Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the problem for c=1 and 2. The time complexities of the algorithms are O((k+m)log n) for c=1 and O(m^4log n+k^3+n(k^2+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|.
The MW-0-CPEMP is an extension of the longest common subsequence problem on given two sequences @cite_16 . It has an application in the sequence alignment problem that appears in bioinformatics @cite_12 and in natural language processing @cite_18 .
@cite_18 @cite_16 @cite_12
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The longest common subsequence problem is examined from the point of view of parameterized computational complexity. There are several different ways in which parameters enter the problem, such as the number of sequences to be analyzed, the length of the common subsequence, and the size of the alphabet. Lower bounds on the complexity of this basic problem imply lower bounds on a number of other sequence alignment and consensus problems. An issue in the theory of parameterized complexity is whether a problem which takes input (x, k) can be solved in time ƒ(k) · nα where α is independent of k (termed fixed-parameter tractability). It can be argued that this is the appropriate asymptotic model of feasible computability for problems for which a small range of parameter values covers important applications — a situation which certainly holds for many problems in biological sequence analysis. Our main results show that: 1. (1) The longest common subsequence (LCS) parameterized by the number of sequences to be analyzed is hard for W[t] for all t. 2. (2) The LCS problem, parameterized by the length of the common subsequence, belongs to W[P] and is hard for W[2]. 3. (3) The LCS problem parameterized both by the number of sequences and the length of the common subsequence, is complete for W[1]. All of the above results are obtained for unrestricted alphabet sizes. For alphabets of a fixed size, problems (2) and (3) are fixed-parameter tractable. We conjecture that (1) remains hard. The problem of finding a longest common subsequence of two strings is discussed. This problem arises in data processing applications such as comparing two files and in genetic applications such as studying molecular evolution. The difficulty of computing a longest common subsequence of two strings is examined using the decision tree model of computation, in which vertices represent “equal - unequal” comparisons. It is shown that unless a bound on the total number of distinct symbols is assumed, every solution to the problem can consume an amount of time that is proportional to the product of the lengths of the two strings. A general lower bound as a function of the ratio of alphabet size to string length is derived. The case where comparisons between symbols of the same string are forbidden is also considered and it is shown that this problem is of linear complexity for a two-symbol alphabet and quadratic for an alphabet of three or more symbols. We investigate combinatorial enumeration problems related to subsequences of strings; in contrast to substrings, subsequences need not be contiguous. For a finite alphabet E, the following three problems are solved. (1) Number of distinct subsequences: Given a sequence s ∈ Σ n and a nonnegative integer k ≤ n, how many distinct subsequences of length k does s contain? A previous result by Chase states that this number is maximized by choosing s as a repeated permutation of the alphabet. This has applications in DNA microarray production. (2) Number of ρ-restricted p-generated sequences: Given s ∈ Σ n and integers k ≥ 1 and p > 1, how many distinct sequences in Σ k contain no single nucleotide repeat longer than p and can be written as s[ r1 1 ... s rn n with 0 1, determine the number of sequences in E whose longest strictly increasing subsequence has length k, where 0 < k < K. This has applications to significance computations for chaining algorithms. Abstract In this paper we investigate the complexity of finding various kinds of common super- and subsequences with respect to one or two given sets of strings. We show that Longest Minimal Common Supersequence, Shortest Maximal Common Subsequence, and Shortest Maximal Common Non-Supersequence are MAX SNP-hard over a binary alphabet. Moreover, we show that Shortest Common Supersequence, Longest Common Subsequence, Longest Common Non-Supersequence, Shortest Common Non-Subsequence, and Longest Minimal Common Non-Supersequence are MAX SNP-hard over a binary alphabet if the number of zeros is fixed (by the instance). We show how these problems can be related to finding sequences consistent with respect to two given sets of strings. This leads to a unified approach for characterizing the complexity of such problems. Motivation: Alignment-based methods for sequence analysis have various limitations if large datasets are to be analysed. Therefore, alignment-free approaches have become popular in recent years. One of the best known alignment-free methods is the average common substring approach that defines a distance measure on sequences based on the average length of longest common words between them. Herein, we generalize this approach by considering longest common substrings with k mismatches. We present a greedy heuristic to approximate the length of such k-mismatch substrings, and we describe kmacs, an efficient implementation of this idea based on generalized enhanced suffix arrays. Results: To evaluate the performance of our approach, we applied it to phylogeny reconstruction using a large number of DNA and protein sequence sets. In most cases, phylogenetic trees calculated with kmacs were more accurate than trees produced with established alignment-free methods that are based on exact word matches. Especially on protein sequences, our method seems to be superior. On simulated protein families, kmacs even outperformed a classical approach to phylogeny reconstruction using multiple alignment and maximum likelihood. Availability and implementation: kmacs is implemented in C++, and the source code is freely available at http: kmacs.gobics.de Contact: chris.leimeister@stud.uni-goettingen.de Supplementary information: Supplementary data are available at Bioinformatics online. Abstract Alignment-free approaches are gaining persistent interest in many sequence analysis applications such as phylogenetic inference and metagenomic classification clustering, especially for large-scale sequence datasets. Besides the widely used k-mer methods, the average common substring (ACS) approach has emerged to be one of the well-known alignment-free approaches. Two recent works further generalize this ACS approach by allowing a bounded number k of mismatches in the common substrings, relying on approximation (linear time) and exact computation, respectively. Albeit having a good worst-case time complexity aastex amsbsy amsfonts amssymb bm mathrsfs pifont stmaryrd textcomp portland, xspace amsmath, amsxtra empty 10 9 7 6 @math log ^ ... The longest-common-prefix (LCP) array is an adjunct to the suffix array that allows many string processing problems to be solved in optimal time and space. Its construction is a bottleneck in practice, taking almost as long as suffix array construction. In this paper, we describe algorithms for constructing the permuted LCP (PLCP) array in which the values appear in position order rather than lexicographical order. Using the PLCP array, we can either construct or simulate the LCP array. We obtain a family of algorithms including the fastest known LCP construction algorithm and some extremely space efficient algorithms. We also prove a new combinatorial property of the LCP values. Let S = x"1, x"2, ... x"n be a sequence of n distinct elements from a linearly ordered set. We consider the problem of determining the length of the longest increasing subsequences of S. An algorithm which performs this task is described and is shown to perform n log n-n log log n + O(n) comparisons in its worst case. This worst case behavior is shown to be best possible. The problems of finding shortest common supersequences (SCS) and longest common subsequences (LCS) are two well-known NP -hard problems that have applications in many areas including computational molecular biology, data compression, robot motion planning and scheduling, text editing, etc. A lot of fruitless effort has been spent in searching for good approximation algorithms for these problems. In this paper, we show that these problems are inherently hard to approximate in the worst case. In particular, we prove that (i) SCS does not have a polynomial time linear approximation algorithm, unless P = NP ; (ii) There exists a constant @math such that, if SCS has a polynomial time approximation algorithm with ratio @math , where @math is the number of input sequences, then NP is contained in DTIME @math ; (iii) There exists a constant @math such that, if LCS has a polynomial time approximation algorithm with performance ratio @math , then P = NP . The proofs utilize the recent results of [Proc. 23rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 14-23] on the complexity of approximation problems. In the second part of the paper, we introduce a new method for analyzing the average-case performance of algorithms for sequences, based on Kolmogorov complexity. Despite the above nonapproximability results, we show that near optimal solutions for both SCS and LCS can be found on the average. More precisely, consider a fixed alphabet @math and suppose that the input sequences are generated randomly according to the uniform probability distribution and are of the same length @math . Moreover, assume that the number of input sequences is polynomial in @math . Then, there are simple greedy algorithms which approximate SCS and LCS with expected additive errors @math and @math for any @math , respectively. Incidentally, our analyses also provide tight upper and lower bounds on the expected LCS and SCS lengths for a set of random sequences, solving a generalization of another well-known open question on the expected LCS length for two random sequences [K. Alexander, The rate of convergence of the mean length of the longest common subsequence, 1992, manuscript],[V. Chvatal and D. Sankoff, J. Appl. Probab., 12 (1975), pp. 306-315], [D. Sankoff and J. Kruskall, eds., Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, Addison-Wesley, Reading, MA, 1983]. Alignment free sequence comparison methods are attracting persistent interest, driven by data-intensive applications in genome-wide molecular taxonomy and phylogentic reconstruction. Among the methods based on substring composition, the Average Common Substring ( ( ACS )) measure proposed by (RECOMB 2005) admits a straightforward linear time sequence comparison algorithm, while yielding impressive results in multiple applications. An important direction of research is to extend the approach to permit a bounded edit hamming distance between substrings, so as to reflect more accurately the evolutionary process. To date, however, algorithms designed to incorporate (k 1 ) mismatches have (O(kn^2) ) worst-case complexity, worse than the (O(n^2) ) alignment algorithms they are meant to replace. On the other hand, accounting for mismatches does show to lead to much improved classification, while heuristics can improve practical performance. In this paper, we close the gap by presenting the first provably efficient algorithm for the (k ) -mismatch average common string ( ( ACS _k )) problem that takes (O(n) ) space and (O(n ^ k+1 n) ) time in the worst case for any constant (k ). Our method extends the generalized suffix tree model to incorporate a carefully selected bounded set of perturbed suffixes, and can be applicable to other complex approximate sequence matching problems. We consider the complexity of computing a longest increasing subsequence (LIS) parameterised by the length of the output. Namely, we show that the maximal length k of an increasing subsequence of a permutation of the set of integers 1,2,...,n can be computed in time O(nloglogk) in the RAM model, improving the previous 30-year bound of O(nlogk). The algorithm also improves on the previous O(nloglogn) bound. The optimality of the new bound is an open question. Reducing the computation of a longest common subsequence (LCS) between two strings to an LIS computation leads to a simple O(rloglogk)-time algorithm for two sequences having r pairs of matching symbols and an LCS of length k. The problem of finding a longest common subsequence of two strings has been solved in quadratic time and space. An algorithm is presented which will solve this problem in quadratic time and in linear space. A natural problem in extremal combinatorics is to maximize the number of distinct subsequences for any length- @math string over a finite alphabet @math ; this value grows exponentially, but slower than @math . We use the probabilistic method to determine the maximizing string, which is a cyclically repeating string. The number of distinct subsequences is exactly enumerated by a generating function, from which we also derive asymptotic estimates. For the alphabet @math , @math has the maximum number of distinct subsequences, namely @math . We also consider the same problem with sub strings in lieu of sub sequences . Here, we show that an appropriately truncated de Bruijn word attains the maximum. For both problems, we compare the performance of random strings with that of the optimal ones. It is shown that the multiple alignment problem with SP-score is NP-hard for each scoring matrix in a broad class M that includes most scoring matrices actually used in biological applications. The problem remains NP-hard even if sequences can only be shifted relative to each other and no internal gaps are allowed. It is also shown that there is a scoring matrix M0 such that the multiple alignment problem for M0 is MAX-SNP-hard, regardless of whether or not internal gaps are allowed. Constraint Programming (CP) has proven to be an effective platform for constraint based sequence mining. Previous work has focused on standard frequent sequence mining, as well as frequent sequence mining with a maximum ’gap’ between two matching events in a sequence. The main challenge in the latter is that this constraint can not be imposed independently of the omnipresent frequency constraint. Indeed, the gap constraint changes whether a subsequence is included in a sequence, and hence its frequency. In this work, we go beyond that and investigate the integration of timed events and constraining the minimum maximum gap as well as minimum maximum span. The latter constrains the allowed time between the first and last matching event of a pattern. We show how the three are interrelated, and what the required changes to the frequency constraint are. Key in our approach is the concept of an extension window defined by gap span and we develop techniques to avoid scanning the sequences needlessly, as well as using a backtracking-aware data structure. Experiments demonstrate that the proposed approach outperforms both specialized and CP-based approaches in almost all cases and that the advantage increases as the minimum frequency threshold decreases. This paper is an extension of the original manuscript presented at CPAIOR’17 [5]. In the paper two combinatorial problems for the set Fnq of sequences of length n over the alphabet Fq= 0, 1, ?, q?1 are considered. The maximum size N?q(n, t) of the set of common subsequences of length n?t and the maximum size N+q(n, t) of the set of common supersequences of length n+t of two different sequences of Fnq are found for any nonnegative integers n and t. The number N?q(n, t)+1 (respectively, N+q(n, t)+1) is equal to the minimum number N of different subsequences of length n?t (supersequences of length n+t) of an unknown sequence X?Fnq which are sufficient for its reconstruction. Simple algorithms to recover X?Fnq from N?q(n, t)+1 of its subsequences of length n?t and from N+q(n, t)+1 of its supersequences of length n+t are given. This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA D-M2M with classic MOEA D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA D-M2M performs better. ABSTRACT We study the computational complexity of two popular problems in multiple sequence alignment: multiple alignment with SP-score and multiple tree alignment. It is shown that the first problem is NP-complete and the second is MAX SNP-hard. The complexity of tree alignment with a given phylogeny is also considered. The tree-to-tree correctmn problem Is to determine, for two labeled ordered trees T and T', the distance from T to T' as measured by the mlmmum cost sequence of edit operaUons needed to transform T into T' The edit operations investigated allow changing one node of a tree into another node, deleting one node from a tree, or inserting a node into a tree An algorithm Is presented which solves this problem m time O(V* V'*LZ* L'2), where V and V' are the numbers of nodes respectively of T and T', and L and L' are the maximum depths respectively of T and T' Possible apphcatmns are to the problems of measuring the similarity between trees, automatic error recovery and correction for programming languages, and determining the largest common substructure of two trees The problem of finding a center string that is "close" to everygiven string arises in computational molecular biology and codingtheory. This problem has two versions: the Closest String problemand the Closest Substring problem. Given a set of strings S= s1, s2, ...,sn , each of length m, the Closest Stringproblem is to find the smallest d and a string s of lengthm which is within Hamming distance d to eachsi e S. This problem comes fromcoding theory when we are looking for a code not too far away froma given set of codes. Closest Substring problem, with an additionalinput integer L, asks for the smallest d and a strings, of length L, which is within Hamming distance daway from a substring, of length L, of each si. This problemis much more elusive than the Closest String problem. The ClosestSubstring problem is formulated from applications in findingconserved regions, identifying genetic drug targets and generatinggenetic probes in molecular biology. Whether there are efficientapproximation algorithms for both problems are major open questionsin this area. We present two polynomial-time approximationalgorithms with approximation ratio 1 + e for any smalle to settle both questions.