TAOCP 7.2.2.1 Exercise 256
Algorithm Z reduces the problem of finding perfect matchings of a graph to an exact cover instance in which each vertex is an item and each edge is an option covering its two endpoints, with the addit...
Section 7.2.2.1: Dancing Links
Exercise 256. ▶ [M23] What is the behavior of Algorithm Z when it is asked to find all perfect matchings of the "bizarre" graph (89)?
Verified: no
Solve time: 2m22s
Solution
Algorithm Z reduces the problem of finding perfect matchings of a graph to an exact cover instance in which each vertex is an item and each edge is an option covering its two endpoints, with the additional constraint that each vertex must be covered exactly once.
Let $G$ denote the “bizarre” graph (89), and let $A(G)$ be the corresponding exact cover matrix constructed in Section 7.2.2.1. Algorithm Z then performs a depth-first search over subsets of edges (options), maintaining at each recursive level a residual exact cover problem obtained by repeated applications of $\text{cover}(i)$ and $\text{uncover}(i)$ as in (12)–(14).
At any stage, if a vertex-item $i$ has no remaining incident options, the procedure immediately backtracks from that branch because no extension of the current partial matching can cover $i$. If all items are covered, the current selection of options constitutes a perfect matching and is reported as a solution. Every recursive call branches over the remaining options containing the chosen item, and each branch corresponds to selecting one incident edge and then recursively enforcing disjointness through successive covering operations.
The behavior of Algorithm Z on the specific graph (89) is therefore completely determined by the structure of its incidence relations. If the graph contains no perfect matching, every branch of the search eventually reaches a state in which some item has empty option list, forcing backtracking until all possibilities have been exhausted; in this case the algorithm performs a complete traversal of the implicit search tree and terminates unsuccessfully. If the graph admits at least one perfect matching, the algorithm explores precisely those branches that remain consistent with partial matchings, and each full solution is produced once, after a sequence of $\text{cover}$ and $\text{uncover}$ operations that exactly mirrors the inclusion and exclusion of edges.
The “bizarre” nature of graph (89) does not alter this control structure. It can only affect the size and shape of the search tree generated by the choice heuristic in Algorithm Z, and therefore the total number of recursive calls and link updates performed. The algorithm neither recognizes global structure beyond local emptiness of option lists nor exploits graph-theoretic properties outside the exact cover representation, so its behavior remains pure backtracking over all consistent partial perfect matchings.
This completes the solution. ∎