Combinatorial participatory budgeting, also called indivisible participatory budgeting or budgeted social choice, is a problem in social choice. There are several candidate projects, each of which has a fixed costs. There is a fixed budget, that cannot cover all these projects. Each voter has different preferences regarding these projects. The goal is to find a budget-allocation - a subset of the projects, with total cost at most the budget, that will be funded. Combinatorial participatory budgeting is the most common form of participatory budgeting.

Combinatorial PB can be seen as a generalization of committee voting: committee voting is a special case of PB in which the "cost" of each candidate is 1, and the "budget" is the committee size. This assumption is often called the unit-cost assumption. The setting in which the projects are divisible (can receive any amount of money) is called portioning, fractional social choice, or budget-proposal aggregation.

PB rules have other applications besides proper budgeting. For example:

One class of rules aims to maximize a given social welfare function. In particular, the utilitarian rule aims to find a budget-allocation that maximizes the sum of agents' utilities. With cardinal voting, finding a utilitarian budget-allocation requires solving a knapsack problem, which is NP-hard in theory but can be solved easily in practice. There are also greedy algorithms that attain a constant-factor approximation of the maximum welfare.

There are many possible utility functions for a given rated ballot:

Sreedurga, Bhardwaj and Narahari study the egalitarian rule, which aims to maximize the smallest utility of an agent. They prove that finding an egalitarian budget-allocation is NP-hard, but give pseudo-polynomial time and polynomial-time algorithms when some natural paramerters are fixed. They propose an algorithm that achieves an additive approximation for restricted spaces of instances, and show that it gives exact optimal solutions on real-world datasets. They also prove that the egalitarian rule satisfies a new fairness axiom, which they call maximal coverage.

Annick Laruelle studies welfare maximization under weak ordinal voting, where a scoring rule is used to translate ranking to utility. She studies a greedy approximation to the utilitarian welfare and for the Chamberlin-Courant welfare. She tests three algorithms on real data from the PB in Portugalete in 2018; the results show that the algorithm including project costs in the ballot performs better than the other two.

Fluschnik, Skowron, Triphaus and Wilker study maximization of utilitarian welfare, Chamberlin-Courant welfare, and Nash welfare, assuming cardinal utilities.