Coalition structure generation over graphs

We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph

Multi-type Fair Resource Allocation for Distributed Multi-Robot Systems

Fair resource allocation is essential to ensure that all resource requesters acquire adequate resources and accomplish tasks. We propose solutions to the fairness problem in multi-type resource allocation for multi-robot systems that have multiple resource requesters. We apply the dominant resource fairness (DRF) principle in our solutions to two different systems: single-tasking robots with multi-robot tasks (STR-MRT) and multi-tasking robots with single-robot tasks (MTR-SRT). In STR-MRT, each robot can perform only one task at a time, tasks are divisible, and accomplishing each task requires one or more robots. In MTR-SRT, each robot can perform multiple tasks at a time, tasks are not divisible, and accomplishing each task requires only one robot. We present centralized solutions to the fairness problem in STR-MRT. Meanwhile, we model the decentralized resource allocation in STR-MRT as a coordination game between the robots. Each robot subgroup is formed by robots that strategically select the same resource requester. For a requester associated with a specific subgroup, a consensus-based team formation algorithm further chooses the minimal set of robots to accomplish the task. We leverage the Deep Q-learning Network (DQN) to support requester selection. The results suggest that the DQN outperforms the commonly used Q-learning. Finally, we propose two decentralized solutions to promote fair resource allocation in MTR-SRT, as a centralized solution already exists. We first propose a task-forwarding solution in which the robots need to negotiate the placement of each task. In our second solution, each robot first selects resource requesters and then independently allocates resources to tasks that arrive from the selected requesters. The resource-requester selection phase of the latter solution models a coordination game that is solved by reinforcement learning. The experimental results suggest that both approaches outperform their baselines