3 research outputs found
Cost allocation in connection and conflict problems on networks: a cooperative game theoretic approach
This thesis examines settings where multiple decision makers with conflicting interests
benefit from cooperation in joint combinatorial optimisation problems. It draws on cooperative game theory, polyhedral theory and graph theory to address cost sharing in
joint single-source shortest path problems and joint weighted minimum colouring problems.
The primary focus of the thesis are problems where each agent corresponds to a
vertex of an undirected complete graph, in which a special vertex represents the common supplier. The joint combinatorial optimisation problem consists of determining the
shortest paths from the supplier to all other vertices in the graph. The optimal solution
is a shortest path tree of the graph and the aim is to allocate the cost of this shortest
path tree amongst the agents. The thesis defines shortest path tree problems, proposes
allocation rules and analyses the properties of these allocation rules. It furthermore introduces shortest path tree games and studies the properties of these games. Various core
allocations for shortest path tree games are introduced and polyhedral properties of the
core are studied. Moreover, computational results on finding the core and the nucleolus
of shortest path tree games for the application of cost allocation in Wireless Multihop
Networks are presented.
The secondary focus of the thesis are problems where each agent is interested in
having access to a number of facilities but can be in conflict with other agents. If two
agents are in conflict, then they should have access to disjoint sets of facilities. The
aim is to allocate the cost of the minimum number of facilities required by the agents
amongst them. The thesis models these cost allocation problems as a class of cooperative
games called weighted minimum colouring games, and characterises total balancedness
and submodularity of this class of games using the properties of the underlying graph
On the properties of weighted minimum colouring games
A weighted minimum colouring (WMC) game is induced by an undirected graph and a positive weight vector on its vertices. The value of a coalition in a WMC game is determined by the weighted chromatic number of its induced subgraph. A graph G is said to be globally (respectively, locally) WMC totally balanced, submodular, or PMAS-admissible, if for all positive integer weight vectors (respectively, for at least one positive integer weight vector), the corresponding WMC game is totally balanced, submodular or admits a population monotonic allocation scheme (PMAS). We show that a graph G is globally WMC totally balanced if and only if it is perfect, whereas any graph G is locally WMC totally balanced. Furthermore, G is globally (respectively, locally) WMC submodular if and only if it is complete multipartite (respectively, (2 K2, P4) -free). Finally, we show that G is globally PMAS-admissible if and only if it is (2 K2, P4) -free, and we provide a partial characterisation of locally PMAS-admissible graphs