Computational Complexity of Unweighted Coalitional Manipulation

In this chapter, we will study the computational complexity of manipuation for three common voting rules. We will prove that the unweighted coalitional manipulation problem (Definition 3.1.2) is NP-complete for maximin (Section 4.1) and ranked pairs (Section 4.2), and we will give a polynomial-time algorithm for UCM for Bucklin (Section 4.3). 4.1 Manipulation for Maximin is NP-complete In this section, we prove that the UCM problem for maximin is NP-complete. The proof uses a reduction from the two vertex disjoint paths in directed an-tisymmetric graph problem, which is known to be NP-complete (Fortune et al., 1980). Definition 4.1.1. The two vertex disjoint paths in directed graph problem is defined as follows. We are given a directed graph G and two disjoint pairs of vertices Ôu, u ½ Õ and Ôv, v ½ Õ, where u, u ½ , v, v ½ are all different from each other. We 48