minimum fill in inapproximability and almost tight lower bounds abstract given an n n sparse symmetric matrix with m nonzero entries performing gaussian elimination may turn some zeroes into nonzero values so called fill ins the minimum fill in problem asks whether it is possible to perform the elimination with at most k fill ins we exclude the existence of polynomial time approximation schemes for this problem assuming p np and the existence of 2 o n 1 time approximation schemes for any positive assuming the exponential time hypothesis we also give a 2 o k 1 2 n o 1 parameterized lower bound all these results come as corollaries of a new reduction from vertex cover to the minimum fill in problem which might be of its own interest all previous reductions for similar problems start from some kind of graph layout problems and hence have limited use in understanding their fine grained complexity

fast dynamic programming for locally checkable vertex subset and vertex partitioning problems given a graph g we provide dynamic programming algorithms for many locally checkable vertex subset and vertex partitioning problems their runtime is polynomial in the number of equivalence classes of problem specific equivalence relations on subsets of vertices defined on a given decomposition tree of g using these algorithms all these problems become solvable in polynomial time for many well known graph classes like interval graphs and permutation graphs belmonte and vatshelle 2013 1 given a decomposition of boolean width k we show that the algorithms will have runtime o n 42 o k 2 providing the first large class of problems solvable in fixed parameter single exponential time in boolean width

independent domination in finitely defined classes of graphs we study the independent dominating set problem restricted to graph classes defined by finitely many forbidden induced subgraphs the main result is two sufficient conditions for the problem to be np hard in a finitely defined class of graphs we conjecture that those conditions are also necessary and describe several classes of graphs verifying the conjecture

a complete complexity classification of the role assignment problem in social network theory a society is often represented by a simple graph g where vertices stand for individuals and edges represent relationships between those individuals the description of the social network is tried to be simplified by assigning roles to the individuals such that the neighborhood relation is preserved formally for a fixed graph r we ask for a vertex mapping r vg vr such that r ng u nr r u for all u vg if such a mapping exists the graph g is called r role assignable and the corresponding decision problem is called the r role assignment problem kristiansen and telle conjectured that the r role assignment problem is an np complete problem for any simple connected graph r on at least three vertices in this paper we prove their conjecture in addition we determine the computational complexity of the role assignment problem for nonsimple and disconnected role graphs as these are considered in social network theory as well

approximation algorithms for node deletion problems on bipartite graphs with finite forbidden subgraph characterization in this paper we develop approximation algorithms for a few node deletion problems when the input is restricted to be a bipartite graph we look at node deletion problems for non trivial properties which can be characterized by forbidden structure which has a bounded intersection with both the bipartitions the approximation factors obtained directly depend upon the size of the largest such intersection special instances of this general problem include problems such as the minimum chain vertex deletion minimum dissociation vertex deletion minimum bipartite claw vertex deletion minimum bi complement vertex deletion and minimum bipartite threshold vertex deletion problems the algorithms are based upon the techniques of linear programming and iterative rounding we also use the node deletion algorithms to marginally improve the trivial approximation factor for complementary problem of determining the size of the maximum sized vertex induced subgraph lying in the given graph class and prove the apx completeness of all of these problems

extracting embedded generalized networks from linear programming problems if a linear program lp possesses a large generalized network gn submatrix this structure can be exploited to decrease solution time the problems of finding maximum sets of gn constraints and finding maximum embedded gn submatrices are shown to be np complete indicating that reliable efficient solution of these problems is difficult therefore efficient heuristic algorithms are developed for identifying such structure and are tested on a selection of twenty three real world problems the best of four algorithms for identifying gn constraint sets finds a set which is maximum in twelve cases and averages 99 1 of maximum on average the gn constraints identified comprise more than 62 3 of the total constraints in these problems the algorithm for identifying embedded gn submatrices finds submatrices whose sizes rows plus columns average 96 8 of an lp upper bound over 91 3 of the total constraint matrix was identified as a gn submatrix in these problems on average

an example of algebraic cycles with nontrivial abel jacobi images abstract in this paper we will study a family of quintic hypersurfaces and its abel jacobi images we will construct codimension 2 algebraic cycles on the blowing up of certain dimension 3 quintic hypersurfaces defined over number fields and show that they are homologically trivial but have nontrivial images under the 3 adic abel jacobi map

partitioning a graph into small pieces with applications to path transversal given a graph g v e and an integer k in mathbb n we study k vertex separator resp k edge separator where the goal is to remove the minimum number of vertices resp edges such that each connected component in the resulting graph has less than k vertices we also study k path transversal where the goal is to remove the minimum number of vertices such that there is no simple path of length k our main results are the following improved approximation algorithms

a width parameter useful for chordal and co comparability graphs abstract belmonte and vatshelle tcs 2013 used mim width a graph width parameter bounded on interval graphs and permutation graphs to explain existing algorithms for many domination type problems on those graph classes we investigate new graph classes of bounded mim width strictly extending interval graphs and permutation graphs the graphs k t k t and k t s t are graphs obtained from the disjoint union of two cliques of size t and one clique of size t and one independent set of size t respectively by adding a perfect matching we prove that interval graphs are k 3 s 3 free chordal graphs and k t s t free chordal graphs have mim width at most t 1 permutation graphs are k 3 k 3 free co comparability graphs and k t k t free co comparability graphs have mim width at most t 1 chordal graphs and co comparability graphs have unbounded mim width in general we obtain several algorithmic consequences for instance while minimum dominating set is np complete on chordal graphs it can be solved in time n o t on k t s t free chordal graphs the third statement strengthens a result of belmonte and vatshelle stating that either those classes do not have constant mim width or a decomposition with constant mim width cannot be computed in polynomial time unless p n p we generalize these ideas to bigger graph classes we introduce a new width parameter sim width of stronger modeling power than mim width by making a small change in the definition of mim width we prove that chordal graphs and co comparability graphs have sim width at most 1 we investigate a way to bound mim width for graphs of bounded sim width by excluding k t k t and k t s t as induced minors or induced subgraphs and give algorithmic consequences lastly we show that circle graphs have unbounded sim width and thus also unbounded mim width

2 role assignments on triangulated graphs if g is a graph a k role assignment is a function mapping each vertex into a role a positive integer 1 2 k so that if x and y have the same role then the sets of roles assigned to their neighbors are the same a graph is called a triangulated graph if it contains no chord less cycle of four or more vertices one interesting type of triangulated graph is the indifference graph that is a graph for which we can find a function on its vertex set so that if x and y are adjacent then their assigned function values are close we study 2 role assignments for triangulated graphs we provide a greedy algorithm for finding a 2 role assignment on a connected nonbipartite triangulated graph with at most one pendant vertex we characterize indifference graphs that have a 2 role assignment

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

faster algorithms for vertex partitioning problems parameterized by clique width abstract many np hard problems such as dominating set are fpt parameterized by clique width for graphs of clique width k given with a k expression dominating set can be solved in 4 k n o 1 time however no fpt algorithm is known for computing an optimal k expression for a graph of clique width k if we rely on known algorithms to compute a 2 3 k 1 expression via rank width and then solving dominating set using the 2 3 k 1 expression the above algorithm will only give a runtime of 4 2 3 k n o 1 there have been results which overcome this exponential jump the best known algorithm can solve dominating set in time 2 o k 2 n o 1 by avoiding constructing a k expression bui xuan et al 2013 7 we improve this to 2 o k log k n o 1 indeed we show that for a graph of clique width k a large class of domination and partitioning problems lc vsp including dominating set can be solved in 2 o k log k n o 1 our main tool is a variant of rank width using the rank of a 0 1 matrix over the rational field instead of the binary field

an optimal algorithm for finding a maximum independent set of a circular arc graph a new algorithm is presented for finding a maximum independent set of a circular arc graph when the graph is given in the form of a family of n arcs our algorithm requires only o n cdot log n time and o n space furthermore if the endpoints of the arcs are already sorted it runs in o n time this algorithm is time and space optimal to within a constant factor

on computing the minimum 3 path vertex cover and dissociation number of graphs the dissociation number of a graph g is the number of vertices in a maximum size induced subgraph of g with vertex degree at most 1 a k path vertex cover of a graph g is a subset s of vertices of g such that every path of order k in g contains at least one vertex from s the minimum 3 path vertex cover is a dual problem to the dissociation number for this problem we present an exact algorithm with a running time of o 1 5171 n on a graph with n vertices we also provide a polynomial time randomized approximation algorithm with an expected approximation ratio of 2311 for the minimum 3 path vertex cover

an incremental linear time algorithm for recognizing interval graphs the fastest known algorithm for recognizing interval graphs s booth and s lucker j comput system sci 13 1976 pp 335 379 iteratively manipulates the system of all maximal cliques of the given graph in a rather complicated way in order to construct a consecutive arrangement more precisely a tree representation of all possible consecutive arrangements this paper presents a much simpler algorithm using a related but much more informative tree representation of interval graphs this tree is constructed in an incremental fashion by adding vertices to the graph in a predefined order such that adding a vertex u takes o operatorname adj u 1 amortized time

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

an o n m time algorithm for finding a minimum weightdominating set in a permutation graph farber and keil algorithmica 4 1989 pp 221 236 presented an o n 3 time algorithm for finding a minimum weight dominating set in permutation graphs this result was improved to o n 2 log 2n by tsai and hsu sigal 90 algorithms lecture notes in computer science springer verlag new york 1990 pp 109 117 and to o n n m by the authors of this paper inform process lett 37 1991 pp 219 224 respectively in this paper we introduce a new faster algorithm that takes only o n m time to solve the same problem where m is the number of edges in a graph of n vertices

stochastic collocation methods on unstructured grids in high dimensions via interpolation in this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes to accomplish this we present the framework of least orthogonal interpolation which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions these interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space this feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes we present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm numerical benchmark problems are also presented to demonstrate the efficacy of the method

a hybrid collocation galerkin scheme for convective heat transfer problems with stochastic boundary conditions summary we present a numerical method to study convective heat transfer in a high reynolds number incompressible flow around a cylinder subject to uncertain boundary conditions we exploit the one way coupling of the energy and momentum transport to derive a semi intrusive uncertainty propagation scheme which combines galerkin and collocation approaches for computing approximate statistics of the stochastic temperature field the hybrid scheme converges rapidly and dramatically reduces the overall computational cost compared with the conventional uncertainty propagation schemes copyright q 2009 john wiley sons ltd

o m n algorithms for the recognition and isomorphism problems on circular arc graphs circular arc graphs have a rich combinatorial structure the circular endpoint sequence of arcs in a model for a circular arc graph is usually far from unique we present a natural restriction on these models to make it meaningful to define the unique representations for circular arc graphs we characterize those circular arc graphs which have unique restricted models and give an o m cdot n algorithm for recognizing circular arc graphs we think a more careful implementation could reduce the complexity to o n 2 our approach is to reduce the recognition problem of circular arc graphs to that of circle graphs this approach has the following advantages it is conceptually simpler than tucker s o n 3 recognition algorithm it exploits the similarity between circle graphs and circular arc graphs in a natural fashion it yields an isomorphism algorithm a main contribution of this result is an illustration of the transformed decomposition technique the decomposition tree developed for circular arc graphs generalizes the concept of the pq tree which is a data structure that keeps track of all possible interval representations of a given interval graph as a consequence our approach also yields an o m cdot n isomorphism algorithm for circle graphs

stochastic polynomial interpolation for uncertainty quantification with computer experiments multivariate polynomials are increasingly being used to construct emulators of computer models for uncertainty quantification for deterministic computer codes interpolating polynomial metamodels should be used instead of noninterpolating ones for logical consistency and prediction accuracy however available methods for constructing interpolating polynomials only provide point predictions there is no known method that can provide probabilistic statements about the interpolation error furthermore there are few alternatives to grid designs and sparse grids for constructing multivariate interpolating polynomials a significant disadvantage of these designs is the large gaps between allowable design sizes this article proposes a stochastic interpolating polynomial sip that seeks to overcome the problems discussed above a bayesian approach in which interpolation uncertainty is quantified probabilistically through the posterior distribution of the output is employed this allows assessment of the effe

computing role assignments of chordal graphs in social network theory a simple graph g is called k role assignable if there is a surjective mapping that assigns a number from 1 k called a role to each vertex of g such that any two vertices with the same role have the same sets of roles assigned to their neighbors the decision problem whether such a mapping exists is called the k role assignment problem this problem is known to be np complete for any fixed k 2 in this paper we classify the computational complexity of the k role assignment problem for the class of chordal graphs we show that for this class the problem can be solved in linear time for k 2 but remains np complete for any k 3 this generalizes earlier results by sheng and answers her open problem

new algorithms for weighted k domination and total k domination problems in proper interval graphs abstract given a positive integer k a k dominating set in a graph g is a set of vertices such that every vertex not in the set has at least k neighbors in the set a total k dominating set is a set of vertices such that every vertex of the graph has at least k neighbors in the set the problems of finding the minimum size of a k dominating respectively total k dominating set in a given graph are referred to as k domination respectively total k domination these generalizations of the classical domination and total domination problems are known to be np hard in the class of chordal graphs and more specifically even in the classes of split graphs both problems and undirected path graphs in the case of total k domination on the other hand it follows from previous works by bui xuan et al 2013 8 and by belmonte and vatshelle 2013 3 that these two families of problems are solvable in time o v g 3 k 4 in the class of interval graphs we develop faster algorithms for k domination and total k domination in the class of proper interval graphs by means of reduction to a single shortest path computation in a derived directed acyclic graph with o v g 2 k nodes and o v g 4 k arcs we show that a suitable implementation which avoids constructing all arcs of the digraph leads to a running time of o v g 3 k the algorithms are also applicable to the weighted case

constructing nested nodal sets for multivariate polynomial interpolation we present a robust method for choosing multivariate polynomial interpolation nodes our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity a variant of a weighted lebesgue function nodes are therefore chosen that tend to control oscillations in the resulting interpolant this method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space our method is therefore flexible nested nodal sets are produced in spaces of arbitrary dimensions and the number of nodes added at each stage can be arbitrary the algorithm produces a nodal set given a probability measure on the input space thus parameterizing interpolants with respect to finite measures we present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics the lebesgue constant the vandermonde determinant and the vandermonde condition number we also show that a nongreedy version of the

graph classes with structured neighborhoods and algorithmic applications given a graph in any of the following graph classes trapezoid graphs circular permutation graphs convex graphs dilworth k graphs k polygon graphs circular arc graphs and complements of k degenerate graphs we show how to compute decompositions with the number of d neighborhoods bounded by a polynomial of the input size combined with results of bui xuan telle and vatshelle 2013 1 this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes the boolean width of a graph is related to the number of 1 neighborhoods and our results imply that any of these graph classes have boolean width o logn

semitotal domination new hardness results and a polynomial time algorithm for graphs of bounded mim width abstract a semitotal dominating set of a graph g with no isolated vertex is a dominating set d of g such that every vertex in d is within distance two of another vertex in d the minimum size t 2 g of a semitotal dominating set of g is squeezed between the domination number g and the total domination number t g semitotal dominating set is the problem of finding given a graph g a semitotal dominating set of g of size t 2 g in this paper we continue the systematic study on the computational complexity of this problem when restricted to special graph classes in particular we show that it is solvable in polynomial time for the class of graphs of bounded mim width by a reduction to total dominating set and we provide several approximation lower bounds for subclasses of subcubic graphs moreover we obtain complexity dichotomies in monogenic classes for the decision versions of semitotal dominating set and total dominating set finally we show that it is np complete to recognise the graphs such that t 2 g t g and those such that g t 2 g even if restricted to be planar and with maximum degree at most 4 and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

a linear time algorithm for paired domination on circular arc graphs in a graph g a vertex subset s v g is said to be a dominating set of g if every vertex not in s is adjacent to a vertex in s a dominating set s of a graph g is called a paired dominating set if the induced subgraph g s contains a perfect matching the paired domination problem involves finding a minimum paired dominating set of g for this problem chen et al j comb optim 19 4 2010 457 470 proposed an o n m time algorithm on interval graphs and cheng et al discrete appl math 155 16 2007 2077 2086 designed an o m n m time algorithm on circular arc graphs in this paper we strengthen the results of cheng et al by showing an o n m time algorithm moreover the algorithm can be completed in o n time if an intersection model of a circular arc graph g with sorted endpoints is given since interval graphs are circular arc graphs we also obtain a linear time algorithm on interval graphs

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

boolean width of graphs abstract we introduce the graph parameter boolean width related to the number of different unions of neighborhoods boolean sums of neighborhoods across a cut of a graph for many graph problems this number is the runtime bottleneck when using a divide and conquer approach for an n vertex graph given with a decomposition tree of boolean width k we solve maximum weight independent set in time o n 2 k 2 2 k and minimum weight dominating set in time o n 2 n k 2 3 k with an additional n 2 factor in the runtime we can also count all independent sets and dominating sets of each cardinality boolean width is bounded on the same classes of graphs as clique width boolean width is similar to rank width which is related to the number of g f 2 sums of neighborhoods instead of the boolean sums used for boolean width we show for any graph that its boolean width is at most its clique width and at most quadratic in its rank width we exhibit a class of graphs the hsu grids having the property that a hsu grid on n 2 vertices has boolean width log n and rank width clique width tree width and branch width n

computing role assignments of split graphs a k role assignment of a simple graph g is a surjective function r v g 1 k where r u u n u r v v n v for every pair u v v g with r u r v this concept appears in the context of social networks it is known that the problem of finding a 2 role assignment of chordal graphs can be solved in linear time and that deciding whether a graph of this class admits a k role assignment for any fixed k 3 is np complete in this work we investigate this problem on split graphs a subclass of chordal graphs we characterize the split graphs admitting a 3 role assignment such result leads to a linear time algorithm for this case furthermore we prove that deciding whether a split graph has a k role assignment is np complete for any fixed k 4

minimal dominating sets in graph classes combinatorial bounds and enumeration abstract the number of minimal dominating sets that a graph on n vertices can have is known to be at most 1 715 9 n this upper bound might not be tight since no examples of graphs with 1 570 5 n or more minimal dominating sets are known for several classes of graphs we substantially improve the upper bound on the number of minimal dominating sets at the same time we give algorithms for enumerating all minimal dominating sets where the running time of each algorithm is within a polynomial factor of the proved upper bound for the graph class in question in several cases we provide examples of graphs containing the maximum possible number of minimal dominating sets for graphs in that class thereby showing the corresponding upper bounds to be tight

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

subset feedback vertex set on graphs of bounded independent set size abstract the weighted subset feedback vertex set problem is a generalization of the classical feedback vertex set problem and asks for a vertex set of minimum weight size that intersects all cycles containing a vertex of a predescribed set of vertices although subset feedback vertex set and feedback vertex set exhibit different computational complexity on split graphs no similar characterization is known on other classes of graphs towards the understanding of the complexity difference between the two problems it is natural to study the importance of structural graph parameters here we consider graphs of bounded independent set number for which it is known that weighted feedback vertex set can be solved in polynomial time we provide a dichotomy result with respect to the size of a maximum independent set in particular we show that weighted subset feedback vertex set can be solved in polynomial time for graphs with 3 whereas we prove that the problem remains np hard for graphs with 4 moreover we show that the unweighted subset feedback vertex set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time n o to complement our results we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size node multiway cut is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals based on our findings for subset feedback vertex set we settle the complexity of node multiway cut as well as its variants where nodes are weighted and or the terminals are deletable for every value of the given independent set number

simultaneous consecutive ones submatrix and editing problems classical complexity and fixed parameter tractable results abstract a binary matrix m has the consecutive ones property c1p for rows resp columns if there exists a permutation of its columns resp rows that arranges the ones consecutively in all the rows resp columns if m has the c1p for rows and the c1p for columns then m is said to have the simultaneous consecutive ones property sc1p in this article we consider the classical complexity and fixed parameter tractability of a few variants of decision problems related to the sc1p given a binary matrix m and a positive integer d we focus on problems that decide whether there exists a set of rows columns and rows as well as columns respectively of size at most d in m whose deletion results in a matrix with the sc1p we also consider problems that decide whether there exists a set of 0 entries 1 entries and 0 entries as well as 1 entries respectively of size at most d in m whose flipping results in a matrix with the sc1p in this paper we show that all the above mentioned problems are np complete we could also prove that all these problems are fixed parameter tractable with respect to solution size as the parameter except for two variants flipping 1 entries and flipping 0 1 entries we also give improved fpt algorithms for certain problems on restricted binary matrices

efficient algorithms for the domination problems on interval and circular arc graphs this paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants i e the weighted independent connected and total domination problems on interval graphs given an interval model with endpoints sorted these algorithms run in time o n or o n log log n where n is the number of vertices the results are then extended to solve the same problems on circular arc graphs in o n m time where m is the number of edges of the input graph

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

a sparse composite collocation finite element method for elliptic spdes this work presents a stochastic collocation method for solving elliptic pdes with random coefficients and forcing term which are assumed to depend on a finite number of random variables the method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution s statistics the selection of collocation points is based on a smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable a sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed resulting in a substantial reduction of overall degrees of freedom like in the monte carlo approach the algorithm results in solving a number of uncoupled purely deterministic elliptic problems which allows the integration of existing fast solvers for elliptic pdes numerical examples on two dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the full composite collocation finite element method and the monte carlo method

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

multi fidelity stochastic collocation method for computation of statistical moments abstract we present an efficient numerical algorithm to approximate the statistical moments of stochastic problems in the presence of models with different fidelities the method extends the multi fidelity approximation method developed in 18 26 by combining the efficiency of low fidelity models and the accuracy of high fidelity models our method exhibits fast convergence with a limited number of high fidelity simulations we establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi fidelity algorithm

the complexity of dominating set in geometric intersection graphs abstract we study the parameterized complexity of the dominating set problem in geometric intersection graphs in one dimension we investigate intersection graphs induced by translates of a fixed pattern q that consists of a finite number of intervals and a finite number of isolated points we prove that dominating set on such intersection graphs is polynomially solvable whenever q contains at least one interval and whenever q contains no intervals and for any two point pairs in q the distance ratio is rational the remaining case where q contains no intervals but does contain an irrational distance ratio is shown to be np complete and contained in fpt when parameterized by the solution size in two and higher dimensions we prove that dominating set is contained in w 1 for intersection graphs of semi algebraic sets with constant description complexity so far this was only known for unit squares finally we establish w 1 hardness for a large class of intersection graphs

the k path vertex cover in cartesian product graphs and complete bipartite graphs abstract for a graph g and a positive integer k a subset s of vertices of g is called a k path vertex cover if s intersects all paths of order k in g the cardinality of a minimum k path vertex cover is denoted by k g and called the k path vertex cover number of g in this paper we study some cartesian product graphs and give several estimations and the exact values of k g

node deletion problems on bipartite graphs a set of problems which has attracted considerable interest recently is the set of node deletion problems the general node deletion problem can be stated as follows given a graph find the minimum number of nodes whose deletion results in a subgraph satisfying property r in ly this problem was shown to be np complete for a large class of properties the class of properties that are hereditary on induced subgraphs using a small number of reduction schemes from the node cover problem since the node cover problem becomes polynomial on bipartite graphs it might be hoped that this is the case with other node deletion problems too in this paper we characterize those properties for which the bipartite restriction of the node deletion problem is polynomial and those for which it remains np complete similar results follow for analogous problems on other structures such as families of sets hypergraphs and 0 1 matrices for example in the case of matrices our result states that ifm is a class of 0 1 matrices which is closed under permutation and deletion of rows and columns then finding the largest submatrix inm of a matrix is polynomial if the matrices ofm have bounded rank and np complete otherwise 1 introduction a common approach when faced with an np complete problem c k gj is to restrict its input domain with the hope that the problem may become solvable in polynomial time under the restriction in fact it turns out that sometimes this is the case for example the node cover problem np complete on general graphs k can be solved efficiently when restricted to bipartite graphs this follows from k6nig s theorem that the node covering number of a bipartite graph equals the number of edges in a maximum matching and moreover a minimum node cover can be found efficiently from a maximum matching when the graph is bipartite see for example la in other cases however the node cover problem continues to be np complete under restriction for example the node cover problem on planar graphs gjs much work has been done recently kd ly ldl on a class of problems called the node deletion or maximum subgraph problems the general node deletion problem can be stated as follows given a graph or digraph g find a set of nodes of minimum cardinality whose deletion results in a graph or digraph satisfying a property r several of the known np complete problems such as the node cover the max clique the feedback node set k as well as some polynomial problems such as the connectivity of a graph e it can be formulated in an obvious way as node deletion problems by specifying appropriately the property r

rbf loi augmenting radial basis functions rbfs with least orthogonal interpolation loi for solving pdes on surfaces abstract we present a new method for the solution of pdes on manifolds m r d of co dimension one using stable scale free radial basis function rbf interpolation our method involves augmenting polyharmonic spline phs rbfs with polynomials to generate rbf finite difference rbf fd formulas these polynomial basis elements are obtained using the recently developed least orthogonal interpolation technique loi on each rbf fd stencil to obtain local restrictions of polynomials in r 3 to stencils on m the resulting rbf loi method uses cartesian coordinates does not require any intrinsic coordinate systems or projections of points onto tangent planes and our tests illustrate robustness to stagnation errors we show that our method produces high orders of convergence for pdes on the sphere and torus and present some applications to reaction diffusion pdes motivated by biology

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

a domain adaptive stochastic collocation approach for analysis of mems under uncertainties this work proposes a domain adaptive stochastic collocation approach for uncertainty quantification suitable for effective handling of discontinuities or sharp variations in the random domain the basic idea of the proposed methodology is to adaptively decompose the random domain into subdomains within each subdomain a sparse grid interpolant is constructed using the classical smolyak construction s smolyak quadrature and interpolation formulas for tensor products of certain classes of functions soviet math dokl 4 1963 240 243 to approximate the stochastic solution locally the adaptive strategy is governed by the hierarchical surpluses which are computed as part of the interpolation procedure these hierarchical surpluses then serve as an error indicator for each subdomain and lead to subdivision whenever it becomes greater than a threshold value the hierarchical surpluses also provide information about the more important dimensions and accordingly the random elements can be split along those dimensions the proposed adaptive approach is employed to quantify the effect of uncertainty in input parameters on the performance of micro electromechanical systems mems specifically we study the effect of uncertain material properties and geometrical parameters on the pull in behavior and actuation properties of a mems switch using the adaptive approach we resolve the pull in instability in mems switches the results from the proposed approach are verified using monte carlo simulations and it is demonstrated that it computes the required statistics effectively

on the inapproximability of independent domination in 2p3 free perfect graphs we consider the complexity of approximation for the independent dominating set problem in 2p3 free graphs i e graphs that do not contain two disjoint copies of the chordless path on three vertices as an induced subgraph we show that if p np the problem cannot be approximated for 2p3 free graphs in polynomial time within a factor of n1 e for any constant e 0 where n is the number of vertices in the graph moreover we show that the result holds even if the 2p3 free graph is restricted to being weakly chordal and thereby perfect

efficient algorithms for the conditional covering problem we consider the conditional covering problem in an undirected network in which each vertex represents a demand point that must be covered by a facility as well as a potential facility site each facility can cover all vertices within a given coverage radius except the vertex at which the facility is located the objective is to locate facilities to cover all vertices such that the total facility location cost is minimized in this paper new upper bounds are proposed for the conditional covering problem on paths cycles extended stars and trees in particular we provide an o nlogn time algorithm for paths an o n 2logn time algorithm for cycles an o n 1 5logn time algorithm for extended stars and an o n 3 time algorithm for trees our algorithms for paths extended stars and trees improve the previous upper bounds from o n 2 o n 2 and o n 4 respectively

efficient localization of discontinuities in complex computational simulations surrogate models for computational simulations are input output approximations that allow computationally intensive analyses such as uncertainty propagation and inference to be performed efficiently when a simulation output does not depend smoothly on its inputs the error and convergence rate of many approximation methods deteriorate substantially this paper details a method for efficiently localizing discontinuities in the input parameter domain so that the model output can be approximated as a piecewise smooth function the approach comprises an initialization phase which uses polynomial annihilation to assign function values to different regions and thus seed an automated labeling procedure followed by a refinement phase that adaptively updates a kernel support vector machine representation of the separating surface via active learning the overall approach avoids structured grids and exploits any available simplicity in the geometry of the separating surface thus reducing the number of model

weighted discrete least squares polynomial approximation using randomized quadratures we discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation the problem stems from uncertainty quantification uq where the independent variables of the functions are random variables with specified probability measure we propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product gaussian quadrature points we analyze the stability properties of this method and prove that the method is asymptotically stable provided that the number of points scales linearly up to a logarithmic factor with the cardinality of the polynomial space specific results in both bounded and unbounded domains are obtained along with a convergence result for chebyshev measure numerical examples are provided to verify the theoretical results

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

on pairwise compatibility graphs having dilworth number two a graph g v e is called a pairwise compatibility graph pcg if there exists a tree t a positive edge weight function w on t and two non negative real numbers d m i n and d m a x d m i n d m a x such that v coincides with the set of leaves of t and there is an edge u v e if and only if d m i n d t w u v d m a x where d t w u v is the sum of the weights of the edges on the unique path from u to v in t when the constraints on the distance between the pairs of leaves concern only d m a x or only d m i n the two subclasses lpgs leaf power graphs and mlpgs minimum leaf power graphs are defined the dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another it is known that lpg mlpg is not empty and that threshold graphs i e dilworth one graphs are contained in it in this paper we prove that dilworth two graphs belong to the set lpg mlpg too our proof is constructive since we show how to compute all the parameters t w d m a x and d m i n exploiting the usual representation of dilworth two graphs in terms of node weight function and thresholds for graphs with dilworth number two that are also split graphs i e split permutation graphs we provide another way to compute t w d m i n and d m a x when these graphs are given in terms of their intersection model

a bi fidelity method for the multiscale boltzmann equation with random parameters abstract in this paper we study the multiscale boltzmann equation with multi dimensional random parameters by a bi fidelity stochastic collocation sc method developed in 52 70 71 by choosing the compressible euler system as the low fidelity model we adapt the bi fidelity sc method to combine computational efficiency of the low fidelity model with high accuracy of the high fidelity boltzmann model with only a small number of high fidelity asymptotic preserving solver runs for the boltzmann equation the bi fidelity approximation can capture well the macroscopic quantities of the solution to the boltzmann equation in the random space a priori estimate on the accuracy between the high and bi fidelity solutions together with a convergence analysis is established finally we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method

mim width iii graph powers and generalized distance domination problems abstract we generalize the family of problems and locally checkable vertex partition problems to their distance versions which naturally captures well known problems such as distance r dominating set and distance r independent set we show that these distance problems are in xp parameterized by the structural parameter mim width and hence polynomial time solvable on graph classes where mim width is bounded and quickly computable such as k trapezoid graphs dilworth k graphs circular permutation graphs interval graphs and their complements convex graphs and their complements k polygon graphs circular arc graphs complements of d degenerate graphs and h graphs if given an h representation we obtain these results by showing that taking any power of a graph never increases its mim width by more than a factor of two to supplement these findings we show that many classes of problems are w 1 hard parameterized by mim width solution size we show that powers of graphs of tree width w 1 or path width w and powers of graphs of clique width w have mim width at most w these results provide new classes of bounded mim width we prove a slight strengthening of the first statement which implies that surprisingly leaf power graphs which are of importance in the field of phylogenetic studies have mim width at most 1