A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

The NP-complete Permutation Pattern Matching problem asks whether a $k$-permutation $P$ is contained in a $n$-permutation $T$ as a pattern. This is the case if there exists an order-preserving embedding of $P$ into $T$. In this paper, we present a fixed-parameter algorithm solving this problem with a worst-case runtime of $\mathcal{O}(1.79^{\mathsf{run}(T)}\cdot n\cdot k)$, where $\mathsf{run}(T)$ denotes the number of alternating runs of $T$. This algorithm is particularly well-suited for instances where $T$ has few runs, i.e., few ups and downs. Moreover, since $\mathsf{run}(T)<n$, this can be seen as a $\mathcal{O}(1.79^{n}\cdot n\cdot k)$ algorithm which is the first to beat the exponential $2^n$ runtime of brute-force search. Furthermore, we prove that under standard complexity theoretic assumptions such a fixed-parameter tractability result is not possible for $\mathsf{run}(P)$

Kernelization lower bound for Permutation Pattern Matching

A permutation $\pi$ contains a permutation $\sigma$ as a pattern if it contains a subsequence of length $|\sigma|$ whose elements are in the same relative order as in the permutation $\sigma$. This notion plays a major role in enumerative combinatorics. We prove that the problem does not have a polynomial kernel (under the widely believed complexity assumption \mbox{NP} \not\subseteq \mbox{co-NP}/\mbox{poly}) by introducing a new polynomial reduction from the clique problem to permutation pattern matching

Estimating Genome Reversal Distance by Genetic Algorithm

Sorting by reversals is an important problem in inferring the evolutionary relationship between two genomes. The problem of sorting unsigned permutation has been proven to be NP-hard. The best guaranteed error bounded is the 3/2- approximation algorithm. However, the problem of sorting signed permutation can be solved easily. Fast algorithms have been developed both for finding the sorting sequence and finding the reversal distance of signed permutation. In this paper, we present a way to view the problem of sorting unsigned permutation as signed permutation. And the problem can then be seen as searching an optimal signed permutation in all n2 corresponding signed permutations. We use genetic algorithm to conduct the search. Our experimental result shows that the proposed method outperform the 3/2-approximation algorithm

Pattern matching in (213,231)(213,231)-avoiding permutations

Given permutations $\sigma \in S_k$ and $\pi \in S_n$ with $k<n$, the \emph{pattern matching} problem is to decide whether $\pi$ matches $\sigma$ as an order-isomorphic subsequence. We give a linear-time algorithm in case both $\pi$ and $\sigma$ avoid the two size-$3$ permutations $213$ and $231$. For the special case where only $\sigma$ avoids $213$ and $231$, we present a $O(max(kn^2,n^2\log(\log(n)))$ time algorithm. We extend our research to bivincular patterns that avoid $213$ and $231$ and present a $O(kn^4)$ time algorithm. Finally we look at the related problem of the longest subsequence which avoids $213$ and $231$

Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations

Given a set $\Pi$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(\Pi)$, the set of permutations of length at most $n$ avoiding the patterns in $\Pi$, in time $O(|S_{\le n - 1}(\Pi)| \cdot k + |S_{n}(\Pi)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $\Pi$ in each permutation in $S_n$. Surprisingly, when $|\Pi| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutation analyzed, all of our algorithms take time at most polynomial per outputted permutation. If we want to solve only the enumerative variant of each problem, computing $|S_{\le n}(\Pi)|$ or tallying permutations according to $\Pi$-patterns, rather than to store information about every permutation, then all of our algorithms can be implemented in $O(n^{k+1}k)$ space. Using our algorithms, we generated $|S_5(\Pi)|, \ldots, |S_{16}(\Pi)|$ for each $\Pi \subseteq S_4$ with $|\Pi| > 4$, and analyzed OEIS matches. We obtained a number of potentially novel pattern-avoidance conjectures. Our algorithms extend to considering permutations in any set closed under standardization of subsequences. Our algorithms also partially adapt to considering vincular patterns

Robust Multimodal Graph Matching: Sparse Coding Meets Graph Matching

Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in sparsity-related techniques. We cast the problem, resembling group or collaborative sparsity formulations, as a non-smooth convex optimization problem that can be efficiently solved using augmented Lagrangian techniques. The method can deal with weighted or unweighted graphs, as well as multimodal data, where different graphs represent different types of data. The proposed approach is also naturally integrated with collaborative graph inference techniques, solving general network inference problems where the observed variables, possibly coming from different modalities, are not in correspondence. The algorithm is tested and compared with state-of-the-art graph matching techniques in both synthetic and real graphs. We also present results on multimodal graphs and applications to collaborative inference of brain connectivity from alignment-free functional magnetic resonance imaging (fMRI) data. The code is publicly available.Comment: NIPS 201

The computational landscape of permutation patterns

In the last years, different types of patterns in permutations have been studied: vincular, bivincular and mesh patterns, just to name a few. Every type of permutation pattern naturally defines a corresponding computational problem: Given a pattern P and a permutation T (the text), is P contained in T? In this paper we draw a map of the computational landscape of permutation pattern matching with different types of patterns. We provide a classical complexity analysis and investigate the impact of the pattern length on the computational hardness. Furthermore, we highlight several directions in which the study of computational aspects of permutation patterns could evolve.Comment: 23 pages, to appear in Journal of Pure and Applied Mathematics, Special Issue for Permutation Patterns 201

A General System for Heuristic Solution of Convex Problems over Nonconvex Sets

We describe general heuristics to approximately solve a wide variety of problems with convex objective and decision variables from a nonconvex set. The heuristics, which employ convex relaxations, convex restrictions, local neighbor search methods, and the alternating direction method of multipliers (ADMM), require the solution of a modest number of convex problems, and are meant to apply to general problems, without much tuning. We describe an implementation of these methods in a package called NCVX, as an extension of CVXPY, a Python package for formulating and solving convex optimization problems. We study several examples of well known nonconvex problems, and show that our general purpose heuristics are effective in finding approximate solutions to a wide variety of problems.Comment: 39 pages, 7 figure

CLEAR: A Consistent Lifting, Embedding, and Alignment Rectification Algorithm for Multi-View Data Association

Many robotics applications require alignment and fusion of observations obtained at multiple views to form a global model of the environment. Multi-way data association methods provide a mechanism to improve alignment accuracy of pairwise associations and ensure their consistency. However, existing methods that solve this computationally challenging problem are often too slow for real-time applications. Furthermore, some of the existing techniques can violate the cycle consistency principle, thus drastically reducing the fusion accuracy. This work presents the CLEAR (Consistent Lifting, Embedding, and Alignment Rectification) algorithm to address these issues. By leveraging insights from the multi-way matching and spectral graph clustering literature, CLEAR provides cycle consistent and accurate solutions in a computationally efficient manner. Numerical experiments on both synthetic and real datasets are carried out to demonstrate the scalability and superior performance of our algorithm in real-world problems. This algorithmic framework can provide significant improvement in the accuracy and efficiency of existing discrete assignment problems, which traditionally use pairwise (but potentially inconsistent) correspondences. An implementation of CLEAR is made publicly available online

Innovative Non-parametric Texture Synthesis via Patch Permutations

In this work, we present a non-parametric texture synthesis algorithm capable of producing plausible images without copying large tiles of the exemplar. We focus on a simple synthesis algorithm, where we explore two patch match heuristics; the well known Bidirectional Similarity (BS) measure and a heuristic that finds near permutations using the solution of an entropy regularized optimal transport (OT) problem. Innovative synthesis is achieved with a small patch size, where global plausibility relies on the qualities of the match. For OT, less entropic regularization also meant near permutations and more plausible images. We examine the tile maps of the synthesized images, showing that they are indeed novel superpositions of the input and contain few or no verbatim copies. Synthesis results are compared to a statistical method, namely a random convolutional network. We conclude by remarking simple algorithms using only the input image can synthesize textures decently well and call for more modest approaches in future algorithm design