  The aim of this paper is to introduce models and algorithms for the
Participatory Budgeting problem when projects can interact with each other. In
this problem, the objective is to select a set of projects that fits in a given
budget. Voters express their preferences over the projects and the goal is then
to find a consensus set of projects that does not exceed the budget. Our goal
is to detect such interactions thanks to the preferences expressed by the
voters. Through the projects selected by the voters, we detect positive and
negative interactions between the projects by identifying projects that are
consistently chosen together. In presence of project interactions, it is
preferable to select projects that interact positively rather than negatively,
all other things being equal. We introduce desirable properties that utility
functions should have in presence of project interactions and we build a
utility function which fulfills the desirable properties introduced. We then
give axiomatic properties of aggregation rules, and we study three classical
aggregation rules: the maximization of the sum of the utilities, of the product
of the utilities, or of the minimal utility. We show that in the three cases
the problems solved by these rules are NP-hard, and we propose a branch and
bound algorithm to solve them. We conclude the paper by experiments.
