Measuring satisfaction in societies with opinion leaders and mediators

An opinion leader-follower model (OLF) is a two-action collective decision-making model for societies, in which three kinds of actors are considered:Preprin

Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions

A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $\mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{\mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $\mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $\mathsf{poly}(k)$ vertices.Comment: Extended abstract appears in the proceedings of IPEC 202

Measuring satisfaction and power in influence based decision systems

We introduce collective decision-making models associated with influence spread under the linear threshold model in social networks. We define the oblivious and the non-oblivious influence models. We also introduce the generalized opinion leader–follower model (gOLF) as an extension of the opinion leader–follower model (OLF) proposed by van den Brink et al. (2011). In our model we allow rules for the final decision different from the simple majority used in OLF. We show that gOLF models are non-oblivious influence models on a two-layered bipartite influence digraph. Together with OLF models, the satisfaction and the power measures were introduced and studied. We analyze the computational complexity of those measures for the decision models introduced in the paper. We show that the problem of computing the satisfaction or the power measure is #P-hard in all the introduced models even when the subjacent social network is a bipartite graph. Complementing this result, we provide two subfamilies of decision models in which both measures can be computed in polynomial time. We show that the collective decision functions are monotone and therefore they define an associated simple game. We relate the satisfaction and the power measures with the Rae index and the Banzhaf value of an associated simple game. This will allow the use of known approximation methods for computing the Banzhaf value, or the Rae index to their practical computation.Peer ReviewedPostprint (author's final draft