Overlapping Coalition Formation: Charting the Tractability Frontier. Y. Zick , G. Chalkiadakis and E. Elkind (AAMAS 2012). Framework. Each p layer has a divisible resource (in our model, an integer weight) .
Y. Zick, G. Chalkiadakis and E. Elkind
Each player has a divisible resource (in our model, an integer weight).
A coalition is formed by agents contributing some of their weight to a certain collaborative task.
OCF Games [Chalkiadakis et. al, 2010]:
Goal: find an optimal coalition structure; divide coalitional payoffs in a stable manner.
Finding optimal coalition structures/stable payoff allocations is known to be NP-hard:
The objective of our work is to identify conditions that make optimization and stability tractable.
Agents may form coalitions of at most size 2:
The problem can be modeled as a graph
Agents are weighted nodes
computing an optimal allocation is NP-hard even for a single agent (the KNAPSACKproblem).
One agent with large weight – find the optimal set of tasks to complete.
computing an optimal allocation for a constant # of agents can be done in poly(W+1) time, where Wis the maximal weight of any agent.
Computational complexity: even when weights are at most 3, complex interactions cause NP-hardness (the X3C problem).
We assume that:
Suppose that the interaction graph is a tree
if the interaction graph is a tree, an optimal allocation can be computed in time linear in the # of agents and polynomial in (W+1).
Optimal resource allocation
Is (CS,x) in the core?
Arbitration functions: Given a set’s deviation from an outcome, how much will it get from surviving agreements with non-deviators?
if there is an efficient algorithm to compute the most one can get from global arbitration functions, then P = NP.
if the arbitration function is local, and the interaction graph is a tree, then one can verify if an outcome is stable in poly(n,W+1) time.
Bounded hyper-treewidth: Our results can be extended to graphs with bounded hyper-treewidth.
If the graph is “tree-like”we can still obtain efficient algorithms.
Computational Issues: A major obstacle in OCF games.
But:if interactions are (somewhat) local, both for values and arbitration functions, we can obtain poly-time algorithms.