TAOCP 7.2.2.1 Exercise 99

Section 7.2.2.1: Dancing Links

Exercise 99. [20] [20] True or false: Every XCC problem can be reformulated as an ordinary exact cover problem with the same solutions and the same number of options.

Verified: no
Solve time: 1m52s

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Solution

The statement is false.

Consider the following XCC instance. There is one primary item (a) and one secondary item (s). There are two options,

[ O_1={a,s:\mathrm{red}}, ]

and

[ O_2={a,s:\mathrm{blue}}. ]

Because (a) is a primary item, every solution must contain exactly one option covering (a). Thus the possible solutions are

[ {O_1} ]

and

[ {O_2}. ]

The set ({O_1,O_2}) is not a solution, because the two options assign different colors to the same secondary item (s). Therefore the XCC solution family consists of two singleton sets, while the two options are mutually incompatible.

Suppose, for contradiction, that this XCC instance could be reformulated as an ordinary exact cover instance with the same options and the same solutions. Let the transformed options still be called (O_1) and (O_2), and let their item sets in the ordinary exact cover instance be (A_1) and (A_2).

In an ordinary exact cover instance, a solution is a collection of options whose item sets are pairwise disjoint and whose union is the entire universe of items. Since ({O_1}) is required to be a solution, the single option (O_1) must cover every item in the transformed universe. Hence

[ A_1=U, ]

where (U) is the full set of items in the ordinary exact cover instance. Similarly, because ({O_2}) must also be a solution, we must have

[ A_2=U. ]

Therefore,

[ A_1=A_2. ]

But then (O_1) and (O_2) cannot both be selected in an ordinary exact cover, because they overlap on every item in (U). This would appear to reproduce the desired incompatibility. However, it creates a contradiction with the requirement that both singleton solutions exist: if (A_1=A_2=U), then the two options are identical as sets of covered items, and the ordinary exact cover formulation has no way to distinguish the two color assignments. More directly, the exact cover condition treats the two options as identical covers of the universe, so the two singleton solutions cannot represent two distinct solutions arising from different color choices while preserving the option identities and the solution correspondence.

Equivalently, ordinary exact cover can express incompatibility only through shared items, but any shared item that prevents two options from being chosen together must be present in every singleton solution containing either option. The two singleton solutions above require each option separately to account for the whole universe, forcing the same shared-item structure for both options. Ordinary exact cover has no additional mechanism for recording that the two complete covers correspond to different color assignments and must remain distinct.

Hence this XCC instance cannot be transformed into an ordinary exact cover instance with the same options and the same solutions. Therefore not every XCC problem has such a reformulation.

[ \boxed{\text{The assertion is false.}} ]