votes relative to c
′, but also helps d gain votes relative to c. Thus, candidate d **enforces our limit of adding**
voters, q.

In a constructive control by deleting candidates, there are a number of challenges to overcome in the addition of buffer voters and candidates. First, we must ensure that we cannot cheat by deleting either c
′
or d**. One way to overcome this is to clone both** c
′ and d**, making copies of these candidates, all with the**
same function, such that we cannot delete all of the copies of either. Also, to extend to k-approval or k**-veto**
by adding buffer candidates, we must also ensure that a valid control cannot involve deleting such buffer candidates. This can also be ensured by cloning each of the buffer candidates.

Theorem 22 k**-approval elections are computationally resistant to constructive control by adding or deleting**
candidates.

Proof: Consider an instance of Hitting Set, where we are given a set S = {s1, . . . , sm}, n **subsets of** S,
T1, . . . , Tn**, and positive integer 1** ≤ q ≤ m.

In our election, the candidate set will consist of

$$\begin{array}{r c l}{{C=\{c\}}}&{{\cup}}&{{\{c_{1}^{\prime},\ldots,c_{m-q+1}^{\prime}\}}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{\cup}}&{{\{d_{1},\ldots,d_{m-q+1}\}}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{}}\end{array}$$

and the voter set will consist of

V = {(c ≻ c ′ 1 ≻ · · · ≻ c ′ k−1 ≻ · · ·),(c ′ k ≻ · · · ≻ c ′ 2k−1 ≻ · · ·), . . . , (c ′ m−q−k+2 ≻ · · · ≻ c ′ m−q+1 ≻ · · ·) | 1 ≤ i ≤ (n + 2m)(m − q + 1) − m + q} ∪ {(d1 ≻ · · · ≻ dk ≻ · · ·),(dk+1 ≻ · · · ≻ d2k ≻ · · ·), . . . , (dm−q−k+1 ≻ · · · ≻ c ′ m−q+1 ≻ · · ·) | 1 ≤ i ≤ (n + 2m)(m − q + 1)} ∪ {(Ti ≻ x1 ≻ · · · ≻ xk−1 ≻ c ′ j ≻ · · ·) | 1 ≤ i ≤ n, 1 ≤ j ≤ m − q + 1} ∪ {(si ≻ x1 ≻ · · · ≻ xk−1 ≻ c ≻ · · ·) | 1 ≤ i ≤ m} ∪ {(si ≻ x1 ≻ · · · ≻ xk−1 ≻ c ′ j ≻ · · ·) | 1 ≤ i ≤ m, 1 ≤ j ≤ m − q + 1}.
$$\mathbf{}V$$
As seen in [16], adding a set of candidates corresponding to a hitting set of the established candidates C − S makes c **a winner and vice versa, as this is the same election with additional buffer candidates and**