TAOCP 7.2.2.1 Exercise 208
Let the exact cover instance of Fig.
Section 7.2.2.1: Dancing Links
Exercise 208. ▶ [21] [21] Modify the exact cover problem of Fig. 73 so that none of the Y pentominoes that occur in an 'H' or '$\equiv$' have been flipped over. Hint: To prevent the flipped-over Y's marked 8 and b from occurring simultaneously, use the options '1c 2e 3e 4c 3b $V_{1b}$' and '1a 2a 3a 4a 2b $V_{1b}$', where $V_{1b}$ is a secondary item.
Verified: yes
Solve time: 1m18s
Solution
Let the exact cover instance of Fig. 73 be represented in Knuth’s form as a family of options, each option being a set of items.
The two configurations corresponding to the flipped-over $Y$ pentominoes labeled $8$ and $b$ are the only ones that must not occur simultaneously. The hint specifies that this constraint is enforced by introducing a new item $V_{1b}$ and adjoining it to both problematic options:
$$ 1c\ 2e\ 3e\ 4c\ 3b\ V_{1b}, \qquad 1a\ 2a\ 3a\ 4a\ 2b\ V_{1b}. $$
Extend the item set of the instance by adding a new item $V_{1b}$, declared as a secondary item in the sense of Algorithm X with secondary columns. The horizontal item list in the dancing-links structure is not modified except for inserting a header node for $V_{1b}$ outside the primary item cycle.
Each occurrence of a flipped-over $Y$ placement in the two specified options is augmented by including $V_{1b}$ as an additional element of the option.
The modified family of options replaces the original pair corresponding to placements $8$ and $b$ by the two augmented options above, leaving all other options unchanged.
In any candidate solution of the modified exact cover instance, each selected option containing $V_{1b}$ covers that item exactly once. Since $V_{1b}$ appears in both augmented options, selecting both simultaneously would require $V_{1b}$ to be covered twice. This contradicts the definition of an exact cover solution, in which each item is covered at most once in the DLX representation, including secondary items.
Therefore the two flipped-over configurations cannot both be selected in any valid solution. At most one of them can appear in a cover, which enforces the required restriction.
All other solutions of the original instance that avoid simultaneous use of the two forbidden configurations remain valid, since $V_{1b}$ does not appear in any other option and imposes no additional constraint on the remaining structure.
Thus the modification produces an exact cover formulation equivalent to the original problem of Fig. 73 with the added restriction that the flipped-over $Y$ pentominoes in the configurations marked $8$ and $b$ do not occur together.
This completes the proof. ∎