allocate the cost of a minimum cost spanning tree by defining a claims problem and using claims rules, then providing easy and intuitive ways to distribute this cost. Depending on the starting point that we consider, we define two models. On the one hand, the benefit-sharing model considers individuals’ costs to the source as the starting point, and then the benefit of building the efficient tree is shared by the agents. On the other hand, the costs-sharing model starts from the individuals’ minimum connection costs (the cheapest connection they can use), and the additional cost, if any, is then allocated. As we prove, both approaches provide the same family of allocations for every minimum cost spanning tree problem. These models can be understood as a central planner who decides the best way to connect the agents (the efficient tree) and also establishes the amount each agent has to pay. In so doing, the central planner takes into account the maximum and minimum amount they should pay and some equity criteria given by a particular (claims) rule. We analyze some properties of this family of cost allocations, specially focusing in coalitional stability (core selection), a central concern in the literature on cost allocation.

A review of cooperative rules and their associated algorithms for minimum-cost spanning tree problems

Minimum-cost spanning tree problems are well-known problems in the operations research literature. Some agents, located at different geographical places, want a service provided by a common supplier. Agents will be served through costly connections. Some part of the literature has focused, mainly, in studying how to allocate the connection cost among the agents. We review the papers that have addressed the allocation problem using cooperative game theory. We also relate the rules defined through cooperative games with rules defined directly from the problem, either through algorithms for computing a minimal tree, either through a cone-wise decomposition.

Cost additive rules in minimum cost spanning tree problems with multiple sources

In this paper, we introduce a family of rules in minimum cost spanning tree problems with multiple sources called Kruskal sharing rules. This family is characterized by cone-wise additivity and independence of irrelevant trees. We also investigate some subsets of this family and provide axiomatic characterizations of them. The first subset is obtained by adding core selection. The second is obtained by adding core selection and equal treatment of source costs.

Minimum-cost spanning tree games: Bird rule revisited

Minimum-cost spanning tree (MCST) game is a classical model in the theory of cooperative games. The MCST game has been drawing continuous attention since it was raised by Claus and Kleitman in 1973. The MCST game is closely related to the MCST problem, a fundamental model in graph theory and combinatorial optimization, and has been widely applied in cost allocations for constructions of water networks, power networks, road networks and railway networks, etc. Due to its simplicity and intuitiveness, the Bird rule is one of the most famous solutions in the MCST game and has been extensively studied in the literature. Based on the linear programming formulation of the MCST problem by Edmonds, we provide a new equivalent closed-form formula for the Bird rule. We also study a generalization of the MCST game with weighted nodes. We show that the core is nonempty in the generalized model, and in particular, a modified Bird rule is proved to be still applicable.

An egalitarian approach for sharing the cost of a spanning tree.

A minimum cost spanning tree problem analyzes the way to efficiently connect individuals to a source. Hence the question is how to fairly allocate the total cost among these agents. Our approach, reinterpreting the spanning tree cost allocation as a claims problem defines a simple way to allocate the optimal cost with two main criteria: (1) each individual only pays attention to a few connection costs (the total cost of the optimal network and the cost of connecting himself to the source); and (2) an egalitarian criteria is used to share costs. Then, using claims rules, we define an egalitarian solution so that the total cost is allocated as equally as possible. We show that this solutions could propose allocations outside the core, a counter-intuitive fact whenever cooperation is necessary. Then we propose a modification to get a core selection, obtaining in this case an alternative interpretation of the Folk solution.

Sharing the cost of maximum quality optimal spanning trees

Minimum cost spanning tree problems have been widely studied in operation research and economic literature. Multi-objective optimal spanning trees provide a more realistic representation of different actual problems. Once an optimal tree is obtained, how to allocate its cost among the agents defines a situation quite different from what we have in the minimum cost spanning tree problems. In this paper, we analyze a multi-objective problem where the goal is to connect a group of agents to a source with the highest possible quality at the cheapest cost. We compute optimal networks and propose cost allocations for the total cost of the project. We analyze properties of the proposed solution; in particular, we focus on coalitional stability (core selection), a central concern in the literature on minimum cost spanning tree problems.

A characterization of the folk rule for multi-source minimal cost spanning tree problems

In this paper we provide an axiomatic characterization of the folk rule for minimum cost spanning tree problems with multiple sources. The properties we need are: cone-wise additivity, cost monotonicity, symmetry, isolated agents, and equal treatment of source costs.

The folk rule through a painting procedure for minimum cost spanning tree problems with multiple sources

We consider minimum cost spanning tree problems with multiple sources. We propose a cost allocation rule based on a painting procedure. Agents paint the edges on the paths connecting them to the sources. We prove that the painting rule coincides with the folk rule.

Upgrading nodes in tree-shaped hub location

Upgrading a node in a graph means to improve the quality of the edges that have it as an extreme. The first problem where upgrading was considered is the Minimum Cost Spanning Tree Problem. There, upgrading one or two of the extremes of an edge reduces progressively its cost and makes the edge more likely to be in the optimal tree. In this paper we analyze upgradings of nodes in a discrete location problem, the so-called Tree of Hubs Location Problem. Hubs are transshipment points through which a product is sent from origins to destinations. In addition to locate the hubs, to determine a tree that connects the hubs and to allocate non-hub nodes to hubs, i.e., the decisions to be made in the Tree of Hubs Location Problem, a decision has to be made about which of the hubs will be upgraded to reduce the overall cost of the distribution network. We provide two different Mixed Integer Linear Programming formulations for the problem, tighten the formulations and generate several families of valid inequalities for them. A computational study is presented showing the improvements attained with the strengthening of the formulations and comparing them.

The degree and cost adjusted folk solution for minimum cost spanning tree games

In this paper two cost sharing solutions for minimum cost spanning tree problems are introduced, the degree adjusted folk solution and the cost adjusted folk solution. These solutions overcome the problem of the classical reductionist folk solution as they have considerable strict ranking power, without breaking established axioms. As such they provide an affirmative answer to an open question, put forward in Bogomolnaia and Moulin (2010).

Characterizing rules in minimum cost spanning tree problems

A theorem of Bergantiños and Vidal-Puga (2015) states that in minimum cost spanning tree problems a rule satisfies separability and reductionism if and only if it is induced by an extra-costs function. In this article, we point out that the positivity of the extra-costs function implies the positivity of the rule; moreover, we prove that the rule further satisfies continuity if and only if the extra-costs function is continuous, and it satisfies symmetry if and only if the extra-costs function is symmetric.

Investigation on irreducible cost vectors in minimum cost arborescence problems

We study cost allocation rules in minimum cost arborescence problems, where agents need to build a network to a source in order to obtain some resource. Provided a vector of costs of edges (agent/source pairs), the agents cooperate to construct a minimum cost arborescence rooted at the source in order to reduce the total building cost. Minimum cost arborescence problems are extensions of well-studied minimum cost spanning tree problems to deal with asymmetric edge costs. Regarding cost allocation rules in minimum cost arborescence problems, Dutta and Mishra (2012) extended the folk rule, which is one of the most important rules in minimum cost spanning tree problems, based on the problem with the vector of the most reduced costs, called irreducible form. In minimum cost spanning tree problems, several axiomatic characterizations of the folk rule have been proposed. However, it is difficult to extend them in minimum cost arborescence problems. One of the reasons is that strong and reasonable axioms in minimum cost spanning tree problems, which imply irreducible-form dependence of cost allocation rules, are not satisfied by the folk rule in minimum cost arborescence problems. Hence, we search for other axioms which imply irreducible-form dependence. For this purpose, we investigate irreducible cost vectors in minimum cost arborescence problems, and characterize the irreducible form.

A monotonic and merge-proof rule in minimum cost spanning tree situations

We present a new model for cost sharing in minimum cost spanning tree problems to allow planners to identify how many agents merge. Under this new framework, in contrast to the traditional model, there are rules that satisfy the property of Merge-proofness. Furthermore, strengthening this property and adding some others, such as Population Monotonicity and Solidarity, makes it possible to define a unique rule that coincides with the weighted Shapley value of an associated cost game.

Strategic sharing of a costly network

We study minimum cost spanning tree problems for a set of users connected to a source. Prim’s algorithm provides a way of finding the minimum cost tree m. This has led to several definitions in the literature, regarding how to distribute the cost. These rules propose different cost allocations, which can be understood as compensations and/or payments between players, with respect to the status quo point: each user pays for the connection she uses to be linked to the source. In this paper we analyze the rationale behind a distribution of the minimum cost by defining an a priori transfer structure. Our first result states the existence of a transfer structure such that no user is willing to choose a different tree from the minimum cost tree. Moreover, given a transfer structure, we implement the above solution as a subgame perfect equilibrium outcome of a game where players decide sequentially with whom to connect. Finally, we obtain that the existence of a transfer structure supporting an allocation characterizes the core of the monotone cooperative game associated with a minimum cost spanning tree problem. This transfer structure is called social transfer structure. Therefore, the minimum cost spanning tree emerges as both a social and individual solution.