Bribery Can Get Harder in Structured Multiwinner Approval Election

We study the complexity of constructive bribery in the context of structured multiwinner approval elections. Given such an election, we ask whether a certain candidate can join the winning committee by adding, deleting, or swapping approvals, where each such action comes at a cost and we are limited by a budget. We assume our elections to either have the candidate interval or the voter interval property, and we require the property to hold also after the bribery. While structured elections usually make manipulative attacks significantly easier, our work also shows examples of the opposite behavior. We conclude by presenting preliminary insights regarding the destructive variant of our problem.

💡 Research Summary

This paper investigates the computational complexity of constructive bribery in multi‑winner approval elections when the underlying preference profile exhibits a structured property: either the Candidate Interval (CI) property or the Voter Interval (VI) property. In such elections each voter’s approval set (or each candidate’s set of approving voters) forms a contiguous interval with respect to a given linear order of candidates or voters, respectively. The authors consider the classic approval‑based multi‑winner rule AV, which selects all size‑k committees with maximal total approval score, and they ask whether a distinguished candidate p can be forced to belong to at least one winning committee by performing a limited number of elementary bribery actions.

Three elementary actions are studied, following the model of Faliszewski et al. (2017b):

1. AddApprovals – add a candidate to a voter’s approval set.
2. DelApprovals – delete a candidate from a voter’s approval set.
3. SwapApprovals – replace one approved candidate by another in a voter’s set.

For each action two cost models are examined: (i) a uniform unit cost for every operation, and (ii) an individual cost that may depend on the specific voter–candidate pair (denoted by a preceding ‘$’). The bribery problem asks whether, given a budget B (or a bound on the number of unit operations), one can achieve the desired outcome while preserving the CI or VI structure after the modifications. The preservation requirement is standard in the literature on structured elections and reflects real‑world scenarios where the underlying order (e.g., a left‑right political spectrum or a chronological timeline) cannot be altered by the manipulator.

The main contribution is a near‑complete complexity classification for all six combinations (three actions × two cost models) under both CI and VI domains. The results are summarized in Table 1 of the paper and can be described as follows:

• AddApprovals is polynomial‑time solvable (P) in every setting—unrestricted, CI, and VI—regardless of the cost model. The interval structure actually simplifies the problem because one can greedily add approvals to the target candidate while maintaining contiguity.

• DelApprovals with uniform unit cost remains in P for both CI and VI, matching the unrestricted case. However, when each deletion has an individual cost, the problem becomes NP‑complete for CI elections while staying polynomial for VI elections. The hardness proof reduces from Cubic Independent Set: vertices correspond to candidates, and the budget encodes a selection of vertices that form an independent set. The interval constraint forces deletions to respect contiguity, which mirrors the independence condition.

• SwapApprovals where every swap must involve the preferred candidate p (i.e., swapping some other approved candidate for p) is NP‑complete in the unrestricted setting but drops to P for both CI and VI. The interval order fixes the relative positions of candidates, allowing a dynamic‑programming algorithm that decides which voters to convert to approve p without breaking the interval property.

• SwapApprovals with unrestricted swaps (any candidate may be swapped for any other) stays NP‑complete in all three environments, even when the cost is uniform. The authors give reductions from Restricted Exact Cover by 3‑Sets (RX3C), encoding each 3‑set as a voter’s interval and forcing a selection of swaps that exactly covers the universe. The interval constraints do not alleviate the combinatorial explosion because the choice of which candidate to replace is still free.

The paper also sketches the destructive variant, where the goal is to keep p out of every winning committee. While destructive bribery is usually easier than its constructive counterpart, the authors observe that for CI elections the destructive case appears more challenging; only limited results for unit‑cost models are presented, and a full classification remains open.

Methodologically, the polynomial‑time algorithms exploit the interval structure: for CI elections the candidate axis is fixed, and one can scan the axis while maintaining a feasible set of approvals; for VI elections the voter order is fixed, and analogous scanning is performed over voters. The NP‑hardness proofs carefully construct elections that satisfy the interval property both before and after bribery, ensuring that the reduction respects the structural constraint.

Two concrete motivating examples are provided. In the CI scenario, days of a month are candidates, a hotel’s staffing requirement k is the committee size, and each booking is an interval of days that a guest occupies. Adding, deleting, or moving bookings corresponds to bribery actions, and the cost reflects the likelihood of each change. The VI scenario mirrors a team of archaeologists assigning staff to excavation sites over consecutive days, again fitting the interval model. These examples illustrate how preserving the interval structure is essential for the real‑world interpretation of the problem.

The authors discuss the broader significance of their findings. While structured domains often render otherwise intractable problems easy (e.g., winner determination for many rules becomes polynomial), this work shows that complexity reversals can also occur: a problem that is easy in the unrestricted case can become hard when the domain is restricted, and vice versa. Such reversals are rare but have been observed previously only in a few contexts (e.g., weighted elections with three candidates). Here, the reversal is demonstrated for natural bribery actions and realistic cost models, highlighting that structural restrictions interact non‑trivially with the cost structure of manipulative actions.

The paper concludes with several avenues for future research: extending the analysis to multi‑candidate objectives, exploring other multi‑winner rules (e.g., Chamberlin‑Courant, Monroe), investigating additional structured domains (e.g., two‑dimensional grids), and developing fixed‑parameter or approximation algorithms for the NP‑hard cases identified. Moreover, a complete treatment of the destructive variant, especially under individual costs, is left as an open problem.

In summary, the work provides a comprehensive map of the computational landscape for bribery in structured multi‑winner approval elections, revealing both expected simplifications and surprising hardness results, and it bridges theoretical insights with practical scheduling and resource‑allocation scenarios.