On the core of cost-revenue games: Minimum cost spanning tree games with revenues

In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net revenues that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a nonempty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a nonempty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost).     

On the complexity of testing membership in the core of min-cost spanning tree games

LetN = {1,...,n} be a finite set of players andK _N the complete graph on the node setN∪{0}. Assume that the edges ofK _N have nonnegative weights and associate with each coalitionS∪N of players as costc(S) the weight of a minimal spanning tree on the node setS∪{0}. Using transformation from EXACT COVER BY 3-SETS, we exhibit the following problem to beNP-complete. Given the vectorxε?^itN withx(N) =c(N). decide whether there exists a coalitionS such thatx(S) >c(S).     

Spanning network games

We study fundamental properties of monotone network enterprises which contain public vertices and have positive and negative costs on edges and vertices. Among the properties studied are the nonemptiness of the core, characterization of nonredundant core constraints, ease of computation of the core and the nucleolus, and cases of decomposition of the core and the nucleolus. Received December 1994/Final version March 1998     

On the complexity of the robust spanning tree problem with interval data

This paper studies the complexity of the robust spanning tree problem with interval data (RSTID). It shows that the problem is NP-complete, settling the conjecture of Kouvelis and Yu, and that it remains so for complete graphs or when the intervals are all [0,1]. These results indicate that the difficulty of RSTID stems from both the graph topology and the structure of the cost intervals, suggesting new directions for search algorithms.     

On computational complexity of membership test in flow games and linear production games

Let Γ≡(N,v) be a cooperative game with the player set N and characteristic function v: 2^N→R. An imputation of the game is in the core if no subset of players could gain advantage by splitting from the grand coalition of all players. It is well known that, for the flow game (and equivalently, for the linear production game), the core is always non-empty and a solution in the core can be found in polynomial time. In this paper, we show that, given an imputation x, it is NP-complete to decide x is not a member of the core, for the flow game. And because of the specific reduction we constructed, the result also holds for the linear production game. Received: October 2000/Final version: March 2002     

On the complexity of the one-terminal network design problem

The one-terminal network design problem considered here is to select a subset of the set of potential edges so as to minimize the sum of construction cost plus expected usage cost with discounting. We distinguish between easy and hard cases of this problem.     

Simulated annealing algorithm for the robust spanning tree problem

This paper addresses the robust spanning tree problem with interval data, i.e. the case of classical minimum spanning tree problem when edge weights are not fixed but take their values from some intervals associated with edges. The problem consists of finding a spanning tree that minimizes so-called robust deviation, i.e. deviation from an optimal solution under the worst case realization of interval weights. As it was proven in Kouvelis and Yu (Robust Discrete Optimization and Its Applications, Kluwer Academic, Norwell, 1997), the problem is NP-hard, therefore it is of great interest to tackle it with some metaheuristic approach, namely simulated annealing, in order to calculate an approximate solution for large scale instances efficiently. We describe theoretical aspects and present the results of computational experiments. To the best of our knowledge, this is the first attempt to develop a metaheuristic approach for solving the robust spanning tree problem.     

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in d‐dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 262–286, 2017     

On the core of network synthesis games

We use polynomial formulations to show that several rational and discrete network synthesis games, including the minimum cost spanning tree game, satisfy the assumptions of Owen's linear production game model. We also discuss computational issues related to finding and recognizing core points for these classes of games.     

A tabu search heuristic for the generalized minimum spanning tree problem

This paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem (GMSTP) known to be NP-hard. Given a graph whose vertex set is partitioned into clusters, the GMSTP consists of designing a minimum cost tree spanning all clusters. An attribute based tabu search heuristic employing new neighborhoods is proposed. An extended set of TSPLIB test instances for the GMSTP is generated and the heuristic is compared with recently proposed genetic algorithms. The proposed heuristic yields the best results for all instances. Moreover, an adaptation of the tabu search algorithm is proposed for a variation of the GMSTP in which each cluster must be spanned at least once.     

Strongly essential coalitions and the nucleolus of peer group games

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespecified collection of size polynomial in the number of players. We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore. As an application, we consider peer group games, and show that they admit at most 2n−1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2ⁿ⁻¹. We propose an algorithm that computes the nucleolus of an n-player peer group game in time directly from the data of the underlying peer group situation.Research supported in part by OTKA grant T030945. The authors thank a referee and the editor for their suggestions on how to improve the presentation     

Allocation processes in cooperative games

This paper deals with a temporal aspect of cooperative games. A solution of the game is reached through an allocation process. At each stage of the allocation process of a cooperative game a budget of fixed size is distributed among the players. In the first part of this paper we study a type of process that, at any stage, endows the budget to a player whose contribution to the total welfare, according to some measurements, is maximal. It is shown that the empirical distribution of the budget induced by each process of the family converges to a least square value of the game, one such value being the Shapley value. Other allocation processes presented here converge to the core or to the least core. Received: January 2001/Revised: July 2002 I am grateful to the Associate Editor and to the two anonymous referees of International Journal of Game Theory. This research was partially supported by the Israel Science Foundation, grant no. 178/99     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

Characterization sets for the nucleolus

. We introduce the concept of a characterization set for the nucleolus of a cooperative game and develop sufficient conditions for a collection of coalitions to form a characterization set thereof. Further, we formalize Kopelowitz's method for computing the nucleolus through the notion of a sequential LP process, and derive a general relationship between the size of a characterization set and the complexity of computing the nucleolus. Received May 1994/Revised version May 1997/Final version February 1998     

Stability and largeness of core for symmetric games

Largeness of the core is sufficient for stability of the core. In general the necessity is not known. In this paper we answer affirmatively the necessity for symmetric games. We also prove its equivalence to n specified vectors being imputations and also to the convexity of the lower boundary of the set of all acceptable pay-off vectors of the game. In this paper we establish the equivalence of a condition given by Shapley to the newly evolved condition, thereby give an alternate proof to Shapley’s condition. Received: January 1997/final version: March 1999     

Degree sum conditions for the existence of homeomorphically irreducible spanning trees

In 1990, Albertson, Berman, Hutchinson, and Thomassen proved a theorem which gives a minimum degree condition for the existence of a spanning tree with no vertices of degree 2. Such a spanning tree is called a homeomorphically irreducible spanning tree (HIST). In this paper, we prove that every graph of order $n$ ($n\ge 8$) contains a HIST if $d(u)+d(v)\ge n?1$ for any nonadjacent vertices $u$ and $v$. The degree sum condition is best possible.     

Algorithms for the optimum communication spanning tree problem

Optimum Communication Spanning Tree Problem is a special case of the Network Design Problem. In this problem given a graph, a set of requirements r _ij and a set of distances d _ij for all pair of nodes (i,j), the cost of communication for a pair of nodes (i,j), with respect to a spanning tree T is defined as r _ij times the length of the unique path in T, that connects nodes i and j. Total cost of communication for a spanning tree is the sum of costs for all pairs of nodes of G. The problem is to construct a spanning tree for which the total cost of communication is the smallest among all the spanning trees of G. The problem is known to be NP-hard. Hu (1974) solved two special cases of the problem in polynomial time. In this paper, using Hu’s result the first algorithm begins with a cut-tree by keeping all d _ij equal to the smallest d _ij . For arcs (i,j) which are part of this cut-tree the corresponding d _ij value is increased to obtain a near optimal communication spanning tree in pseudo-polynomial time. In case the distances d _ij satisfy a generalised triangle inequality the second algorithm in the paper constructs a near optimum tree in polynomial time by parametrising on the r _ij .     

Random minimal spanning tree and percolation on the N-cube

The N-cube is a graph with 2^N vertices and N2^N−1 edges. Suppose independent uniform random edge weights are assigned and let T be the spanning tree of minimal (total) weight. Then the weight of T is asymptotic to N⁻¹2^N ∑^∞_i=1 i⁻³ as N→∞. Asymptotics are also given for the local structure of T and for the distribution of its kth largest edge weight, k fixed. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 63–82, 1998