Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules

@inproceedings{Baumeister2010TakingTF,
  title={Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules},
  author={Dorothea Baumeister and J{\"o}rg Rothe},
  booktitle={Inf. Process. Lett.},
  year={2010}
}
Towards a dichotomy for the Possible Winner problem in elections based on scoring rules
The Complexity of the Possible Winner Problem over Partitioned Preferences
TLDR
This work considers the computational complexity of Possible-Winner under the assumption that the voter preferences are partitioned, and proves NP-hardness for a class of rules that contain all voting rules that produce scoring vectors with at least $4$ distinct values.
Computing the Extremal Possible Ranks with Incomplete Preferences
TLDR
It is established that the minimal and maximal positions are hard to compute (NP-hard) for every positional scoring rule, pure or not, and none of the tractable variants of necessary/possible winner determination remain tractable for extremal position determination.
A Control Dichotomy for Pure Scoring Rules
TLDR
The first dichotomy theorem for pure scoring rules for a control problem is obtained, which shows that CCAV is solvable in polynomial time for constructive control by adding voters (CCAV), and every pure scoring rule in which only the two top-rated candidates gain nonzero scores.
Determining Possible and Necessary Winners Given Partial Orders
TLDR
The complexity of possible/necessary winner problems for the following common voting rules are completely characterized: a class of positional scoring rules (including Borda), Copeland, maximin, Bucklin, ranked pairs, voting trees, and plurality with runoff.
Computational complexity of two variants of the possible winner problem
TLDR
The main result is that it is NP-complete to determine whether there is a scoring vector that makes c win the election, if the set of possible scoring vectors for an m-candidate election is restricted to those of the form (α1,..., αm−4, x1, x2, x3, 0).
The Possible Winner Problem with Uncertain Weights
TLDR
This work introduces a novel variant of this problem in which not some of the voters' preferences are uncertain but some of their weights, and presents a general framework to study this problem, both for integer and rational weights.
The Possible Winner Problem with Uncertain Weights 1
TLDR
This work introduces a novel variant of this problem in which not some of the voters’ preferences are uncertain but some of their weights, and presents a general work to study this problem, both for integer and rational weights.
The Complexity of Possible Winners on Partial Chains
TLDR
This work investigates the PW problem on partial chains, i.e., partial orders that are a total order on a subset of their domains, and establishes that the problem is NP-complete for all pure positional scoring rules other than the plurality and veto rules.
Classifying the Complexity of the Possible Winner Problem on Partial Chains
TLDR
It is shown that the PW problem on partial chains is NP-complete for all pure positional scoring rules other than the plurality rule and the veto rule, while, of course, for the latter two rules this problem remains in P.
...
...

References

SHOWING 1-10 OF 30 REFERENCES
Towards a dichotomy for the Possible Winner problem in elections based on scoring rules
Towards a Dichotomy of Finding Possible Winners in Elections Based on Scoring Rules
TLDR
It is shown that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.
Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders
TLDR
It is proved that for Copeland, maximin, Bucklin, and ranked pairs, the possible winner problem is NP-complete; also, a sufficient condition on scoring rules is given for the possiblewinner problem to be NP- complete (Borda satisfies this condition).
Computational complexity of two variants of the possible winner problem
TLDR
The main result is that it is NP-complete to determine whether there is a scoring vector that makes c win the election, if the set of possible scoring vectors for an m-candidate election is restricted to those of the form (α1,..., αm−4, x1, x2, x3, 0).
Voting procedures with incomplete preferences
TLDR
It is shown that the possible and necessary Condorcet winners for a partial preference profile can be computed in polynomial time as well and point out connections to vote manipulation and elicitation.
Unweighted Coalitional Manipulation under the Borda Rule Is NP-Hard
TLDR
This work settles the open problem of can one add a certain number of additional votes to an election such that a distinguished candidate becomes a winner and shows NP-hardness even for two manipulators and three input votes.
Possible winners when new alternatives join: new results coming up!
TLDR
It is shown that the PcWNA problems are NP-complete for the Bucklin, Copeland0, and maximin (a.k.a. Simpson) rule, even when the number of new alternatives is no more than a constant.
Complexity of unweighted coalitional manipulation under some common voting rules
TLDR
The main result is that UCM is NP-complete under the maximin rule; this resolves an enigmatic open question and provides an extreme hardness-of-approximation result for an optimization version of UCM under ranked pairs.
Dichotomy for voting systems
Manipulation of copeland elections
TLDR
The complexity of Copelandα-manipulation for each rational α e [0, 1] for the case of irrational voters is resolved and the problem remains NP-complete for α e {0,1}.
...
...