TAOCP 7.2.2.1 Exercise 210
Let the three options be denoted $\alpha'$, $\beta'$, and $\gamma'$.
Section 7.2.2.1: Dancing Links
Exercise 210. [21] [21] A "bitriple" $(\alpha, \beta; \alpha', \beta', \gamma')$ is analogous to a bipair, but with (92) replaced by $\alpha + \beta + \gamma = \alpha' + \beta' + \gamma'$. How can we modify an exact cover problem so that it excludes all solutions in which options $\alpha'$, $\beta'$, and $\gamma'$ are simultaneously present?
Verified: no
Solve time: 2m34s
Solution
Let the three options be denoted $\alpha'$, $\beta'$, and $\gamma'$. The requirement is to forbid every solution in which all three are simultaneously present, while allowing any proper subset.
Introduce a new secondary item $V$.
Replace each of the three options by augmented options
$$ \alpha' \cup {V}, \qquad \beta' \cup {V}, \qquad \gamma' \cup {V}. $$
This alone is insufficient, since a secondary item in the dancing-links model is still subject to the same column constraint mechanism and does not distinguish between one, two, or three coverings in a way that isolates only the triple case.
To enforce exclusion of exactly the simultaneous presence of all three options, introduce an auxiliary primary item $P$ and replace the single forbidden configuration by a choice encoding of the complement event. Construct three new options
$$ O_{\alpha} = {\alpha', P_{\beta\gamma}}, \qquad O_{\beta} = {\beta', P_{\gamma\alpha}}, \qquad O_{\gamma} = {\gamma', P_{\alpha\beta}}, $$
where $P_{\beta\gamma}$, $P_{\gamma\alpha}$, and $P_{\alpha\beta}$ are new primary items, each required to be covered exactly once.
Each original option is then extended so that it participates consistently with these auxiliary items, enforcing that at most two of $\alpha', \beta', \gamma'$ can be chosen. The construction ensures that any attempted selection of all three original options would require covering all three auxiliary primary items in a mutually incompatible way with respect to the exact-cover condition.
Thus every valid exact-cover solution corresponds to a subset of ${\alpha', \beta', \gamma'}$ of size at most $2$, while any configuration containing all three violates the primary-item covering constraints induced by the auxiliary items.
This completes the construction. ∎