Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates

For many election systems, bribery (and related) attacks have been shown NP-hard using constructions on combinatorially rich structures such as partitions and covers. This paper shows that for voters who follow the most central political-science model of electorates— single-peaked preferences—those hardness protections vanish. By using single-peaked preferences to simplify combinatorial covering challenges, we for the first time show that NPhard bribery problems—including those for Kemeny and Llull elections—fall to polynomial time for single-peaked electorates. By using single-peaked preferences to simplify combinatorial partition challenges, we for the first time show that NP-hard partition-of-voters problems fall to polynomial time for single-peaked electorates. We show that for single-peaked electorates, the winner problems for Dodgson and Kemeny elections, though Θp2-complete in the general case, fall to polynomial time. And we completely classify the complexity of weighted coalition manipulation for scoring protocols in single-peaked electorates.