gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

curvature estimation along noisy digital contours by approximate global optimization in this paper we introduce a new curvature estimator along digital contours which we called global min curvature gmc estimator as opposed to previous curvature estimators it considers all the possible shapes that are digitized as this contour and selects the most probable one with a global optimization approach the gmc estimator exploits the geometric properties of digital contours by using local bounds on tangent directions defined by the maximal digital straight segments the estimator is then adapted to noisy contours by replacing maximal segments with maximal blurred digital straight segments experiments on perfect and damaged digital contours are performed and in both cases comparisons with other existing methods are presented

additivity obstructions for integral matrices and pyramids in discrete tomography there are two related notions of interest h uniqueness and h additivity of finite subsets of n m which are defined for certain finite sets h of linear subspaces of r m one knows complete sets of obstructions for h uniqueness bad h configurations and for h additivity weakly bad h configurations the classical case when h is the set of coordinate axes in r 2 is well known let h m denote the set of the m coordinate hyperplanes of r m the following question was raised in p c fishburn j c lagarias j a reeds l a shepp sets uniquely determined by projections on axes ii discrete case discrete math 91 1991 149 159 is there an upper bound on the weights of the bad h m configurations one needs to consider to determine h m uniqueness m 3 of an arbitrary set in n m this question can be asked for other sets h of linear subspaces and also for h additivity the answer to this question in the case of uniqueness is known when h is a set of lines in this paper we answer this question for uniqueness and additivity in the case of h 3 we show that there is no upper bound on the weights of the bad configurations resp weakly bad configurations one needs to consider to determine h 3 uniqueness resp h 3 additivity

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

reconstructing convex polyominoes from horizontal and vertical projections in 1 we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections we defined an algorithm that establishes the existence of a convex polyomino whose horizontal and vertical projections are equal to a pair of assigned vectors h v with h m and v n its computational cost is o n4m4 in this paper we introduce some operations for recontructing convex polyominoes by means of vectors h s and v s partial sums these operations allows us to define a new algorithm whose complexity is less than o n2m2

permutohedra and minimal matrices the notions of minimality uniqueness and additivity originated in discrete tomography they have applications to kronecker products of characters of the symmetric group and arise as the optimal solutions of quadratic transportation problems here we introduce the notion of real minimality and give geometric characterizations of all these notions for a matrix a by considering the intersection of the permutohedron determined by a with the transportation polytope in which a lies we also study the computational complexity of deciding if the properties of being additive real minimal unique and minimal hold for a given matrix and show how to efficiently construct some matrix with any of these properties

approximating binary images from discrete x rays we study the problem of approximating binary images that are accessible only through few evaluations of their discrete x ray transform i e through their projections counted with multiplicity along some lines this inverse discrete problem belongs to a class of generalized set partitioning problems and allows natural packing and covering relaxations for these mathbb np hard optimization problems we present various approximation algorithms and provide estimates for their worst case performance further we report on computational results for various variants of these algorithms in particular the corresponding integer programs are solved with only small absolute error for instances up to 250 000 binary variables

a full row rank system matrix generated by the strip based projection model in discrete tomography abstract let c u k be an underdetermined linear system generated by the strip based projection model in discrete tomography where c is row rank deficient in the case of one scanning direction the linear dependency of the rows of c is studied in this paper an index set h is specified such that if all rows of c with row indices in h are deleted then the rows of resultant matrix f are maximum linearly independent rows of c therefore the corresponding system f u k is equivalent to c u k and consequently the cost of an image reconstruction from f u k is reduced

on the comptational complexity of determining polyatomic structures by x rays the problem of recovering the structure of crystalline materials from their discrete x rays is of fundamental interest in many practical applications an important special case concerns determining the position of atoms of several different types in the integer lattice given the number of each type lying on each line parallel to some lattice directions we show that the corresponding consistency problem is np complete for any two or more different fixed directions when six or more types of atoms are involved

knot segmentation in 3d ct images of wet wood this paper proposes a fully automatic method to segment wood knots from images obtained by an x ray computed tomography scanner wood knot segmentation is known to be a difficult problem in the presence of sapwood because of the quite similar density of knots and wet sapwood classical segmentation techniques produce unsatisfactory results due to the very weak distinction between these two intensities to overcome this limitation caused by physical characteristics we propose to exploit the geometric properties of both the knot shapes and knot sapwood interface based on previous work related to automatic knot detection a new segmentation algorithm is proposed that uses a robust curvature estimation of 2d digital contours the segmentation process is fast easily parallelizable and produces better segmentation results than other state of the art algorithms it may be reproduced from the precise description given here or from source code available online

polarized mixed hodge structures and the local monodromy of a variation of hodge structure a variation of polarized hodge structures on the complement of a divisor with normal crossings gives rise locally on the divisor to a commuting set ni of nilpotent endomorphisms of the vector space underlying the variation the logarithms of the unipotent parts of the various picard lefschetz transformations these reflect properties of the singularities of the period map associated to the variation cf 5 7 9 and sect 3 for example a variation depending on a single parameter defines asymptotically a mixed hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism n 9 11 this paper is mainly concerned with the properties of the set n arising in the several parameters case in particular theorem 3 3 asserts that all the elements in the open polyhedral cone c spanned over r by the n s determine the same monodromy weight filtration it also describes the relationship between this common filtration and those associated to the faces of the cone this statement was conjectured by deligne based on his analogous result 4 1 9 2 for the q cone associated to those variations which arise from families of polarized non singular algebraic varieties in w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations section2 focuses on the case when each element in such set defines in the sense of 2 4 a polarized mixed hodge structure with a fixed hodge filtration f as is the case for the monodromy cone c of a variation these conditions imply cf 2 16 a generic uniqueness for the monodromy weight filtration we also consider those pola

all linear and integer programs are slim 3 way transportation programs we show that any rational convex polytope is polynomial time representable as a 3 way line sum transportation polytope of slim r c 3 format this universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure we provide a polynomial time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs our construction resolves several standing problems on 3 way transportation polytopes for example it demonstrates that unlike the case of 2 way contingency tables the range of values an entry can attain in any slim 3 way contingency table with specified 2 margins can contain arbitrary gaps our smallest such example has format 6 4 3 our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables for example it automatically provides new proofs for some classical results including a well known real feasible but integer infeasible 6 4 3 transportation polytope of m vlach and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator

regular switching components abstract we consider a problem of discrete tomography which consists of reconstructing a finite lattice set s z 2 from given horizontal and vertical x rays i e with prescribed numbers of points in each row and column without additional requirements the problem can be solved in polynomial time many variants require the solution to be in a chosen class a for instance the problem is np complete for the class a h v of hv convex lattice sets and it is polynomial for the class a h v p of hv convex polyominoes twenty years after these results the problem s complexity remains unknown for a c the class of c convex lattice sets i e two dimensional lattice polytopes the main difficulty to solve this problem comes from combinatorial structures called switching components switching components are closed path with horizontal and vertical edges such that the solutions s are composed of either the elements of even or of odd indices this binary choice can be encoded by a boolean variable associated with the switching component and the convexity constraints are simply encoded by sat clauses 2 clauses for hv convexity and 3 clauses for c convexity the purpose of the paper is to investigate the properties of the switching components and the consequences of the convexity requirements we divide the switching components in two classes regular if their turning angle is constant irregular otherwise we prove that adjacent regular switching components have the same boolean values this property allows us to merge them into extended switching components we prove that if all switching components are regular then the extended switching components are all independent then the number of solutions with the considered feet is 2 n where n is the number of extended switching components finally we prove that they are geometrically ordered

power watershed a unifying graph based optimization framework in this work we extend a common framework for graph based image segmentation that includes the graph cuts random walker and shortest path optimization algorithms viewing an image as a weighted graph these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework we then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm which we term the power watershed in particular when q 2 the power watershed leads to a multilabel scale and contrast invariant unique global optimum obtained in practice in quasi linear time placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation

on the computational complexity of reconstructing three dimensional lattice sets from their two dimensional x rays abstract a generalization of a classical discrete tomography problem is considered reconstruct three dimensional lattice sets from their two dimensional x rays parallel to three coordinate planes first we prove that this reconstruction problem is np hard then we propose some greedy algorithms that provide approximate solutions of the problem

the complexity of three way statistical tables multiway tables with specified marginals arise in a variety of applications in statistics and operations research we provide a comprehensive complexity classification of three fundamental computational problems on tables existence counting and entry security one outcome of our work is that each of the following problems is intractable already for slim 3 tables with constant number 3 of rows 1 deciding existence of 3 tables with specified 2 marginals 2 counting all 3 tables with specified 2 marginals 3 deciding whether a specified value is attained in a specified entry by at least one of the 3 tables having the same 2 marginals as a given table this implies that a characterization of feasible marginals for such slim tables sought by much recent research is unlikely to exist another consequence of our study is a systematic efficient way of embedding the set of 3 tables satisfying any given 1 marginals and entry upper bounds in a set of slim 3 tables satisfying suitable 2 marginals with no entry bounds this provides a valuable tool for studying multi index transportation problems and multi index transportation polytopes remarkably it enables us to automatically recover a famous example due to vlach of a real feasible integer infeasible collection of 2 marginals for 3 tables of smallest possible size 3 4 6

three dimensional statistical data security problems suppose there is a three dimensional table of cross tabulated nonnegative integer statistics and suppose that all of the row column and file sums are revealed together with the values in some of the individual cells in the table the question arises as to whether as a consequence the values contained in some of the other suppressed cells can be deduced from the information revealed the corresponding problem in two dimensions has been comprehensively studied by gusfield siam j comput 17 1988 pp 552 571 who derived elegant polynomial time algorithms for the identification of any such compromised cells and for calculating the tightest bounds on the values contained in all cells that follow from the information revealed in this note it is shown by contrast that the three dimensional version of the problem is np complete it is also shown that if the suggested row column and file sums for an unknown three dimensional table are given with or without the values in some of the cells the problem of determining whether there exists any table with the given sums is np complete in the course of proving these results the np completeness of some constrained latin square construction problems which are of some interest in their own right is established

frobenius modules and hodge asymptotics we exhibit a direct correspondence between the potential defining the h1 1 small quantum module structure on the cohomology of a calabi yau manifold and the asymptotic data of the a model variation of hodge structure this is done in the abstract context of polarized variations of hodge structure and frobenius modules

sets of uniqueness and minimal matrices abstract in this note we give an algebraic characterization of sets of uniqueness in terms of matrices with non negative integer coefficients and prescribed row and column sums and of the dominance order of partitions or majorization our proof uses some identities involving characters of the symmetric group as an application we describe all sets of uniqueness contained in the box b 2 2 r for all r

reconstructing polyatomic structures from discrete x rays we address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices a polyatomic structure t can be defined as an integer lattice in dimension d 2 whose points may be occupied by c distinct types of atoms to analyze t we conduct measurements that we call discrete x rays a discrete x ray in direction determines the number of atoms of each type on each line parallel to given such non parallel x rays we wish to reconstruct t the complexity of the problem for c 1 one atom type has been completely determined by gardner et al technical report 970 05012 techn univ munchen fak f math 1997 who proved that the problem is np complete for any dimension d 2 and 3 non parallel x rays and that it can be solved in polynomial time otherwise ryser mathematical association of america and quinn boden rahway new jersey 1963 the np completeness result above clearly extends to any c 2 and therefore when studying the polyatomic case we can assume that 2 as shown in another article by the same authors gardner et al theoret comput sci 1997 to appear this problem is also np complete for c 6 atoms even for dimension d 2 and axis parallel x rays gardner et al 1997 conjecture that the problem remains np complete for c 3 4 5 although as they point out the proof idea in gardner et al 1997 does not seem to extend to c 5 we resolve the conjecture from gardner et al 1997 by proving that the problem is indeed np complete for c 3 in 2d even for axis parallel x rays our construction relies heavily on some structure results for the realizations of 0 1 matrices with given row and column sums

on the algorithmic inversion of the discrete radon transform the present paper deals with the computational complexity of the discrete inverse problem of reconstructing finite point sets and more general functionals with finite support that are accessible only through some of the values of their discrete radon transform it turns out that this task behaves quite differently from its well studied companion problem involving 1 dimensional x rays concentrating on the case of coordinate hyperplanes in rd and on functionals zd d with d 0 1 r n0 for some arbitrary but fixed r we show in particular that the problem can be solved in polynomial time if information is available for m such hyperplanes when m d 1 but is np hard for m d and d 0 1 r however for d n0 a case that is relevant in the context of contingency tables the problem is still in p similar results are given for the task of determining the uniqueness of a given solution and for a related counting problem

on the reconstruction of binary and permutation matrices under binary tomographic constraints the paper studies the problem of reconstructing binary matrices constrained by binary tomographic information we prove new np hardness results that sharpen previous complexity results in the realm of discrete tomography but also allow applications to related problems for permutation matrices hence our results can be interpreted in terms of other combinatorial problems including the queens problem

duality of gaus manin systems associated to linear free divisors we investigate differential systems occurring in the study of particular non isolated singularities the so called linear free divisors we obtain a duality theorem for these mathcal d modules taking into account filtrations and deduce degeneration properties of certain frobenius manifolds associated to linear sections of the milnor fibres of the divisor

polyhedral combinatorics of multi index axial transportation problems abstract this paper studies integer points ip and integer vertices iv of the p index axial transportation polytope p atp of order n 1 n 2 n p n 1 n 2 n p 2 p 2 defined by integer vectors as well as noninteger vertices of the 3 atp in particular for the p atp we establish criteria for the minimum and maximum number of ips and describe the class of polytopes for which the number of ips coincides with the number of ivs for the 3 atp of order n n n we prove the theorem on the exponential growth of denominators of fractional components of the polytope vertices three conjectures are stated regarding the maximum number of vertices of the p atp the maximum number of ivs and the structure of the nondegenerate polytopes with the maximum number of ips

a construction of frobenius manifolds with logarithmic poles and applications a construction theorem for frobenius manifolds with logarithmic poles is established this is a generalization of a theorem of hertling and manin as an application we prove a partial generalization of the reconstruction theorem of kontsevich and manin for projective smooth varieties with convergent gromov witten potential a second application is a construction of frobenius manifolds out of a variation of polarized hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled

watershed cuts minimum spanning forests and the drop of water principle we study the watersheds in edge weighted graphs we define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface we first establish the consistency of these watersheds they can be equivalently defined by their catchment basinsrdquo through a steepest descent property or by the dividing linesrdquo separating these catchment basins through the drop of water principle then we prove through an equivalence theorem their optimality in terms of minimum spanning forests afterward we introduce a linear time algorithm to compute them to the best of our knowledge similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm both in theory and in practice finally the defined concepts are illustrated in image segmentation leading to the conclusion that the proposed approach improves on the tested images the quality of watershed based segmentations

random generation of 2 2 2 j contingency tables we propose two markov chains for sampling m 1 dimensional contingency tables indexed by 1 2 m 1 2 n stationary distributions of our chains are the uniform distribution and a conditional multinomial distribution which is equivalent to the hypergeometric distribution if m 1 mixing times of our chains are bounded by n n 1 ln n 2me n n 1 ln dn e where d is the average of the values in the cells and e is a given error bound we use the path coupling method for estimating the mixing times of our chains and showed that our chains mix rapidly

an introduction to periodical discrete sets from a tomographical perspective in this paper we introduce a new class of binary matrices whose entries show periodical configurations and we furnish a first approach to their analysis from a tomographical point of view in particular we propose a polynomial time algorithm for reconstructing matrices with a special periodical behavior from their horizontal and vertical projections we succeeded in our aim by using a reduction involving polyominoes which can be characterized by means of 2 sat formulas

x rays characterizing some classes of discrete sets in this paper we study the problem of determining discrete sets by means of their x rays an x ray of a discrete set f in a direction u counts the number of points in f on each line parallel to u a class f of discrete sets is characterized by the set u of directions if each element in f is determined by its x rays in the directions of u by using the concept of switching component introduced by chang and ryser comm acm 14 1971 21 combinatorial mathematics the carus mathematical monographs no 14 the mathematical association of america rahway 1963 and extended in discrete comput geom 5 1990 223 we prove that there are some classes of discrete sets that satisfy some connectivity and convexity conditions and that cannot be characterized by any set of directions gardner and gritzmann trans amer math soc 349 1997 2271 show that any set u of four directions having cross ratio that does not belong to 4 3 3 2 2 3 4 characterizes the class of convex sets we prove the converse that is if u s cross ratio is in 4 3 3 2 2 3 4 then the hv convex sets cannot be characterized by u we show that if the horizontal and vertical directions do not belong to u gardner and gritzmann s result cannot be extended to hv convex polyominoes if the horizontal and vertical directions belong to u and u s cross ratio is not in 4 3 3 2 2 3 4 we believe that u characterizes the class of hv convex polyominoes we give experimental evidence to support our conjecture moreover we prove that there is no number such that if u then u characterizes the hv convex polyominoes this number exists for convex sets and is equal to 7 see trans amer math soc 349 1997 2271