18 research outputs found

    Splittability and 1-amalgamability of permutation classes

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    A permutation class CC is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations σ\sigma and τ\tau in CC, each with a marked element, we can find a permutation π\pi in CC containing both σ\sigma and τ\tau such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class Av(1423,1342)Av(1423, 1342) is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.Comment: 17 pages, 7 figure

    Long Paths Make Pattern-Counting Hard, and Deep Trees Make It Harder

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    A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes

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    Griddings of Permutations and Hardness of Pattern Matching

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    We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations ? (the "text") and ? (the "pattern"), and the goal is to decide whether ? contains ? as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern ?; this restriction is known as Av(?)-PPM. It has been previously shown that Av(?)-PPM is polynomial for any ? of size at most 3, while it is NP-hard for any ? containing a monotone subsequence of length four. In this paper, we present a new hardness reduction which allows us to show, in a uniform way, that Av(?)-PPM is hard for every ? of size at least 6, for every ? of size 5 except the symmetry class of 41352, as well as for every ? symmetric to one of the three permutations 4321, 4312 and 4231. Moreover, assuming the exponential time hypothesis, none of these hard cases of Av(?)-PPM can be solved in time 2^o(n/log n). Previously, such conditional lower bound was not known even for the unconstrained PPM problem. On the tractability side, we combine the CSP approach of Guillemot and Marx with the structural results of Huczynska and Vatter to show that for any monotone-griddable permutation class ?, PPM is polynomial when the text is restricted to a permutation from ?

    Generalized Coloring of Permutations

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    A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D. We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413

    The Hierarchy of Hereditary Sorting Operators

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    We consider the following general model of a sorting procedure: we fix a hereditary permutation class C\mathcal{C}, which corresponds to the operations that the procedure is allowed to perform in a single step. The input of sorting is a permutation π\pi of the set [n]={1,2,,n}[n]=\{1,2,\dotsc,n\}, i.e., a sequence where each element of [n][n] appears once. In every step, the sorting procedure picks a permutation σ\sigma of length nn from C\mathcal{C}, and rearranges the current permutation of numbers by composing it with σ\sigma. The goal is to transform the input π\pi into the sorted sequence 1,2,,n1,2,\dotsc,n in as few steps as possible. This model of sorting captures not only classical sorting algorithms, like insertion sort or bubble sort, but also sorting by series of devices, like stacks or parallel queues, as well as sorting by block operations commonly considered, e.g., in the context of genome rearrangement. Our goal is to describe the possible asymptotic behavior of the worst-case number of steps needed when sorting with a hereditary permutation class. As the main result, we show that any hereditary permutation class C\mathcal{C} falls into one of five distinct categories. Disregarding the classes that cannot sort all permutations, the number of steps needed to sort any permutation of [n][n] with C\mathcal{C} is either Θ(n2)\Theta(n^2), a function between O(n)O(n) and Ω(n)\Omega(\sqrt{n}), a function betwee O(log2n)O(\log^2 n) and Ω(logn),or\Omega(\log n), or 1$, and for each of these cases we provide a structural characterization of the corresponding hereditary classes

    Bears with Hats and Independence Polynomials

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    Consider the following hat guessing game. A bear sits on each vertex of a graph GG, and a demon puts on each bear a hat colored by one of hh colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess gg colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number μ^\hat{\mu}, arising from the hat guessing game. The parameter μ^\hat{\mu} is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of GG, and to compute the exact value of μ^\hat{\mu} of cliques, paths, and cycles
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