TAOCP 7.1.4 Exercise 212

The flaw in the previous solution is that it tried to define ZDD nodes as states indexed by a subset $X \subseteq U$.

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Section 7.1.4: Binary Decision Diagrams

Exercise 212. ▶ [25] [25] What’s a good way to compute the ZDD for an exact cover problem?

Verified: no
Solve time: 4m49s

The flaw in the previous solution is that it tried to define ZDD nodes as states indexed by a subset $X \subseteq U$. This turns the construction into an exponential dynamic program over $2^{|U|}$ states and is not the intended ZDD construction method. A correct approach must construct the ZDD structurally from the exact cover instance using recursive decomposition over constraints, while ensuring that the resulting structure is a reduced ZDD over a fixed ordering of the sets.

We reformulate the problem in the standard exact cover form. Let $U$ be the set of items and let $\mathcal{S} = {S_1,\dots,S_m}$ be subsets of $U$. An exact cover is a subfamily $\mathcal{T} \subseteq \mathcal{S}$ such that every element of $U$ appears in exactly one set in $\mathcal{T}$, and chosen sets are pairwise disjoint.

We construct a ZDD whose paths represent exactly the valid subfamilies $\mathcal{T}$.

Recursive structure of the construction

We define a recursive function $Z(P)$, where $P$ is an exact cover subproblem consisting of:

  • a current set of remaining items $U_P \subseteq U$,
  • a current set of available sets $\mathcal{S}_P \subseteq \mathcal{S}$, each still contained in $U_P$.

The initial problem is $Z(U, \mathcal{S})$.

The ZDD for $P$ is defined by structural decomposition on an item.

If $U_P = \varnothing$, then all constraints are satisfied and the only solution is the empty choice:

$$ Z(P) = \top. $$

If there exists an item $x \in U_P$ that is contained in no set of $\mathcal{S}_P$, then no exact cover is possible:

$$ Z(P) = \bot. $$

Otherwise, choose an item $x \in U_P$. This choice is a branching point driven by the constraint that $x$ must be covered exactly once. Every solution must select exactly one set $S \in \mathcal{S}_P$ with $x \in S$.

For each such set $S$, define the reduced subproblem $P_S$ as follows:

  • Remove all items in $S$ from $U_P$,
  • Remove from $\mathcal{S}_P$ every set that intersects $S$, since those would violate disjointness,
  • Restrict remaining sets to the reduced universe.

Formally,

$$ U_{P_S} = U_P \setminus S, $$

$$ \mathcal{S}_{P_S} = {T \in \mathcal{S}_P \mid T \cap S = \varnothing}. $$

Then every exact cover of $P$ is obtained uniquely by choosing one such $S$ and combining it with a solution of $P_S$. Hence the solution family satisfies

$$ \mathcal{Z}(P) = \biguplus_{S \in \mathcal{S}_P: x \in S} {, S \cup T : T \in \mathcal{Z}(P_S),}. $$

ZDD construction from the recursion

We now interpret this recursion as a ZDD construction.

Each recursive call $Z(P)$ corresponds to a ZDD node representing the family of partial solutions of $P$. The outgoing edges are determined by the choices of sets $S$ containing the chosen item $x$.

To fit the ZDD model over the fixed ordering $S_1 \prec \cdots \prec S_m$, each choice of $S_i$ is implemented as including variable $S_i$, while all sets that intersect $S_i$ are excluded in the corresponding subproblem. This enforces consistency with the exact cover constraint.

Thus, conceptually:

  • the decision “use $S_i$” corresponds to the 1-edge of the ZDD,
  • all incompatible sets are forced absent in the resulting subproblem,
  • the 0-edge corresponds to skipping $S_i$ when it is not chosen for covering the selected item.

Sharing and canonicity

The crucial computational improvement is that ZDD nodes are not indexed by subsets of $U$, but by the induced residual subproblem $P$. Two recursive calls that yield isomorphic subproblems (same remaining items and same remaining sets up to identity) produce identical ZDD nodes.

Hence we memoize on canonical encodings of $P$ (for example, bitsets for remaining items and remaining sets restricted by earlier decisions). This guarantees that the construction produces a directed acyclic graph rather than a tree.

After memoization, standard ZDD reduction rules apply:

  • elimination of nodes whose 1-child is $\bot$,
  • merging of isomorphic subgraphs,
  • zero-suppression, ensuring that variables (sets) that cannot appear in any solution are omitted.

Correctness

We prove correctness by induction on $|U_P|$.

If $U_P = \varnothing$, the construction returns $\top$, which represents the empty family, matching the unique exact cover of an already satisfied instance.

Otherwise, every solution must choose exactly one set $S$ containing the selected item $x$. The recursion enumerates exactly these choices, and the reduction to $P_S$ correctly removes all conflicting sets and covered items, ensuring that no element can be covered twice.

Conversely, any path from the root to $\top$ corresponds to a sequence of consistent choices of sets, each resolving a distinct item, and the disjoint-union decomposition ensures that every exact cover arises from exactly one such path.

Therefore, the constructed reduced and memoized ZDD represents exactly the family of all exact covers of $U$, and it is obtained by a structural recursion over constraints rather than an explicit enumeration of subsets of $U$.