# Unweighted coalitional manipulation under the Borda rule Is NP-hard

IJCAI, pp. 55-60, 2011.

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Abstract:

The Borda voting rule is a positional scoring rule where, for m candidates, for every vote the first candidate receives m- 1 points, the second m- 2 points and so on. A Borda winner is a candidate with highest total score. It has been a prominent open problem to determine the computational complexity of UNWEIGHTED COALITIONAL MANIPULATION...More

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Introduction

- In their recent overview on “AI’s war on manipulation” Faliszewski and Procaccia [2010] write “An enigmatic open problem is the complexity of Unweighted Coalitional Manipulation under Borda.” Here, we settle this open problem by showing NP-hardness for UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA,1 which we subsequently refer to as BORDA MANIPULATION.
- Zuckerman et al [2009] showed that for BORDA MANIPULATION a greedy algorithm can always find a set of x manipulators if the given input allows x − 1 manipulators to make a distinguished candidate win.
- In other words, this means that the optimization version of BORDA MANIPULATION is polynomial-time approximable with an additive error one.
- WEIGHTED COALITIONAL MANIPULATION UNDER BORDA is known to be NP-hard even for three candidates [Conitzer et al, 2007; Hemaspaandra and Hemaspaandra, 2007]

Highlights

- In their recent overview on “AI’s war on manipulation” Faliszewski and Procaccia [2010] write “An enigmatic open problem is the complexity of Unweighted Coalitional Manipulation under Borda.” Here, we settle this open problem by showing NP-hardness for UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA,1 which we subsequently refer to as BORDA MANIPULATION
- We show that BORDA MANIPULATION remains NP-hard for three input votes and for any other number greater than three
- We investigate the structure of instances resulting from the NP-hardness reduction
- We showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators
- Our NP-hardness proof is of theoretical nature in the sense that it is a purely worst-case result with little impact on practical aspects of solving BORDA MANIPULATION

Conclusion

- The authors showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators.
- The authors' NP-hardness proof is of theoretical nature in the sense that it is a purely worst-case result with little impact on practical aspects of solving BORDA MANIPULATION.
- This motivates the issue of parameterizing NP-hard problems such as BORDA MANIPULATION in the spirit of multivariate algorithmics [Niedermeier, 2010].
- It is of interest whether in case of two manipulators one can solve the problem in less than O(|C|!) time

Summary

## Introduction:

In their recent overview on “AI’s war on manipulation” Faliszewski and Procaccia [2010] write “An enigmatic open problem is the complexity of Unweighted Coalitional Manipulation under Borda.” Here, we settle this open problem by showing NP-hardness for UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA,1 which we subsequently refer to as BORDA MANIPULATION.- Zuckerman et al [2009] showed that for BORDA MANIPULATION a greedy algorithm can always find a set of x manipulators if the given input allows x − 1 manipulators to make a distinguished candidate win.
- In other words, this means that the optimization version of BORDA MANIPULATION is polynomial-time approximable with an additive error one.
- WEIGHTED COALITIONAL MANIPULATION UNDER BORDA is known to be NP-hard even for three candidates [Conitzer et al, 2007; Hemaspaandra and Hemaspaandra, 2007]
## Conclusion:

The authors showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators.- The authors' NP-hardness proof is of theoretical nature in the sense that it is a purely worst-case result with little impact on practical aspects of solving BORDA MANIPULATION.
- This motivates the issue of parameterizing NP-hard problems such as BORDA MANIPULATION in the spirit of multivariate algorithmics [Niedermeier, 2010].
- It is of interest whether in case of two manipulators one can solve the problem in less than O(|C|!) time

- Table1: Two manipulative votes v1 and v2 illustrating the strategy used in the proof of Proposition 4.2

Funding

- NB was supported by the DFG project “PAWS”, NI 369/10

Reference

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Best Paper of IJCAI, 2011

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