matching cutsets in graphs of diameter 2 we say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue and every vertex has at most one neighbour coloured in the opposite colour in this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset in particular we present a polynomial time algorithm which solves this problem for graphs of diameter two

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

calculating the dual gabor window for general sampling sets methods for the calculation of the so called dual gabor window for general sampling sets are developed first we consider the standard situation where the time frequency plane tf plane is over sampled on a grid i e a rectangular lattice is considered with more points than the dimension of the signal space second we show that the proposed method can be easily extended to general subgroups of the tf plane one typical example of such a sampling set is the so called quincunx lattice

the isometric identities and inversion formulas of complex continuous wavelet transforms abstract the reproducing kernel function of the image space of a family of complex wavelet transforms is presented an admissible wavelet is obtained by convolution computation next the correlative characterisation of the image space of the family of complex wavelet transforms is provided when the scale is fixed furthermore the isometric identities and inversion formulas are obtained which provide a theoretic basis for investigating the image space of the general wavelet transform

algorithmic approach to the satisfactory graph partitioning problem abstract in a given graph we want to partition the set of its vertices in two subsets such that each vertex is satisfied in that it has at least as many neighbours in its own subset as in the other by introducing weights for the vertices and the edges we generalize the problem these more general problems are strongly np complete for the unweighted version we present some sufficient conditions for the existence of a solution and propose an exact and a heuristic solution method

max cut and max bisection are np hard on unit disk graphs we prove that the max cut and max bisection problems are np hard on unit disk graphs we also show that l precision graphs are planar for l 1 2 and give a dichotomy theorem for max cut computational complexity on l precision unit disk graphs

an improved kernel for max bisection above tight lower bound abstract we study kernelizations for max bisection above tight lower bound which is to decide if a given graph g v e admits a bisection with at least e 2 k crossing edges the best known kernel for this problem has 16k vertices based on the gallai edmonds decomposition we divide the vertices of g into several categories and study the roles of vertices in each category for obtaining a larger number of crossing edges by making use of the properties of maximum matchings in g graph g is partitioned into a set of blocks and each block in g is closely related to the number of crossing edges of a bisection of g by analyzing the number of crossing edges in blocks an improved kernel of 8k vertices is presented

construction and calculation of reproducing kernel determined by various linear differential operators we presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root further we gave the formula for calculation additionally by studying the recurrence relation of the reproducing kernel with arithmetic latent roots we found that the reproducing kernel with a multi knots interpolation constraint can be concisely represented by one with an initial value constraint

degree constrained 2 partitions of graphs a k1 k2 partition of a graph g is a vertex partition v1 v2 of g satisfying that g vi ki for i 1 2 we determine for all positive integers k1 k2 the complexity of deciding whether a given graph has a k1 k2 partition we also address the problem of finding a function g k1 k2 such that the k1 k2 partition problem is n p complete for the class of graphs of minimum degree less than g k1 k2 and polynomial for all graphs with minimum degree at least g k1 k2 we prove that g 1 k k for k 3 that g 2 2 3 and that g 2 3 if it exists has value 4 or 5

computing reproducing kernels with arbitrary boundary constraints smoothing with l splines involves the use of the penalty term x 2 2 int lx 2 t dt where l is an arbitrary linear differential operator of order m it also involves the choice of m co

algorithms for vertex partitioning problems on graphs with fixed clique width many vertex partitioning problems can be expressed within a general framework introduced by telle and proskurowski they showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree width in this paper we consider a very similar framework in relationship with more general classes of graphs we propose a polynomial time algorithm on classes of graphs with bounded clique width for all the optimization problems in our framework these classes of graphs are more general than the classes of graphs with bounded tree width in the sense that classes of graphs with bounded tree width have also bounded clique width but not necessarily the inverse our framework includes problems such as independent dominating set p dominating set induced bounded degree subgraph induced p regular subgraph perfect matching cut graph k coloring and graph list k coloring with cardinality constraints fixed k this paper thus provides a second distinct framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique width after a first framework called ms1 due to the work of courcelle makowsky and rotics for which they obtained a linear time algorithm

discrete zak transforms polyphase transforms and applications we consider three different versions of the zak 1967 transform zt for discrete time signals namely the discrete time zt the polyphase transform and a cyclic discrete zt in particular we show that the extension of the discrete time zt to the complex z plane results in the polyphase transform an important and well known concept in multirate signal processing and filter bank theory we discuss fundamental properties relations and transform pairs of the three discrete zt versions and we summarize applications of these transforms in particular the discrete time zt and the cyclic discrete zt are important for discrete time gabor 1946 expansion weyl heisenberg frame theory since they diagonalize the weyl heisenberg frame operator for critical sampling and integer oversampling the polyphase representation plays a fundamental role in the theory of filter banks especially dft filter banks simulation results are presented to demonstrate the application of the discrete zt to the efficient calculation of dual gabor windows tight gabor windows and frame bounds

simple local search problems that are hard to solve many algorithms for np hard optimization problems find solutions that are locally optimal in the sense that the solutions cannot be improved by a polynomially computable perturbation very little is known about the complexity of finding locally optimal solutions either by local search algorithms or using other indirect methods johnson papadimitriou and yannakakis j comput system sci 37 1988 pp 79 100 studied this question by defining a complexity class pls that captures local search problems it was proved that finding a partition of a graph that is locally optimal into equal parts with respect to the acclaimed kernighan lin algorithm is pls complete it is shown here that several natural simple local search problems are pls complete and thus just as hard two examples are finding a partition that cannot be improved by a single swap of two vertices and finding a stable configuration for an undirected connectionist network when edges or other objects are unweighted then a local optimum

on stable cutsets in line graphs we answer a question of brandstadt et al by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is np complete conversely the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently the proof of our np completeness result is based on a refinement on a result due to chvatal that recognizing decomposable graphs with maximum degree 4 is an np complete problem here a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v we also discuss some open problems on stable cutsets

fixed parameter complexity in ai and nonmonotonic reasoning many relevant intractable problems become tractable if some problem parameter is fixed however various problems exhibit very different computational properties depending on how the runtime required for solving them is related to the fixed parameter chosen the theory of parameterized complexity deals with such issues and provides general techniques for identifying fixed parameter tractable and fixed parameter intractable problems we study the parameterized complexity of various problems in ai and nonmonotonic reasoning we show that a number of relevant parameterized problems in these areas are fixed parameter tractable among these problems are constraint satisfaction problems with bounded treewidth and fixed domain restricted forms of conjunctive database queries restricted satisfiability problems propositional logic programming under the stable model semantics where the parameter is the dimension of a feedback vertex set of the program s dependency graph and circumscriptive inference from a positive k cnf restricted to models of bounded size we also show that circumscriptive inference from a general propositional theory when the attention is restricted to models of bounded size is fixed parameter intractable and is actually complete for a novel fixed parameter complexity class

efficient algorithms for discrete gabor transforms on a nonseparable lattice the discrete gabor transform dgt is the most commonly used transform for signal analysis and synthesis using a linear frequency scale it turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup most of the literature focuses on separable subgroups in this paper we will survey existing methods for a generalization to arbitrary groups as well as present an improvement on existing methods comparisons are made with respect to the computational complexity and the running time of optimized implementations in the c programming language the new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download by summarizing general background information on the state of the art this article can also be seen as a research survey sharing with the readers experience in the numerical work in gabor analysis

discrete gabor transform a feasible algorithm for implementing the gabor expansion the coefficients of which are computed by the discrete gabor transform dgt is presented for a given synthesis window and sampling pattern computing the auxiliary biorthogonal function of the dgt is nothing more than solving a linear system the dgt presented applies for both finite as well as infinite sequences by exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like dgt is obtained as the discrete fourier transform dft is a discrete realization of the continuous time fourier transform similarly the dgt introduced provides a feasible vehicle to implement the useful gabor expansion

on total functions existence theorems and computational complexity abstract nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between p and np we show that this class which we call tfnp contains a host of important problems whose membership in p is currently not known these include besides factoring local optimization brouwer s fixed points a computational version of sperner s lemma bimatrix equilibria in games and linear complementarity for p matrices

a 2 approximation for the maximum satisfying bisection problem given a graph g v e a satisfying bisection of g is a partition of the vertex set v into two sets v1 v2 such that midv1mid midv2mid and such that every vertex v set membership variant v has at least as many neighbors in its own set as in the other set the problem of deciding whether a graph g admits such a partition is view the mathml source complete in bazgan et al 2008 c bazgan z tuza d vanderpooten approximation of satisfactory bisection problems journal of computer and system sciences 75 5 2008 875 883 the authors present a polynomial time 3 approximation for maximizing the number of satisfied vertices in a bisection it remained an open problem whether one could find a 3 c approximation for c 0 see bazgan et al 2010 c bazgan z tuza d vanderpooten satisfactory graph partition variants and generalizations european journal of operational research 206 2 2010 271 280 in this paper we solve this problem by presenting a polynomial time 2 approximation algorithm for the maximum number of satisfied vertices in a satisfying bisection

solving singular nonlinear second order periodic boundary value problems in the reproducing kernel space abstract in this paper we present a new method for solving a singular nonlinear second order periodic boundary value problem its analytical solution is represented in the form of series in the reproducing kernel space in the mean time the n term approximation u n x to the analytical solution u x is obtained and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

computationally attractive real gabor transforms we present a gabor transform for real discrete signals and present a computationally attractive method for computing the transform for the critically sampled case we derive a biorthogonal function which is very localized in the time domain therefore truncation of this biorthogonal function allows us to compute approximate expansion coefficients with significantly reduced computational requirements further truncation does not degrade the numerical stability of the transform we present a tight upper bound on the reconstruction error incurred due to use of a truncated biorthogonal function and summarize computational savings for example the expense of transforming a length 2048 signal using length 16 blocks is reduced by a factor of 26 over similar fft based methods with at most 0 04 squared error in the reconstruction

dealing with several parameterized problems by random methods abstract in this paper we apply random methods to deal with several parameterized problems for the parameterized weighted p 3 packing problem by randomly partitioning the vertices in given graph a tripartite graph can be obtained we prove that the parameterized weighted p 3 packing problem can be solved in polynomial time on tripartite graphs based on the algorithm on tripartite graphs a randomized parameterized algorithm of running time o 32 k is given for the parameterized weighted p 3 packing problem for the parameterized weighted load coloring problem by randomly partitioning the vertices in given graph into two parts and studying the structure properties of the connected components in two parts a randomized parameterized algorithm of running time o 11 32 k is presented for the parameterized claw free edge deletion problem on diamond free graphs by combining random with branching methods a parameterized algorithm of running time o 2 42 k is given

degree constrained decompositions of graphs bounded treewidth and planarity we study the problem of decomposing the vertex set v of a graph into two nonempty parts v1 v2 which induce subgraphs where each vertex v v1 has degree at least a v inside v1 and each v v2 has degree at least b v inside v2 we give a polynomial time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition and gives such a decomposition if it exists this result and its variants are then applied to designing polynomial time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

parameterized computational complexity of dodgson and young elections abstract we show that the two np complete problems of dodgson score and young score have differing computational complexities when the winner is close to being a condorcet winner on the one hand we present an efficient fixed parameter algorithm for determining a condorcet winner in dodgson elections by a minimum number of switches in the votes on the other hand we prove that the corresponding problem for young elections where one has to delete votes instead of performing switches is w 2 complete in addition we study dodgson elections that allow ties between the candidates and give fixed parameter tractability as well as w 2 completeness results depending on the cost model for switching ties

polynomial time approximation schemes for max bisection on planar and geometric graphs the max bisection and min bisection problems are to find a partition of the vertices of a graph into two equal size subsets that respectively maximizes or minimizes the number of edges with endpoints in both subsets we design the first polynomial time approximation scheme for the max bisection problem on arbitrary planar graphs solving a long standing open problem the method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs in particular max and min bisection which could be of independent interest using a similar method we design also the first polynomial time approximation scheme for max bisection on unit disk graphs which could also be easily extended to other geometrically defined graphs

satisfactory graph partition variants and generalizations the satisfactory partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part this problem was introduced by gerber and kobler m gerber d kobler algorithmic approach to the satisfactory graph partitioning problem european journal of operational research 125 2000 283 291 and studied further by other authors in this paper we first review some applications and related problems then we survey structural complexity and approximation results obtained for satisfactory partition and for some of its variants and generalizations a list of open questions concludes this survey

min max communities in graphs community detection in graphs aims at identifying modules within a network and possibly their hierarchical organization by only using the information encoded in the graph modeling the network generally speaking a community in a network is a subset of its nodes showing higher degree of interconnection with each other than to the remaining nodes this is an informal characterization and different formal definitions of communities have been proposed in the literature also in relation to the available information for most such definitions the problem of detecting a proper partition of the given network into a prefixed number of community is np hard in this paper we consider the case in which a weight is associated to each edge of the graph measuring the amount of interconnection between the corresponding pair of nodes under this hypothesis we introduce and explore a new definition of community that is min max communities to model highly connected sets of nodes a min max community is a set of vertices such that the weakest minimal relation inside the community is stronger than the strongest maximal relation outside by analyzing the structural properties induced by this definition we prove that the problem of partitioning a complete weighted graph into k 0 communities is tractable we also show that a slight modification to this framework results into an np complete problem

a 699 approximation algorithm for max bisection we present a 699 approximation algorithm for max bisection i e partitioning the nodes of a weighted graph into two blocks of equal cardinality so as to maximize the weights of crossing edges this is an improved result from the 651 approximation algorithm of frieze and jerrum and the semidefinite programming relaxation of goemans and williamson