Finding and Counting Permutations via CSPs

Abstract

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is $$n^{k/4 + o(k)}$$ n k / 4 + o ( k ) , and a polynomial-space algorithm whose running time is the better of $$O(1.6181^n)$$ O ( 1 . 6181 n ) and $$O(n^{k/2 + 1})$$ O ( n k / 2 + 1 ) . These results improve the earlier best bounds of $$n^{0.47k + o(k)}$$ n 0.47 k + o ( k ) and $$O(1.79^n)$$ O ( 1 . 79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $$k \in \varOmega (\log {n})$$ k ∈ Ω ( log n ) . We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $$f(k) \cdot n^{o(k/\log {k})}$$ f ( k ) · n o ( k / log k ) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing (4321-avoiding) and 3-decreasing (1234-avoiding) permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes.

Four Shorts Stories on Surprising Algorithmic Uses of Treewidth

“…Theorem 6 (Berendsohn et al [6]). Given a length-k permutation π and length-n permutation σ, the number of occurrences of π in σ can be counted in time n k/3+o(k) .…”

“…The problem is often called Permutation Pattern Matching and is known to be NP-hard [16], but of course can be solved in time O(n k ) by brute force. Albert et al [3] improved this to O(n 2/3k+1 ) time, Ahal and Rabinovich [2] further improved it to n 0.47k+o(k) time, and Berendsohn et al [6] gave an n 0.25k+o(k) time algorithm. Guillemot and Marx [49] showed that Permutation Pattern Matching can be solved in time 2 O(k 2 log k) • n, that is, it is fixed-parameter tractable (FPT) parameterized by the length of π.…”

“…FPT algorithms for the problem, we need different type of algorithms to understand the complexity of the problem in the regime where k is large. Second, the n ck time algorithms [2,3,6] can be easily modified to count the total number of solutions, while the FPT algorithm of Guillemot and Marx [49] returns only a single solution. This is not just a shortcoming of the presentation [49]: the FPT algorithm contains a step where a certain structure is discovered that guarantees that every permutation of length k appears in σ.…”

“…Then the algorithm stops and does not look for any further occurences of π. Furthermore, it is unlikely that the algorithm can be extended to a counting version: Berendsohn et al [6] proved that the counting problem is #W[1]-hard.…”

“…The n ck algorithms for Permutation Pattern Matching [2,3,6] are implicitly or explicitly based on dynamic programming on a certain tree decomposition. Here we follow the presentation of Berendsohn et al [6], where it is shown how high-level arguments and previous results on treewidth can be combined to obtain an n k/3+o(k) time in a very clean way (a further improvement, based on a technical idea of Cygan et al [24], reduces the running time to n 0.25k+o(k) [6]).…”