Exact algorithms for weighted and unweighted Borda manipulation problems

We propose O⁎((m⋅2m)t+1)-time and O⁎((t+m−1t)⋅(t+1)m)-time algorithms for the weighted and the unweighted Borda manipulation problems, respectively, where t is the number of manipulators and m is the number of candidates. In particular, for t=2, our algorithm for the unweighted case runs in O⁎(3m) time, affirmatively resolving one open problem posed by Betzler et al. [2]. As a byproduct of our results, we show that the unweighted Borda manipulation problem admits an algorithm with running time O⁎(29m2log⁡m), based on an integer linear programming technique. Finally, we prove that the unweighted Borda manipulation problem is polynomial-time solvable in single-peaked elections, in contrast to the NP-hardness in the general case.