We formalize a framework for coordinating funding and selecting projects, the costs of which are shared among agents with quasi-linear utility functions and individual budgets. Our model contains the classical discrete participatory budgeting model as a special case, while capturing other useful scenarios. We propose several important axioms and objectives and study how well they can be simultaneously satisfied. We show that whereas welfare maximization admits an FPTAS, welfare maximization subject to a natural and very weak participation requirement leads to a strong inapproximability. This result is bypassed if we consider some natural restricted valuations, namely laminar single-minded valuations and symmetric valuations. Our analysis for the former restriction leads to the discovery of a new class of tractable instances for the Set Union Knapsack problem, a classical problem in combinatorial optimization.
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