depth first search and kuratowski subgraphs soit g v e un graphe non planaire on montre la methode utilisant les techniques de 1ere profondeur pour extraire sur un sous graphe de kuratowsky en un temps o ivi

on the digraphs of sign solvable linear systems abstract a linear system of equations ax b is sign solvable if it is solvable and both its solvability and the sign pattern of its solution vector x are uniquely determined by the sign patterns of a and b it was proved in l bassett j maybee j quirk economitrica 36 1968 544 563 r a bruali b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 that the sign solvability of a linear system ax b in standard form and with a square coefficient matrix a can be described in terms of the signed digraph s a of the matrix a with a specified vertex subset depending on the sign pattern of the vector b in this paper we define the notion of w sign solvable signed digraphs which represent precisely the sign solvable linear systems in standard form we give necessary and sufficient conditions for the underlying unsigned digraphs of w sign solvable signed digraphs in terms of forbidden subdigraphs we also give several necessary and sufficient conditions for strongly connected w sign solvable signed digraphs

nonsingularity criteria for matrices involving combinatorial considerations abstract we survey related nonsingularity criteria for matrices which involve combinatorial considerations generally involving combinatorial restrictions on the matrices considered and conditions on various graphs of those matrices these criteria include the existence of nonzero diagonal products for matrices over arbitrary fields sign nonsingularity due to j maybee potential nonsingularity and stable nonsingularity due to w anderson all for real matrices and some recent work for matrices over arbitrary fields involving a closure operator on matrices due to the author and d hershkowitz

controlled lagrangians and the stabilization of mechanical systems ii potential shaping for pt i see ibid vol 45 p 2253 70 2000 we extend the method of controlled lagrangians cl to include potential shaping which achieves complete state space asymptotic stabilization of mechanical systems the cl method deals with mechanical systems with symmetry and provides symmetry preserving kinetic shaping and feedback controlled dissipation for state space stabilization in all but the symmetry variables potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables the approach also extends the method of controlled lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline

the even cycle problem for directed graphs if each arc in a strongly connected directed graph of minimum in degree and outdegree at least 3 is assigned a weight 0 or 1 then the resulting weighted directed graph has a directed cycle of even total weight this proves a conjecture made by l lovisz in 1975 and has applications to colour critical hypergraphs sign nonsingular matrices and permanents of matrices mathematical institute technical university of denmark dk 2800 lyngby

on the even permutation polytope abstract we consider the convex hull of the even permutations on a set of n elements we define a class of valid inequalities and prove that they induce a large class of distinct facets of the polytope using the inequalities we characterize the polytope for n 4 and we confirm a conjecture of brualdi and liu that unlike the convex hull of all permutations this polytope cannot be described as the solution set of polynomially many linear inequalities we also discuss the difficulty of determining whether a given point is in the polytope

a quadratic identity for the number of perfect matchings of plane graphs we present a quadratic identity on the number of perfect matchings of plane graphs by the method of graphical condensation which generalizes the results found by propp j propp generalized domino shuffling theoret comput sci 303 2003 267 301 kuo e h kuo applications of graphical condensation for enumerating matchings and tilings theoret comput sci 319 2004 29 57 and yan yeh and zhang w g yan y n yeh f j zhang graphical condensation of plane graphs a combinatorial approach theoret comput sci 349 2005 452 461

regular and singular orthants of tridiagonal matrices with respect to a tridiagonal matrix with variable diagonal vector g an orthant is said to be regular singular if the matrix is nonsingular singular for all g in it we give necessary and sufficient conditions for an orthant to be regular or singular our idea is based on observations of a simple two by two matrix and all the results obtained are original and self contained

gadgets approximation and linear programming we present a linear programming based method for finding gadgets i e combinatorial structures reducing constraints of one optimization problem to constraints of another a key step in this method is a simple observation which limits the search space to a finite one using this new method we present a number of new computer constructed gadgets for several different reductions this method also answers a question posed by bellare goldreich and sudan siam j comput 27 1998 pp 804 915 of how to prove the optimality of gadgets linear programming duality gives such proofs the new gadgets when combined with recent results of ha stad proceedings of the 29th acm symposium on theory of computing 1997 pp 1 10 improve the known inapproximability results for max cut and max dicut showing that approximating these problems to within factors of 16 17 epsilon and 12 13 epsilon respectively is np hard for every epsilon 0 prior to this work the best known inapproximability thresholds for both problems were 71 72 m bellare o goldreich and m sudan siam j comput 27 1998 pp 804 915 without using the gadgets from this paper the best possible hardness that would follow from bellare goldreich and sudan and ha s tad is 18 19 we also use the gadgets to obtain an improved approximation algorithm for max3 sat which guarantees an approximation ratio of 801 this improves upon the previous best bound implicit from m x goemans and d p williamson j acm 42 1995 pp 1115 1145 u feige and m x goemans proceedings of the third israel symposium on theory of computing and systems 1995 pp 182 189 of 7704

some results on the structure and spectra of matrix products we consider certain matrix products where successive matrices in the product belong alternately to a particular qualitative class or its transpose the main theorems relate structural and spectral properties of these matrix products to the structure of underlying bipartite graphs one consequence is a characterisation of caterpillars a graph is a caterpillar if and only if all matrix products associated with it have real nonnegative spectrum several other equivalences of this kind are proved the work is inspired by certain questions in dynamical systems where such products arise naturally as jacobian matrices and the results have implications for the existence and stability of equilibria in these systems

augmenting tractable fragments of abstract argumentation we present a new approach to the efficient solution of important computational problems that arise in the context of abstract argumentation our approach makes known algorithms defined for restricted fragments generally applicable at a computational cost that scales with the distance from the fragment thus in a certain sense we gradually augment tractable fragments surprisingly it turns out that some tractable fragments admit such an augmentation and that others do not more specifically we show that the problems of credulous and skeptical acceptance are fixed parameter tractable when parameterized by the distance from the fragment of acyclic argumentation frameworks for most semantics other tractable fragments such as the fragments of symmetrical and bipartite frameworks seem to prohibit an augmentation the acceptance problems are already intractable for frameworks at distance 1 from the fragments for our study we use a broad setting and consider several different semantics for the algorithmic results we utilize recent advances in fixed parameter tractability

when the sign pattern of a square matrix determines uniquely the sign pattern of its inverse abstract we give a precise characterization in terms of parity digraphs of those square matrices a such that for every matrix b with the same sign pattern as a b 1 exists and has the same sign pattern as a 1

on the recognition of s systems abstract a sequence a1 an of nonempty subsets of r m is an s system if for each choice of a1 a1 an an the nullspace of the matrix a1 an is a line that penetrates the strictly positive orthant of r n for the case in which each aj is a sign cone s systems are basic to the study of the sign solvability of systems of linear equations in that case they can be recognized in time o m2 the present paper provides several characterizations of general s systems and describes a polynomial time algorithm based on linear programming for the recognition of such systems when the aj are arbitrary finitely presented convex cones

sign solvable linear complementarity problems abstract this paper presents a connection between qualitative matrix theory and linear complementarity problems lcps an lcp is said to be sign solvable if the set of the sign patterns of the solutions is uniquely determined by the sign patterns of the given coefficients we provide a characterization for sign solvable lcps such that the coefficient matrix has nonzero diagonals which can be tested in polynomial time this characterization leads to an efficient combinatorial algorithm to find the sign pattern of a solution for these lcps the algorithm runs in o time where is the number of the nonzero coefficients

pfaffian orientations and perfect matchings of scale free networks counting perfect matchings is of great interest in theoretical computer science mathematics among other disciplines however explicitly determining the number of perfect matchings of general graphs is a challenge in this paper we present a first analytical study of the enumeration problem of perfect matchings on scale free networks which are ubiquitous in real life systems we focus on two iteratively growing scale free networks one is fractal while the other is non fractal for each network we construct an orientation and prove that it is pfaffian then based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a pfaffian orientation we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations furthermore using the recursive relations we show that for both scale free networks their entropy is zero which is in sharp contrast to the previously obtained results for other networks without scale free behavior but having the same average degree as the studied networks such as square lattice and honeycomb lattice whose entropies are larger than zero our results indicate from another angle that scale free networks have a distinct architecture in comparison with those lattices lacking the power law property

the strong perfect graph theorem a graph g is perfect if for every induced subgraph h the chromatic number of h equals the size of the largest complete subgraph of h and g is berge if no induced subgraph of g is an odd cycle of length at least five or the complement of one the strong perfect graph conjecture berge 1961 asserts that a graph is perfect if and only if it is berge a stronger conjecture was made recently by conforti cornu ejols and vuiskovi c that every berge graph either falls into one of a few basic classes or admits one of a few kinds of separation designed so that a minimum counterexample to berge s conjecture cannot have either of these properties in this paper we prove both of these conjectures

routh reduction and the class of magnetic lagrangian systems in this paper some new aspects related to routh reduction of lagrangian systems with symmetry are discussed the main result of this paper is the introduction of a new concept of transformation that is applicable to systems obtained after routh reduction of lagrangian systems with symmetry so called magnetic lagrangian systems we use these transformations in order to show that under suitable conditions the reduction with respect to a full semi direct product group is equivalent to the reduction with respect to an abelian normal subgroup the results in this paper are closely related to the more general theory of routh reduction by stages

on the hardness of efficiently approximating maximal non l submatrices abstract the sign pattern of a real matrix a is the matrix obtained by replacing each entry of a by its sign a real matrix a is an l matrix if every real matrix with the same sign pattern as a has linearly independent columns l matrices arise naturally in and are essential to the study of sign solvability and related notions in special cases the l matrix property has connections to the even dicycle problem pfaffian orientations and polya s permanent problem unfortunately the problem of recognizing l matrices is known to be co np complete in general we elaborate in this vein by showing a polynomial time inapproximability result for max not l rows a particular optimization version of l matrix recognition by means of an approximation preserving reduction from the previously studied problem max not equal 2 sat

routh s procedure for non abelian symmetry groups we extend routh s reduction procedure to an arbitrary lagrangian system that is one whose lagrangian is not necessarily the difference of kinetic and potential energies with a symmetry group which is not necessarily abelian to do so we analyze the restriction of the euler lagrange field to a level set of momentum in velocity phase space we present a new method of analysis based on the use of quasivelocities we discuss the reconstruction of solutions of the full euler lagrange equations from those of the reduced equations

qualitative comparative statics and audits of model performance abstract it is often difficult to determine if large computer based models actually conform to the mathematical descriptions given in their documentation one way to study this problem is through a sensitivity or comparative statics analysis of the model at issue is the degree to which characteristics of the jacobian matrix associated with a solution to the model can be shown to imply characteristics of the inverse jacobian matrix any necessary characteristics of the inverse jacobian can be tested for via comparative statics sign nonsingularity of the jacobian provides an opportunity to audit a model s mathematical documentation through a comparative statics analysis prototype software was developed that tested for sign nonsingularity and applied to the jacobian matrix corresponding to the u s department of energy s oil market simulation model it was found that the signs of all of the elements of the inverse jacobian matrix could be found this

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

progress on perfect graphs a graph is perfect if for every induced subgraph the chromatic number is equal to the maximum size of a complete subgraph the class of perfect graphs is important for several reasons for instance many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs also the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph

on the complexity of boolean matrix ranks abstract we present a reduction which shows that the fooling set number tropical and determinantal ranks of a boolean matrix are np hard to compute

oriented euler complexes and signed perfect matchings this paper presents oriented pivoting systems as an abstract framework for complementary pivoting it gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign a special case are the nash equilibria of a bimatrix game at the ends of lemke howson paths which have opposite index for euler complexes or oiks an orientation is defined which extends the known concept of oriented abstract simplicial manifolds ordered room partitions for a family of oriented oiks come in pairs of opposite sign for an oriented oik of even dimension this sign property holds also for unordered room partitions in the case of a two dimensional oik these are perfect matchings of an euler graph with the sign as defined for pfaffian orientations of graphs a near linear time algorithm is given for the following problem given a graph with an eulerian orientation with a perfect matching find another perfect matching of opposite sign in contrast the complementary pivoting algorithm for this problem may be exponential

the characteristic polynomial and the matchings polynomial of a weighted oriented graph abstract let g be a weighted oriented graph with skew adjacency matrix s g then g is usually referred as the weighted oriented graph associated to s g denote by g the characteristic polynomial of the weighted oriented graph g which is defined as g det i n s g i 0 n a i g n i in this paper we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph in addition some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given

constrained codes as networks of relations we address the well known problem of determining the capacity of constrained coding systems while the one dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity in contrast computing the exact capacity of a two dimensional constrained coding system is still an elusive research challenge the only known exception in the two dimensional case is an exact however not rigorous solution to the run length limited rll system on the hexagonal lattice furthermore only exponential time algorithms are known for the related problem of counting the exact number of constrained two dimensional information arrays we present the first known rigorous technique that yields an exact capacity of a two dimensional constrained coding system in addition we devise an efficient polynomial time algorithm for counting the exact number of constrained arrays of any given size our approach is a composition of a number of ideas and techniques describing the capacity problem as a solution to a counting problem in networks of relations graph theoretic tools originally developed in the field of statistical mechanics techniques for efficiently simulating quantum circuits as well as ideas from the theory related to the spectral distribution of toeplitz matrices using our technique we derive a closed form solution to the capacity related to the path cover constraint in a two dimensional triangular array the resulting calculated capacity is path cover is a generalization of the well known one dimensional rll constraint for which the capacity is known to be

max algebra and the linear assignment problem max algebra where the classical arithmetic operations of addition and multiplication are replaced by a b max a b and a b a b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation moreover it shows a strong similarity to classical linear algebra for instance it allows a consideration of linear equation systems and the eigenvalue problem the max algebraic permanent of a matrix a corresponds to the maximum value of the classical linear assignment problem with cost matrix a the analogue of van der waerden s conjecture in max algebra is proved moreover the role of the linear assignment problem in max algebra is elaborated in particular with respect to the uniqueness of solutions of linear equation systems regularity of matrices and the minimal dimensional realisation of discrete event systems further the eigenvalue problem in max algebra is discussed it is intimately related to the best principal submatrix problem which is finally investigated given an integer k 1 k n find a k k principal submatrix of the given n n matrix which yields among all principal submatrices of the same size the maximum minimum value for an assignment for k 1 2 n the maximum assignment problem values of the principal k k submatrices are the coefficients of the max algebraic characteristic polynomial of the matrix for a this problem can be used to model job rotations

dimers on the 33 42 lattice in this work we obtain explicit expression of the number of close packed dimers perfect matchings of the 33 42 lattice with cylindrical boundary condition particularly we show that the entropy of 33 42 lattice is the same for cylindrical and toroidal boundary conditions

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

recursive structure of s matrices and an o m2 algorithm for recognizing strong sign solvability abstract an s matrix is an m m 1 real matrix a such that for each matrix a of the same sign pattern as a a s columns are the vertices of an m simplex in r m that has the origin in its interior an s matrix is one that can be made into an s matrix by replacing some columns with their negatives such matrices are of interest because of their fundamental role in the study of sign solvability new results on the recursive structure of these classes of matrices are presented here and are used as the basis of algorithms of time complexity o m 2 for recognizing members of the classes and for testing the strong sign solvability of linear systems

solving linear programs from sign patterns this paper is an attempt to provide a connection between qualitative matrix theory and linear programming a linear program max cx mid ax b x geq 0 is said to be sign solvable if the set of sign patterns of the optimal solutions is uniquely determined by the sign patterns of a b and c it turns out to be np hard to decide whether a given linear program is sign solvable or not we then introduce a class of sign solvable linear programs in terms of totally sign nonsingular matrices which can be recognized in polynomial time for a linear program in this class we devise an efficient combinatorial algorithm to obtain the sign pattern of an optimal solution from the sign patterns of a b and c the algorithm runs in o m time where m is the number of rows of a and is the number of all nonzero entries in a b and c

levy s stochastic area formula in higher dimensions let wt w t 1 w t 2 w t d d 2 be a d dimensional standand brownian motion and let a t be a bounded measurable function from into the space of dxd skew symmetric matrices and x t into d we are concerned with a class of stochastic processes l t a x a particular example of which is p l vy s stochastic area open image in new window

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

a finite difference scheme for option pricing in jump diffusion and exponential levy models we present a finite difference method for solving parabolic partial integro differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a levy process or more generally a time inhomogeneous jump diffusion process we discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the levy measure we propose an explicit implicit finite difference scheme which can be used to price european and barrier options in such models we study stability and convergence of the scheme proposed and under additional conditions provide estimates on the rate of convergence numerical tests are performed with smooth and nonsmooth initial conditions

an approximation of small time probability density functions in a general jump diffusion model we propose a method for approximating probability density functions related to multidimensional jump diffusion processes for small time horizons a closed form approximation of the characteristic function is derived based on the ito taylor expansion the probability density function is then approximated numerically by inverting the characteristic function using fast fourier transform as application we consider a general stochastic volatility model which involves time state dependent drift and diffusion functions as well as jump components we test our approach under the heston model and the bates model and show that our method provides accurate approximations

out degree reducing partitions of digraphs abstract let k be a fixed integer we determine the complexity of finding a p partition v 1 v p of the vertex set of a given digraph such that the maximum out degree of each of the digraphs induced by v i 1 i p is at least k smaller than the maximum out degree of d we show that this problem is polynomial time solvable when p 2 k and np complete otherwise the result for k 1 and p 2 answers a question posed in 3 we also determine for all fixed non negative integers k 1 k 2 p the complexity of deciding whether a given digraph of maximum out degree p has a 2 partition v 1 v 2 such that the digraph induced by v i has maximum out degree at most k i for i 2 it follows from this characterization that the problem of deciding whether a digraph has a 2 partition v 1 v 2 such that each vertex v v i has at least as many neighbours in the set v 3 i as in v i for i 1 2 is np complete this solves a problem from 6 on majority colourings

graph theoretical approach to qualitative solvability of linear systems the structure of sign solvable and strongly sign solvable systems is studied here by a refinement of the graph theoretic approach first suggested by maybee both sign solvability and strong sign solvability are characterized in terms of an associated digraph the problem of recognizing sign solvability is reduced to two subproblems recognizing strong sign solvability and recognizing sign nonsingularity under fairly general conditions on the sign patterns of a it is possible to determine all sign patterns for b which render the system ax b sign solvable

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

sign nonsingular matrices and even cycles in directed graphs a sign nonsingular matrix or l mahix a is a real m x n matrix such that the columns of any real m x n matrix with the same sign pattern as a are linearly independent the problem of recognizing square l matrices is equivalent to that of finding an even cycle in a directed graph in this paper we use graph theoretic methods to investigate l matrices in particular we determine the maximum number of nonzero elements in square l matrices and we characterize completely the semicomplete l matrices i e the square l matrices a j such that at least one of a and ajl is nonzero for any i j and those square l matrices which are combinatorially symmetric i e the main diagonal has only nonzero entries and a j 0 iff a jr 0 we also show that for any n x n l matrix there is an i such that the total number of nonzero entries in the i th row and i th column is less than n unless the matrix has a completely specified structure finally we discuss the algorithmic aspects

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra

permanents pfaffian orientations and even directed circuits given a 0 1 square matrix a when can some of the 1 s be changed toi1 s in such a way that the permanent of a equals the determinant of the modifled matrix when does a real square matrix have the property that every real matrix with the same sign pattern that is the corresponding entries either have the same sign or are both zero is nonsingular when is a hypergraph with n vertices and n hyperedges minimally nonbipartite when does a bipartite graph have a pfa an orientation given a digraph does it have no directed circuit of even length given a digraph does it have a subdivision with no even directed circuit it is known that all of the above problems are equivalent we prove a structural characterization of the feasible instances which implies a polynomial time algorithm to solve all of the above problems the structural characterization says roughly speaking that a bipartite graph has a pfa an orientation if and only if it can be obtained by piecing together in a specifled way planar bipartite graphs and one sporadic nonplanar bipartite graph