the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

some 4 point hurwitz numbers in positive characteristic author s bouw irene i osserman brian abstract in this paper we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p our techniques involve related computations in the case of three branch points and allow us to conclude in many cases that for a particular choice of degeneration all the covers we consider degenerate to separable admissible covers starting from a good understanding of the complex case the proof is centered on the theory of stable reduction of galois covers

learning the optimal kernel for fisher discriminant analysis via second order cone programming kernel fisher discriminant analysis kfda is a popular classification technique which requires the user to predefine an appropriate kernel since the performance of kfda depends on the choice of the kernel the problem of kernel selection becomes very important in this paper we treat the kernel selection problem as an optimization problem over the convex set of finitely many basic kernels and formulate it as a second order cone programming socp problem this formulation seems to be promising because the resulting socp can be efficiently solved by employing interior point methods the efficacy of the optimal kernel selected from a given convex set of basic kernels is demonstrated on uci machine learning benchmark datasets

a new linear time algorithm to compute the genomic distance via the double cut and join distance the genomic distance problem in the hannenhalli pevzner hp theory is the following given two genomes whose chromosomes are linear calculate the minimum number of translocations fusions fissions and inversions that transform one genome into the other this paper presents a new distance formula based on a simple tree structure that captures all the delicate features of this problem in a unifying way and a linear time algorithm for computing this distance

the irreducibility of certain pure cycle hurwitz spaces we study pure cycle hurwitz spaces parametrizing covers of the projective line having only one ramified point over each branch point we start with the case of genus 0 covers using a combination of limit linear series theory and group theory to show that these spaces are always irreducible in the case of four branch points we also compute the associated hurwitz numbers finally we give a conditional result in the higher genus case requiring at least 3g simply branched points these results have equivalent formulations in group theory and in this setting complement results of conway fried parker v olklein

bayesian phylogenetic inference from animal mitochondrial genome arrangements summary the determination of evolutionary relationships is a fundamental problem in evolutionary biology genome arrangement data are potentially more informative than deoxyribonucleic acid sequence data for inferring evolutionary relationships between distantly related taxa we describe a bayesian framework for phylogenetic inference from mitochondrial genome arrangement data using markov chain monte carlo methods we apply the method to assess evolutionary relationships between eight animal phyla

knowledge based proximal support vector machines we propose a proximal version of the knowledge based support vector machine formulation termed as knowledge based proximal support vector machines kbpsvms in the sequel for binary data classification the kbpsvm classifier incorporates prior knowledge in the form of multiple polyhedral sets and determines two parallel planes that are kept as distant from each other as possible the proposed algorithm is simple and fast as no quadratic programming solver needs to be employed effectively only the solution of a structured system of linear equations is needed

the embedding problem over a hilbertian pac field we show that the absolute galois group of a countable hilbertian p seudo a lgebraically c losed field of characteristic 0 is a free profinite group of countably infinite rank theorem a as a consequence g q q is the extension of groups with a fairly simple structure e g 2 sn by a countably free group in addition we characterize those pac fields over which every finite group is a galois group as those with the rg hilbertian property

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

sorting permutations and binary strings by length weighted rearrangements abstract comparative genomics is a line of study which aims at determining dis similarities between genomes one way of doing this is to infer the large scale mutations also called genome rearrangements that are responsible for transforming one genome into another under the parsimony principle numerous efforts have been made over the past years regarding the traditional approach in which all genome rearrangements have the same unit cost however since long rearrangements are more likely to disturb the organism some works considered the length weighted approach in which the cost of each rearrangement is a function of its length the problem is thus modeled as follows given two genomes g 1 and g 2 and a set of allowed genome rearrangements each with a given cost determine their distance i e the total cost of a minimum cost sequence of rearrangements from that transforms g 1 into g 2 in this paper we study length weighted reversals and transpositions mainly considering their prefix variants in which the leftmost part of the genome must always be included we present approximation algorithms and bounds on the distances and diameters i e maximum possible distance over any two genomes of same length for eight problems when genomes are represented as permutations and another eight problems when genomes are represented as binary strings in each case considering two different cost functions

sorting by reversals and the theory of 4 regular graphs abstract we show that the theory of sorting by reversals fits into the well established theory of circuit partitions of 4 regular multigraphs which also involves the combinatorial structures of circle graphs and delta matroids in this way we expose strong connections between the two theories that have not been fully appreciated before we also discuss a generalization of sorting by reversals involving the double cut and join dcj operation finally we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

successor rules for flipping pancakes and burnt pancakes a stack of n pancakes can be rearranged in all n ways by a sequence of n 1 flips and a stack of n burnt pancakes can be rearranged in all 2 n n ways by a sequence of 2 n n 1 flips in both cases a computer program can efficiently generate suitable solutions we approach these tasks instead from a human perspective how can we determine the next flip directly from the current stack how can we flip the minimum or maximum number of burnt pancakes overall what if we are only allowed to flip the top n 2 n 1 or n burnt pancakes we answer the first question with simple successor rules that take worst case o n time and amortized o 1 time then we answer the second question exactly for minimization and provide conjectures for maximization for the third question we prove that solutions almost certainly exist for pancakes and burnt pancakes using only these three flips more broadly we discuss how efficiency and optimality can shape iterative solutions to hamilton cycle problems in highly symmetric graphs

linear programming approaches for multicategory support vector machines abstract we consider linear programming approaches for support vector machines svm the linear programming problems are introduced as an approximation of the quadratic programming problems commonly used in svm when we consider the kernel based nonlinear discriminators the approximation can be viewed as kernel principle component analysis which generates an important subspace from the feature space characterized the kernel function we show that any data points nonlinearly and implicitly projected into the feature space by kernel functions can be approximately expressed as points lying a low dimensional euclidean space explicitly which enables us to develop linear programming formulations for nonlinear discriminators we also introduce linear programming formulations for multicategory classification problems we show that the same maximal margin principle exploited in svm can be involved into the linear programming formulations moreover considering the low dimensional feature subspace extraction we can generate nonlinear multicategory discriminators by solving linear programming problems numerical experiments on real world datasets are presented we show that the fairly low dimensional feature subspace can achieve a reasonable accuracy and that the linear programming formulations calculate discriminators efficiently we also discuss a sampling strategy which might be crucial for huge datasets

robust kernel discriminant analysis using fuzzy memberships linear discriminant analysis lda is a simple but widely used algorithm in the area of pattern recognition however it has some shortcomings in that it is sensitive to outliers and limited to linearly separable cases to solve these problems in this paper a non linear robust variant of lda called robust kernel fuzzy discriminant analysis rkfda is proposed rkfda uses fuzzy memberships to reduce the effect of outliers and adopts kernel methods to accommodate non linearly separable cases there have been other attempts to solve the problems of lda including attempts using kernels however rkfda encompassing previous methods is the most general one furthermore theoretical analysis and experimental results show that rkfda is superior to other existing methods in solving the problems

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

connected components of affine primitive permutation groups abstract for a finite group g the hurwitz space h r g i n g is the space of genus g covers of the riemann sphere with r branch points and the monodromy group g in this paper we give a complete list of primitive genus one systems of affine type that is we assume that g is a primitive group of affine type under this assumption we determine the braid orbits on the suitable nielsen classes which is equivalent to finding connected components in h r 1 i n g furthermore we give a new algorithm for computing large braid orbits on nielsen classes this algorithm utilizes a correspondence between the components of h r 1 i n g and h r 1 i n m where m is the point stabilizer in g

whole genome duplications and contracted breakpoint graphs the genome halving problem motivated by the whole genome duplication events in molecular evolution was solved by el mabrouk and sankoff in the pioneering paper siam j comput 32 2003 pp 754 792 the el mabrouk sankoff algorithm is rather complex inspiring a quest for a simpler solution an alternative approach to the genome halving problem based on the notion of the contracted breakpoint graph was recently proposed in m a alekseyev and p a pevzner ieee acm trans comput biol bioinformatics 4 2007 pp 98 107 this new technique reveals that while the el mabrouk sankoff result is correct in most cases it does not hold in the case of unichromosomal genomes this raises a problem of correcting a flaw in the el mabrouk sankoff analysis and devising an algorithm that deals adequately with all genomes in this paper we efficiently classify all genomes into two classes and show that while the el mabrouk sankoff theorem holds for the first class it is incorrect for the second class the crux of our analysis is a new combinatorial invariant defined on duplicated permutations using this invariant we were able to come up with a full proof of the genome halving theorem and a polynomial algorithm for the genome halving problem

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

an efficient algorithm for one sided block ordering problem under block interchange distance abstract in this work we study the one sided block ordering problem under block interchange distance given two signed permutations and of size n where represents a partially assembled genome consisting of several blocks i e contigs and represents a completely assembled genome the one sided block ordering problem under block interchange distance is to order i e assemble the blocks of such that the block interchange distance between the assembly of and is minimized the one sided block ordering problem is useful in genome resequencing because its algorithms can be used to assemble the contigs of partially assembled resequencing genomes based on their completely assembled genomes by using permutation groups in algebra we design an efficient algorithm to solve the one sided block ordering problem under block interchange distance in o n log n time moreover we show that the assembly of can be done in o n time and its block interchange distance from can also be calculated in advance in o n time

a faster and simpler algorithm for sorting signed permutations by reversals we give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation our algorithm is faster than the previous algorithm of hannenhalli and pevzner and its faster implementation by berman and hannenhalli the algorithm is conceptually simple and does not require special data structures our study also considerably simplifies the combinatorial structures used by the analysis

on the complexity of unsigned translocation distance translocation is one of the basic operations for genome rearrangement translocation distance is the minimum number of translocations required to transform one genome into the other in this paper we show that computing the translocation distance for unsigned genomes is np hard moreover we show that approximating the translocation distance for unsigned genomes within ratio 1 00017 is np hard

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

1 ɛ approximation of sorting by reversals and transpositions gu et al gave a 2 approximation for computing the minimal number of inversions and transpositions needed to sort a permutation there is evidence that from the point of view of computational molecular biology a more adequate objective function is obtained if transpositions are given double weight we present a 1 e approximation for this problem based on the exact algorithm of hannenhalli and pevzner for sorting by reversals only

the evolution of the random reversal graph genomes are typically represented as signed permutations where each integer corresponds to a single gene and its sign indicates the orientation of the gene in this paper we study genome rearrangements via reversals sankoff and blanchette 1999 20 flipping entire segments of genes and changing their orientations this abstraction leads to the notion of the reversal graph in which two signed permutations are neighbors if they differ by one reversal we identify the evolution of multiple genomes within the random reversal graph i e the probability space consisting of subgraphs over signed permutations obtained by selecting edges reversals with independent probability n which is an important framework for the analysis of genome rearrangements in the sense of relating the reversal rate n with the global evolution of genomes our main result shows that the structure of the random reversal graph changes dramatically at n 1 n 1 2 for n 1 n 1 2 the random graph consists of components of size at most o n ln n a s and for 1 n 1 2 there emerges a unique largest component of size 2 n n a s where is the survival probability of a certain branching process this giant component is furthermore dense in the reversal graph

polynomial time sortable stacks of burnt pancakes pancake flipping a famous open problem in computer science can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible in that context a prefix reversal of length k reverses the order of the first k elements of the permutation the burnt variant of pancake flipping involves permutations of signed integers and reversals in that case not only reverse the order of elements but also invert their signs although three decades have now passed since the first works on these problems neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known in this work we prove a new lower bound for sorting burnt pancakes and show that an important class of permutations known as simple permutations can be optimally sorted in polynomial time