Enumeration Complexity of Permutation Classes
Manuel Bodirsky, Mohamed Elsheikh
We study classes of permutations that are defined by avoidance of forbidden patterns. For each fixed finite set $B$ of forbidden patterns, we study the \emph{avoider enumeration problem}: given an integer $n$, output all $B$-avoiding permutations of length $n$. Avoider enumeration is output-polynomial for every fixed $B$. However, our main notion of efficiency is polynomial delay. We use the insert-maximum tree as a search tree, in which permutations are built by repeatedly inserting a new maximum while preserving pattern avoidance. This leads to the \emph{insert-maximum extension problem}, which asks if a given permutation has an avoiding descendant at a specified depth. An algorithm for this problem can prune branches that cannot reach the target length. Hence, if this problem is in P, we obtain polynomial-delay enumeration. To investigate this problem, we introduce and study \emph{pockets}, the maximal finite subtrees of the insert-maximum tree. For fixed $B$, if there is a polynomial $p$ such that every pocket rooted at $ρ$ has at most $p(|ρ|)$ nodes, then the extension problem is solvable in polynomial time and enumeration has polynomial delay. In particular, a bound on pocket depth depending only on $B$ suffices. We prove such a bound if the class is well-quasi-ordered, has a regular insertion encoding, or if $B$ omits one of the three maximum-position types: leftmost, interior, or rightmost. We also prove that extension is in NP for every fixed $B$, even with binary-encoded target depth. On the negative side, we construct a fixed set of $55$ forbidden patterns for which extension is NP-complete. Consequently, enumerating valid extensions from a given starting permutation cannot have polynomial delay unless P=NP. This also answers a question of Hanika by showing that the forbidden permutation constraint satisfaction problem can be NP-complete for a fixed forbidden set.
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