Šimon Schierreich

Dušan Knop, Šimon Schierreich, Ondřej Suchý

Inspired by the famous Target Set Selection problem, we propose a new discrete model to simultaneously spread two opinions within a social network and perform an initial study of its complexity. Here, we are given a social network, a seed-set of agents for each opinion, two thresholds for each agent, a budget, and a number of rounds. The first threshold represents the willingness of an agent to adopt an opinion if the agent has no opinion at all, while the second threshold states the willingness to acquire a second opinion if the agent already has one. The goal is to add at most budget-many agents to the initial seed-sets such that the process started with these extended seed-sets stabilizes within the given number of rounds, with each agent having either both opinions or none. That is, our goal is to ensure that the spread of opinions is balanced. We show that the problem is NP-hard, and thus we study the problem from the perspective of parameterized complexity. In particular, we show that the problem is FPT when parameterized by the number of rounds, the maximum threshold, and the treewidth combined. This algorithm also applies to the combined parameter, the treedepth and the maximum threshold. Finally, we show that the problem is FPT when parameterized by the vertex cover number, the $3$-path vertex cover number, or the vertex integrity of the input network alone. To complement our tractability results, we show that the problem is W[1]-hard with respect to a) the sizes of the initial seed-sets and the feedback-vertex set number combined, even if all thresholds are bounded by a constant, and b) the budget, the 4-path vertex cover number, and the feedback-vertex set number combined, even if every activation process stabilizes in at most 4 rounds.

Šimon Schierreich, Ildikó Schlotter

Majority illusion is an undesirable phenomenon in social networks in which agents incorrectly perceive a minority opinion as dominant. This can severely distort collective behavior and decision-making. We study the fundamental question of detecting whether a social network allows for a majority illusion. Formally, in the $q$-Majority Illusion problem, we ask whether there exists a binary labeling of agents in which at least a $q$-fraction of agents have the majority of neighbors with the minority label. We investigate how various structural properties of the underlying social network influence the tractability of this question, and provide a detailed map of its computational complexity.

Katarína Cechlárová, Jörg Rothe, Šimon Schierreich et al.

We study the Possible President problem and the Necessary President problem for Schulze voting, a rule that, due to its many desirable axiomatic properties, is popular in practice. In both problems, we are given an election with the candidates partitioned into a set of parties, and we are interested in questions about a given distinguished party. In the Possible President problem, we ask whether it is possible for the parties to each nominate exactly one candidate such that the nominee of the distinguished party is a Schulze winner of the resulting election with only the nominees running. In the Necessary President problem, we ask whether the distinguished party's nominee is a Schulze winner of the resulting election, irrespective of the nomination from the other parties. Rothe and Woitaschik have shown that Possible President is NP-complete and Necessary President is coNP-complete for Schulze elections. We complement and improve their results by a more fine-grained analysis: we determine the parameterized complexity of both problems with respect to all possible parameterizations, where we consider each of three natural parameters -- the number of voters, the maximum party size, and the number of parties -- to be either a constant, a parameter, or unbounded. In particular, we obtain dichotomies regarding the number of voters for both problems.