Counting 4-Patterns in Permutations Is Equivalent to Counting 4-Cycles in Graphs

摘要

Permutation ?f appears in permutation ?? if there exists a subsequence of ?? that is order-isomorphic to ?f. The natural algorithmic question is to check if ?f appears in ??, and if so count the number of occurrences. Only since very recently we know that for any fixed length k, we can check if a given pattern of length k appears in a permutation of length n in time linear in n, but being able to count all such occurrences in f(k)a. n^o(k/log k) time would refute the exponential time hypothesis (ETH). Together with practical applications in statistics, this motivates a systematic study of the complexity of counting occurrences for different patterns of fixed small length k. We investigate this question for k = 4. Very recently, Even-Zohar and Leng [arXiv 2019] identified two types of 4-patterns. For the first type they designed an e?'aìf(n) time algorithm, while for the second they were able to provide an e?'aìf(n^1.5) time algorithm. This brings up the question whether the permutations of the second type are inherently harder than the first type. We establish a connection between counting 4-patterns of the second type and counting 4-cycles (not necessarily induced) in a sparse undirected graph. By designing two-way reductions we show that the complexities of both problems are the same, up to polylogarithmic factors. This allows us to leverage the work done on the latter to provide a reasonable argument for why there is a difference in the complexities for counting 4-patterns of the first and the second type. In particular, even for the seemingly simpler problem of detecting a 4-cycle in a graph on m edges, the best known algorithm works in e?'a(m^{4/3}) time. Our reductions imply that an e?'a(n^{4/3-?μ}) time algorithm for counting occurrences of any 4-pattern of the second type in a permutation of length n would imply an exciting breakthrough for counting (and hence also detecting) 4-cycles. In the other direction, by plugging in the fastest known algorithm for counting 4-cycles, we obtain an algorithm for counting occurrences of any 4-pattern of the second type in e?'a(n^1.48) time.