  In an indivisible participatory budgeting (PB) framework, we have a limited
budget that is to be distributed among a set of projects, by aggregating the
preferences of voters for the projects. All the prior work on indivisible PB
assumes that each project has only one possible cost. In this work, we let each
project have a set of permissible costs, each reflecting a possible degree of
sophistication of the project. Each voter approves a range of costs for each
project, by giving an upper and lower bound on the cost that she thinks the
project deserves. The outcome of a PB rule selects a subset of projects and
also specifies their corresponding costs. We study different utility notions
and prove that the existing positive results when every project has exactly one
permissible cost can also be extended to our framework where a project has
several permissible costs. We also analyze the fixed parameter tractability of
the problem. Finally, we propose some important and intuitive axioms and
analyze their satisfiability by different PB rules. We conclude by making some
crucial remarks.
