randomized divide and conquer improved path matching and packing algorithms we propose a randomized divide and conquer technique that leads to improved randomized and deterministic algorithms for np hard path matching and packing problems for the parameterized max path problem our randomized algorithm runs in time o 4 k k 2 7 m and polynomial space where m is the number of edges in the input graph improving the previous best randomized algorithm for the problem that runs in time o 5 44 k km and exponential space our randomized algorithms for the parameterized max r d matching and max r set packing problems run in time 4 r 1 k n o 1 and polynomial space improving the previous best algorithms for the problems that run in time 10 88 rk n o 1 and exponential space moreover our randomized algorithms can be derandomized to result in significantly improved deterministic algorithms for the problems and they can be extended to solve other matching and packing problems

experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

ordering anisotropy and factored sparse approximate inverses we consider ordering techniques to improve the performance of factored sparse approximate inverse preconditioners concentrating on the ainv technique of m benzi and m t r u ma several practical existing unweighted orderings are considered along with a new algorithm minimum inverse penalty mip that we propose we show how good orderings such as these can improve the speed of preconditioner computation dramatically and also demonstrate a fast and fairly reliable way of testing how good an ordering is in this respect our test results also show that these orderings generally improve convergence of krylov subspace solvers but may have difficulties particularly for anisotropic problems we then argue that weighted orderings which take into account the numerical values in the matrix will be necessary for such systems after developing a simple heuristic for dealing with anisotropy we propose several practical algorithms to implement it while these show promise we conclude a better heuristic is required for robustness

parameterized complexity of max lifetime target coverage in wireless sensor networks max lifetime target coverage can be viewed as a family of problems where the task is to partition the sensors into groups and assign their time slots such that the coverage lifetime is maximized while satisfying some coverage requirement unfortunately these problems are np hard to gain insight into the source of the complexity we initiate a systematic parameterized complexity study of two types of max lifetime target coverage max min target coverage and max individual target coverage we first prove that both problems remain np hard even in the special cases where each target is covered by at most two sensors or each sensor can cover at most two targets by contrast restricting the number of targets reduces the complexity of the considered problems in other words they are both fixed parameter tractable fpt with respect to the parameter number of targets moreover we extend our studies to the structural parameter number k of sensors covering at least two targets positively both problems are in fpt with respect to k finally we show that max min target coverage is in fpt with respect to the combined parameters number of groups and number of targets covered by each group

hitting forbidden subgraphs in graphs of bounded treewidth we study the complexity of a generic hitting problem h subgraph hitting where given a fixed pattern graph h and an input graph g we seek for the minimum size of a set x v g that hits all subgraphs of g isomorphic to h in the colorful variant of the problem each vertex of g is precolored with some color from v h and we require to hit only h subgraphs with matching colors standard techniques e g courcelle s theorem show that for every fixed h and the problem is fixed parameter tractable parameterized by the treewidth of g however it is not clear how exactly the running time should depend on treewidth for the colorful variant we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of g is tightly governed by h the maximum size of a minimal vertex separator in h that is we show for every fixed h that on a graph of treewidth t the colorful problem can be solved in time 2 mathcal o t mu h cdot v g but cannot be solved in time 2 o t mu h cdot v g o 1 assuming the exponential time hypothesis eth furthermore we give some preliminary results showing that in the absence of colors the parameterized complexity landscape of h subgraph hitting is much richer

scheduling and fixed parameter tractability fixed parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions however even though fixed parameter algorithms are appealing for many reasons no such algorithms are known for many fundamental scheduling problems in this paper we present the first fixed parameter algorithms for classical scheduling problems such as makespan minimization scheduling with job dependent cost functions one important example being weighted flow time and scheduling with rejection to this end we identify crucial parameters that determine the problems complexity in particular we manage to cope with the problem complexity stemming from numeric input values such as job processing times which is usually a core bottleneck in the design of fixed parameter algorithms we complement our algorithms with mathsf w 1 w 1 hardness results showing that for smaller sets of parameters the respective problems do not allow fixed parameter algorithms in particular our positive and negative results for scheduling with rejection explore a research direction proposed by daniel marx

faster deterministic parameterized algorithm for k path abstract in the k path problem the input is a directed graph g and an integer k 1 and the goal is to decide whether there is a simple directed path in g with exactly k vertices we give a deterministic algorithm for k path with time complexity o 2 554 k this improves the previously best deterministic algorithm for this problem of zehavi esa 2015 whose time complexity is o 2 597 k the technique used by our algorithm can also be used to obtain faster deterministic algorithms for k tree r dimensional k matching graph motif and partial cover

parameterized and approximation algorithms for finding two disjoint matchings we first present a randomized fixed parameter algorithm for the np hard problem of deciding if there are two matchings m 1 and m 2 in a given graph g such that m 1 m 2 is a given number k the algorithm runs in o 2 k k m n expected time and can be derandomized to run in o 2 2 k 12 log 2 2 k k n m n time where n respectively m is the number of vertices respectively edges in g we then extend the algorithm to the weighted version of the problem we further present a combinatorial approximation algorithm for the np hard problem of finding two disjoint matchings in a given edge weighted graph g so that their total weight is maximized the algorithm achieves an approximation ratio close to 0 76 and runs in o m n 3 n time where is the inverse ackermann function

the parameterized complexity of the k biclique problem given a graph g and an integer k the k b iclique problem asks whether g contains a complete bipartite subgraph with k vertices on each side whether there is an f k vgvo 1 time algorithm solving k b iclique for some computable function f has been a longstanding open problem we show that k b iclique is w 1 hard which implies that such an f k vgvo 1 time algorithm does not exist under the hypothesis w 1 fpt from parameterized complexity theory to prove this result we give a reduction which for every n vertex graph g and small integer k constructs a bipartite graph h e lc r e in time polynomial in n such that if g contains a clique with k vertices then there are k k 1 2 vertices in l with n 1 k common neighbors otherwise any k k 1 2 vertices in l have at most k 1 e common neighbors an additional feature of this reduction is that it creates a gap on the right side of the biclique such a gap might have further applications in proving hardness of approximation results assuming a randomized version of exponential time hypothesis we establish an f k vgvo sk time lower bound for k biclique for any computable function f combining our result with the work of bulatov and marx 2014 we obtain a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems

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

parameterized maximum path coloring we study the well known max path coloring problem from a parameterized point of view focusing on trees and low treewidth networks we observe the existence of a variety of reasonable parameters for the problem such as the maximum degree and treewidth of the network graph the number of available colors and the number of requests one seeks to satisfy or reject in an effort to understand the impact of each of these parameters on the problem s complexity we study various parameterized versions of the problem deriving fixed parameter tractability and hardness results both for undirected and bi directed graphs

parameterized approximation algorithms for packing problems parameterized approximation is a topic of considerable interest in the field of parameterized complexity in the past decade new color coding related techniques including the breakthrough representative sets technique have been proven extremely powerful in the design of fast parameterized algorithms furthermore packing problems which are often solved via color coding related techniques have enjoyed a race towards obtaining the fastest parameterized algorithms that solve them therefore it is natural to ask whether packing problems admit efficient parameterized approximation algorithms in this paper we answer this question affirmatively we present tradeoffs that improve the running times of algorithms for well known special cases of the 3 setk packing problem at the cost of their accuracy consider a packing problem for which there is no known algorithm with approximation ratio and a parameter k if the value of an optimal solution is at least k we seek a solution of value at least k otherwise we seek an arbitrary solution clearly if the best known parameterized algorithm that finds a solution of value t runs in time o f t for some function f we are interested in running times better than o f k our main contribution lies in the adaptation of notions fundamental to the representative sets technique to the design of approximation algorithms we introduce the definition of approximate lopsided universal sets combine partial executions of representative sets based algorithms with approximation algorithms and adapt the iterative expansion framework in the context of representative sets to the design of parameterized approximation algorithms our ideas are intuitive and may be relevant to the design of other parameterized approximation algorithms

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

multiresolution approximate inverse preconditioners we introduce a new preconditioner for elliptic pdes on unstructured meshes using a wavelet inspired basis we compress the inverse of the matrix allowing an effective sparse approximate inverse by solving the sparsity vs accuracy conflict the key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh the true boundary conditions and even the pde coefficients we also show how this gives a new perspective on multiresolution algorithms such as multigrid interpreting the new preconditioner as a variation on node nested multigrid in particular we hope the new preconditioner will combine the best of both worlds fast convergence when multilevel methods can succeed but with robust performance for more difficult problems the rest of the paper discusses the core issues for the preconditioner ordering and construction of a factored approximate inverse in the multiresolution basis robust interpolation on unstructured meshes automatic mesh coarsening and purely algebraic alternatives some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems including discontinuous anisotropic coefficients strong convection and indefinite reaction problems on unstructured meshes with scalability like hierarchical basis methods achieved

edge disjoint packing of stars and cycles abstract we study the parameterized complexity of two graph packing problems edge disjoint k packing of s stars and edge disjoint k packing of s cycles with respect to the choice of parameters we show that although the two problems are fpt with both k and s as parameters they are unlikely to be fixed parameter tractable when parameterized by only k or only s in terms of kernelization complexity we show that edge disjoint k packing of s stars has a kernel with size polynomial in both k and s but in contrast unless np conp poly edge disjoint k packing of s cycles does not have a kernel with size polynomial in both k and s and moreover does not have a kernel with size polynomial in s for any fixed k we also show that edge disjoint k packing of 4 cycles admits a 96 k 2 kernel in general graphs and a 96 k kernel in planar graphs

parameterized coloring problems on chordal graphs in the precoloring extension problem prext a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper coloring of the graph with the given number of colors two parameterized versions of the problem are studied in the paper either the number of precolored vertices or the number of colors used in the precoloring is restricted to be at most k we show that for chordal graphs these problems are polynomial time solvable for every fixed k but w 1 hard if k is the parameter for a graph class f let f ke resp f kv denote those graphs that can be made to be a member of f by deleting at most k edges resp vertices we investigate the connection between prext in f with the two parameters defined above and the coloring of f ke f kv graphs with k being the parameter answering an open question of leizhen cai parameterized complexity of vertex colouring discrete appl math 127 2003 415 429 we show that coloring chordal ke graphs is fixed parameter tractable

parameterized tractability of the maximum duo preservation string mapping problem in this paper we investigate the parameterized complexity of the maximum duo preservation string mapping problem the complementary of the minimum common string partition problem we show that this problem is fixed parameter tractable when parameterized by the number k of conserved duos by first giving a parameterized algorithm based on the color coding technique and then presenting a reduction to a kernel of size o k 6

clustering with local restrictions we study a family of graph clustering problems where each cluster has to satisfy a certain local requirement formally let m be a function on the subsets of vertices of a graph g in the m p q partition problem the task is to find a partition of the vertices into clusters where each cluster c satisfies the requirements that 1 at most q edges leave c and 2 m c

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

on the parameterized complexity of vertex cover and edge cover with connectivity constraints abstract we investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems namely vertex cover and edge cover specifically we impose the additional requirement that each connected component of a solution have at least t vertices resp edges from the solution for a fixed positive integer t and call the problem t total vertex cover resp t total edge cover in both cases the parameter k is the size of the solution we show that both problems remain fixed parameter tractable with these restrictions with running times of the form o c k for some constant c 0 in each case where the o notation hides polynomial factors for each fixed t 2 t total vertex cover has no polynomial kernel unless conp np poly for each fixed t 2 t total edge cover has a linear vertex kernel of size t 1 t k these results significantly improve earlier work on these problems we illustrate the utility of the technique used to solve t total vertex cover by applying it to derive an o c k time fpt algorithm for the t total edge dominating set problem our no poly kernel result for t total vertex cover and the known np hardness result for t total edge cover are in stark contrast to the fact that vertex cover has a 2 k vertex kernel and that edge cover is solvable in polynomial time this illustrates how even the slightest connectivity requirement results in a drastic change in the tractability of problems the curse of connectivity

nonlinear boundary stabilization for timoshenko beam system abstract this paper is concerned with the existence and decay of solutions of the following timoshenko system u t u 1 i 1 n v x i 0 in 0 v v 2 i 1 n u x i 0 in 0 subject to the nonlinear boundary conditions u v 0 in 0 0 u h 1 x u 0 in 1 0 v h 2 x v x u 0 in 1 0 and the respective initial conditions at t 0 here is a bounded open set of r n with boundary constituted by two disjoint parts 0 and 1 and x denotes the exterior unit normal vector at x 1 the functions h i x s i 1 2 are continuous and strongly monotone in s r the existence of solutions of the above problem is obtained by applying the galerkin method with a special basis the compactness method and a result of approximation of continuous functions by lipschitz continuous functions due to strauss the exponential decay of energy follows by using the multiplier and the zuazua method

parameterized counting matching and packing a family of hard problems that admit fptras abstract in the field of parameterized counting complexity the problems that are w 1 hard and admit fixed parameter tractable randomized approximation scheme fptras have attracted much attention in recent years in this paper we focus on the problems on parameterized counting matching and packing these problems include counting set packing counting matching and counting subgraph packing including both vertex disjoint and edge disjoint versions we study the parameterized complexity on these problems on the basis of some results for counting graph matchings we show that a series of problems are w 1 hard furthermore by extending the previous algorithm for counting 3 d matching we obtain fptras for each considered problem respectively our results indicate that the problems on parameterized counting matching and packing form a large family of problems that are w 1 hard and admit fptras

improved kernel results for some fpt problems based on simple observations abstract in this paper we study the kernelization algorithms for several parameterized problems including parameterized co path set problem parameterized path contractibility problem and parameterized connected dominating set on g 7 graphs problem based on simple observations we give simple kernelization algorithms with kernel sizes 4 k 3 k 4 and o k 2 respectively which improves the previous best results 6 k 5 k 3 and o k 3 respectively

parameterized top k algorithms we study algorithmic techniques that produce the best k solutions to an instance of a parameterized np hard problem whose solutions are associated with a scoring function our parameterized top k algorithms proceed in two stages the first stage is a structure algorithm that on a problem instance constructs a structure of feasible size and the second stage is an enumerating algorithm that produces the k best solutions to the instance based on the structure we show that many algorithm design techniques for parameterized algorithms such as branch and search color coding and bounded treewidth can be adopted for designing efficient structure algorithms we then develop new techniques that support efficient enumerating algorithms in particular we show that for a large class of well known np optimization problems there are parameterized top k algorithms that produce the best k solutions for the problems in feasible amount of average time per solution when the parameter value is small finally we investigate the relation between fixed parameter tractability and parameterized top k algorithms

refining an approximate inverse direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades the advances in graph theoretic algorithms have not received enough attention from the iterative methods community for example we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner for very sparse preconditioners however a reformulation of the algorithm with outer products can exploit even more zeros to good advantage we also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings the article ends by proposing a block version of the algorithm for further gains in efficiency and robustness

on the parameterized complexity of multiple interval graph problems multiple interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it many applications of interval graphs also generalize to multiple interval graphs often allowing for more robustness in the modeling of the specific application with this motivation in mind a recent systematic study of optimization problems in multiple interval graphs was initiated in this sequel we study multiple interval graph problems from the perspective of parameterized complexity the problems under consideration are k independent set k dominating set and k clique which are all known to be w 1 hard for general graphs and np complete for multiple interval graphs we prove that k clique is in fpt while k independent set and k dominating set are both w 1 hard we also prove that k independent dominating set a hybrid of the two above problems is also w 1 hard our hardness results hold even when each vertex is associated with at most two intervals and all intervals have unit length furthermore as an interesting byproduct of our hardness results we develop a useful technique for showing w 1 hardness via a reduction from the k multicolored clique problem a variant of k clique we believe this technique has interest in its own right as it should help in simplifying w 1 hardness results which are notoriously hard to construct and technically tedious

existence and asymptotic stability for the semilinear wave equation with boundary damping and source term in this paper we consider the semilinear wave equation with boundary conditions this work is devoted to prove the existence of solutions and uniform decay rates of the wave equation with boundary damping and source term

finding approximate and constrained motifs in graphs one of the most relevant topics in the analysis of biological networks is the identification of functional motifs inside a network a recent approach introduced in literature called graph motif represents the network as a vertex colored graph and the motif m as a multiset of colors an occurrence of a motif m in a vertex colored graph g is a connected induced subgraph of g whose vertex set is colored exactly as m in this paper we investigate three different variants of the graph motif problem the first two variants minimum adding motif min add graph motif and minimum substitution motif min sub graph motif deal with approximate occurrences of a motif in the graph while the third variant constrained graph motif cgm constrains the motif to contain a given set of vertices we investigate the computational and parameterized complexity of the three problems we show that min add graph motifand min sub graph motifare both np hard even when m is a set and the graph is a tree with maximum degree 4 in which each color appears at most twice then we show that min sub graph motifis fixed parameter tractable when parameterized by the size of m finally we consider the parameterized complexity of the cgmproblem we give a fixed parameter algorithm for graphs of bounded treewidth and show that the problem is w 2 hard when parameterized by m even if the input graph has diameter 2

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

parameterized complexity of the anchored k core problem for directed graphs we consider the directed anchoredk core problem where the task is for a given directed graph g and integers b k and p to find an induced subgraph h with at least p vertices the core such that all but at most b vertices the anchors of h have in degree at least k we undertake a systematic analysis of the computational complexity of the directed anchoredk core problem

stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics the solution of linear systems arising in the finite element analysis of shells and solids by the preconditioned conjugate gradient method is considered stabilized and block versions of the ainv factorized approximate inverse preconditioner are presented and tested on a variety of difficult problems comparisons with other preconditioning methods are also included

the role of planarity in connectivity problems parameterized by treewidth for some years it was believed that for connectivity problems such as hamiltonian cycle algorithms running in time 2 o tw n o 1 called single exponential existed only on planar and other topologically constrained graph classes where tw stands for the treewidth of the n vertex input graph this was recently disproved by cygan et al 3 focs 2011 bodlaender et al 1 icalp 2013 and fomin et al 11 soda 2014 who provided single exponential algorithms on general graphs for most connectivity problems that were known to be solvable in single exponential time on topologically constrained graphs in this article we further investigate the role of planarity in connectivity problems parameterized by treewidth and convey that several problems can indeed be distinguished according to their behavior on planar graphs known results from the literature imply that there exist problems like cycle packing that cannot be solved in time 2 o tw log tw n o 1 on general graphs but that can be solved in time 2 o tw n o 1 when restricted to planar graphs our main contribution is to show that there exist natural problems that can be solved in time 2 o tw log tw n o 1 on general graphs but that cannot be solved in time 2 o tw log tw n o 1 even when restricted to planar graphs furthermore we prove that planar cycle packing and planar disjoint pathscannot be solved in time 2 o tw n o 1 the mentioned negative results hold unless the eth fails we feel that our results constitute a first step in a subject that can be further exploited

approximating the maximum clique minor and some subgraph homeomorphism problems we consider the minor and homeomorphic analogues of the maximum clique problem i e the problems of determining the largest h such that the input graph on n vertices has a minor isomorphic to k h or a subgraph homeomorphic to k h respectively as well as the problem of finding the corresponding subgraphs we term them as the maximum clique minor problem and the maximum homeomorphic clique problem respectively we observe that a known result of kostochka and thomason supplies an o n bound on the approximation factor for the maximum clique minor problem achievable in polynomial time we also provide an independent proof of nearly the same approximation factor with explicit polynomial time estimation by exploiting the minor separator theorem of plotkin et al next we show that another known result of bollobas and thomason and of komlos and szemeredi provides an o n bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time on the other hand we show an w n 1 2 o 1 l o g n c lower bound for some constant c unless email protected zptime 2 l o g n o 1 on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem nearly matching our upper bound finally we derive an interesting trade off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth

editing to a graph of given degrees we consider the editing to a graph of given degrees problem that asks for a graph g non negative integers d k and a function v g 1 d whether it is possible to obtain a graph g from g such that the degree of v is v for any vertex v by at most k vertex or edge deletions or edge additions we construct an fpt algorithm for editing to a graph of given degrees parameterized by d k we complement this result by showing that the problem has no polynomial kernel unless np conp poly

arboricity and subgraph listing algorithms in this paper we introduce a new simple strategy into edge searching of a graph which is useful to the various subgraph listing problems applying the strategy we obtain the following four algorithms the first one lists all the triangles in a graph g in o a g m time where m is the number of edges of g and a g the arboricity of g the second finds all the quadrangles in o a g m time since a g is at most three for a planar graph g both run in linear time for a planar graph the third lists all the complete subgraphs k l of order l in o la g l 2 m time the fourth lists all the cliques in o a g m time per clique all the algorithms require linear space we also establish an upper bound on a g for a graph g a g leqq lceil 2m n 1 2 rceil where n is the number of vertices in g

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

looking at the stars the problem of packing k vertex disjoint copies of a graph h into another graph g is np complete if h has more than two vertices in some connected component in the framework of parameterized complexity we analyze a particular family of instances of this problem namely the packing of stars we give a quadratic kernel for packing k copies of h k1 s when we consider the special case of s 2 i e h being a star with two leaves we give a linear kernel and an algorithm running in time o 25 301kk2 5 n3

comparing incomplete sequences via longest common subsequence abstract inspired by scaffold filling a recent approach for genome reconstruction from incomplete data we consider a variant of the well known longest common subsequence problem for the comparison of two sequences the new problem called longest filled common subsequence aims to compare a complete sequence with an incomplete one i e with some missing elements longest filled common subsequence lfcs given a complete sequence a an incomplete sequence b and a multiset m of symbols missing in b asks for a sequence b obtained by inserting the symbols of m into b so that b induces a common subsequence with a of maximum length we investigate the computational and approximation complexity of the problem and we show that it is np hard and apx hard when a contains at most two occurrences of each symbol and we give a polynomial time algorithm when the input sequences are over a constant size alphabet we give a 3 5 approximation algorithm for the longest filled common subsequence problem finally we present a fixed parameter algorithm for the problem when it is parameterized by the number of symbols inserted in b that match symbols of a

on the tractability of finding disjoint clubs in a network abstract we study a variant of the problem of finding a collection of disjoint s clubs in a given network given a graph the problem asks whether there exists a collection of at most r disjoint s clubs that covers at least k vertices of the network an s club is a connected graph that has diameter bounded by s for a positive integer s we demand that each club is non trivial that is it has order at least t 2 for some positive integer t we prove that the problem is apx hard even when the input graph has bounded degree s 2 t 3 and r v moreover we show that the problem is polynomial time solvable when s 4 t 3 and r v and when s 3 t 2 and r v finally for s 2 we present a fixed parameter algorithm for the problem when parameterized by the number of covered vertices

the k distinct language parameterized automata constructions abstract in this paper we pioneer a study of parameterized automata constructions for languages related to the design of parameterized algorithms we focus on the k distinct language l k k defined as the set of words of length k over an alphabet whose symbols are all distinct this language is implicitly related to several breakthrough techniques developed during the last two decades to design parameterized algorithms for fundamental problems such as k path and r dimensional k matching building upon the color coding divide and color and narrow sieves techniques we obtain the following automata constructions for l k we develop non deterministic automata nfas of sizes 4 k o k n o 1 and 2 e k o k n o 1 where the latter satisfies a bounded ambiguity property relevant to approximate counting as well as a non deterministic xor automaton nxa of size 2 k n o 1 where n we show that our constructions can be used to develop both deterministic and randomized algorithms for k path r dimensional k matching and module motif in a natural manner considering also their approximate counting variants our framework is modular and consists of two parts designing an automaton for k distinct and designing a problem specific automaton as well as an algorithm for deciding whether the intersection automaton s language is empty or for counting the number of accepting paths in it

two edge disjoint paths with length constraints abstract we consider the problem of finding for two pairs s 1 t 1 and s 2 t 2 of vertices in an undirected graph an s 1 t 1 path p 1 and an s 2 t 2 path p 2 such that p 1 and p 2 share no edges and the length of each p i satisfies constraint l i where l i k i k i k i with l i indicating no length constraint on p i we regard k 1 and k 2 as parameters and investigate the parameterized complexity of the above problem when at least one of p 1 and p 2 has a length constraint for the 9 different cases of l 1 l 2 we obtain fpt algorithms for 7 of them by using random partition backed by some structural results on the other hand we prove that the problem admits no polynomial kernel for all 9 cases unless n p c o n p p o l y

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

matching and weighted p2 packing algorithms and kernels parameterized algorithms and kernelization algorithms are presented for the weighted p 2 packing problem which is a generalization of the famous graph matching problem the parameterized algorithms are based on the following new techniques and observations 1 new study on structure relationship between graph matchings in general graphs and p 2 packings in bipartite graphs 2 an effective graph bi partitioning algorithm and 3 a polynomial time algorithm for a constrained weighted p 2 packing problem in bipartite graphs the kernelization algorithms are based on the following new techniques 1 the application of graph matching in kernelization 2 a crown reduction structure for weighted problems these techniques lead to randomized and deterministic parameterized algorithms that significantly improve the previous best upper bounds for the problem for both weighted and unweighted versions for the kernelization algorithm by using a weighted version of crown reduction a kernel of size o k 2 is presented where k is the given parameter of the problem

robust preconditioners for linear elasticity fem analyses summary this paper deals with two forms of preconditioner which can be easily used with a conjugate gradient solver to replace a direct solution subroutine in a traditional engineering nite element package they are tested in such a package final over a range of 2 d and 3 d elasticity problems from geotechnical engineering quadratic basis functions are used a number of modications to the basic incomplete choleski ic 0 factorization preconditioner are considered an algorithm to reduce positive o diagonal entries is shown in numerical experiments to ensure stability but at the expense of slow convergence an alternative algorithm of jennings and malik is more successful and a relaxation parameter is introduced which can make a further signicant improvement in performance while maintaining stability a heuristic for determining a near optimal value of is proposed a second form of preconditioning symmetrically scaled element by element due to bartelt is also shown to perform robustly over a range of problems it does not require assembly of the global stiness matrix and has great potential for parallelization 1997 by john wiley sons ltd

efficient computation of representative families with applications in parameterized and exact algorithms let m e i be a matroid and let s ls1 str be a family of subsets of e of size p a subfamily s s is q representative for s if for every set y e of size at most q if there is a set x s disjoint from y with x y i then there is a set x s disjoint from y with x y i by the classic result of bollobas in a uniform matroid every family of sets of size p has a q representative family with at most p qp sets in his famous two families theorem from 1977 lovasz proved that the same bound also holds for any matroid representable over a field f we give an efficient construction of a q representative family of size at most p qp in time bounded by a polynomial in p qp t and the time required for field operations we demonstrate how the efficient construction of representative families can be a powerful tool for designing single exponential parameterized and exact exponential time algorithms the applications of our approach include the following in the l ong d irected c ycle problem the input is a directed n vertex graph g and the positive integer k the task is to find a directed cycle of length at least k in g if such a cycle exists as a consequence of our 6 75k o k no 1 time algorithm we have that a directed cycle of length at least log n if such a cycle exists can be found in polynomial time in the m inimum e quivalent g raph meg problem we are seeking a spanning subdigraph d of a given n vertex digraph d with as few arcs as possible in which the reachability relation is the same as in the original digraph d we provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many connectivity problems such as h amiltonian c ycle or s teiner t ree can be solved in time 2o t n on n vertex graphs of treewidth at most t for the special case of uniform matroids on n elements we give a faster algorithm to compute a representative family we use this algorithm to provide the fastest known deterministic parameterized algorithms for k p ath k t ree and more generally k s ubgraph i somorphism where the k vertex pattern graph is of constant treewidth

kernel bounds for path and cycle problems connectivity problems like k path and k disjoint paths relate to many important milestones in parameterized complexity namely the graph minors project color coding and the recent development of techniques for obtaining kernelization lower bounds this work explores the existence of polynomial kernels for various path and cycle problems by considering nonstandard parameterizations we show polynomial kernels when the parameters are a given vertex cover a modulator to a cluster graph or a promised max leaf number we obtain lower bounds via cross composition e g for hamiltonian cycle and related problems when parameterized by a modulator to an outerplanar graph

parameterized algorithms for edge biclique and related problems abstract maximum edge biclique and related problems have wide applications in management science bioinformatics etc in this paper we study parameterized algorithms for parameterized edge biclique problem parameterized edge biclique packing problem parameterized biclique edge deletion problem and parameterized bipartite biclique clustering problem for the parameterized edge biclique problem the current best result is of running time o 2 k and we give a parameterized algorithm of running time o n k 2 5 k k where k is the parameter and n is the number of vertices in the given graph for the parameterized edge biclique packing problem based on randomized divide and conquer technique a parameterized algorithm of running time o k k l o g k 4 2 k 1 t is given where k is the parameter and t is the number of bicliques in the solution we study the parameterized biclique edge deletion problem on bipartite graphs and general graphs and give parameterized algorithms of running time o 2 k and o 3 k respectively for the parameterized bipartite biclique clustering problem based on modular decomposition method a kernel of size o k 2 and a parameterized algorithm of running time o 2 42 k n m are presented where k is the parameter n is the number of vertices and m is the number of edges in the given graph

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

existence and boundary stabilization of the semilinear wave equation abstract we consider the semilinear wave equation u t u h u f in a bounded domain q r n 1 with smooth boundary subject to mixed boundary conditions u 0 on 0 and u u 0 on 1 0 1 being a partition on suitably chosen we investigate existence uniqueness and asymptotic behavior as t of solution considering h a real continuous function satisfying a sign assumption and f and under some adequate conditions

on finding rainbow and colorful paths in the colorful path problem we are given a graph g v e and an arbitrary vertex coloring function c v k the goal is to find a colorful path i e a path on k vertices that visits each color this problem has been introduced in the classical work of alon et al 1995 1 and the authors proposed a dynamic programming algorithm that runs in time 2 k n o 1 and uses o 2 k space since then the only progress obtained is reducing the space size to a polynomial at the cost of using randomization in this work we show that a progress in time complexity is unlikely if colorful path can be solved in time 2 e k n o 1 then set cover admits a 2 e n n m o 1 time algorithm the same applies to other versions of the problem when edges are colored instead of vertices or we ask for a walk instead of a path or when the requested path walk has specified endpoints we study also a second very related problem in rainbowst connectivity we are given a k edge colored graph and two vertices s and t the goal is to decide whether there is a rainbow path between s and t that is a path on which no color repeats in its vertex variant rainbow vertexst connectivity the input graph is k vertex colored and a rainbow path is defined analogously uchizawa et al 2011 14 show that both variants can be solved in 2 k n o 1 time and exponential space we show that the space size can be reduced to a polynomial while keeping the same running time in contrast to the polynomial space algorithm for colorful path our algorithm for finding rainbow paths is deterministic

beyond bidimensionality parameterized subexponential algorithms on directed graphs in this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs we develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs we exemplify our approaches with two well studied problems for the first problem k leaf out branching which is to find an oriented spanning tree with at least k leaves we obtain an algorithm solving the problem in time 2 o k l o g k n n o 1 on directed graphs whose underlying undirected graph excludes some fixed graph h as a minor for the special case when the input directed graph is planar the running time can be improved to 2 o k n n o 1 the second example is a generalization of the directed hamiltonian path problem namely k internal out branching which is to find an oriented spanning tree with at least k internal vertices we obtain an algorithm solving the problem in time 2 o k l o g k n o 1 on directed graphs whose underlying undirected graph excludes some fixed apex graph h as a minor finally we observe that on these classes of graphs the k directed path problem is solvable in time o email protected kn f e for any e 0 where f is some function of e our methods are based on non trivial combinations of obstruction theorems for undirected graphs kernelization problem specific combinatorial structures and a layering technique similar to the one employed by baker to obtain ptas for planar graphs

kernel and fast algorithm for dense triplet inconsistency we study the parameterized complexity of inferring supertrees from sets of rooted triplets an important problem in phylogenetics for a set l of labels and a dense set t of triplets distinctly leaf labeled by 3 subsets of l we seek a tree distinctly leaf labeled by l and containing all but at most k triplets from t as homeomorphic subtree our results are the first polynomial kernel for this problem with o k 2 labels and a subexponential fixed parameter algorithm running in time o l 4 2 o k 1 3 l o g k

polynomial fixed parameter algorithms a case study for longest path on interval graphs we study the design of fixed parameter algorithms for problems already known to be solvable in polynomial time the main motivation is to get more efficient algorithms for problems with unattractive polynomial running times here we focus on a fundamental graph problem longest path that is given an undirected graph find a maximum length path in g longest path is np hard in general but known to be solvable in o n4 time on n vertex interval graphs we show how to solve longest path on interval graphs parameterized by vertex deletion number k to proper interval graphs in o k9n time notably longest path is trivially solvable in linear time on proper interval graphs and the parameter value k can be approximated up to a factor of 4 in linear time from a more general perspective we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial time solvable problems similarly to what classical parameterized complexity analysis does for np hard problems

a new preconditioning technique for solving large sparse linear systems abstract a new sparse approximate triangular factorization technique for solving large sparse linear system by iterative methods is proposed the method is based on the description of the triangular factorization of a matrix as a product of elementary matrices and provides a general scheme for constructing incomplete preconditioners for the given matrix in particular the familiar incomplete choleski decomposition can be incorporated into this scheme the algorithm based on choice by value compares favorably with the incomplete choleski preconditioner and equipped with a user controlled parameter is able to tackle extremely ill conditioned problems arising in structural analysis semiconductor simulation oil reservoir modelling and other applications when applied to a positive definite symmetric matrix the algorithm produces a preconditioning matrix preserving that property

a 2 o k n algorithm for k cycle in minor closed graph families abstract let c be a proper minor closed family of graphs we present a randomized algorithm that given a graph g c with n vertices finds a simple cycle of size k in g if exists in 2 o k n time the algorithm applies to both directed and undirected graphs in previous linear time algorithms for this problem the runtime dependence on k is super exponential the algorithm can be derandomized yielding a 2 o k n log n time algorithm

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

fixed parameter tractability definability and model checking in this article we study parameterized complexity theory from the perspective of logic or more specifically descriptive complexity theory we propose to consider parameterized model checking problems for various fragments of first order logic as generic parameterized problems and show how this approach can be useful in studying both fixed parameter tractability and intractability for example we establish the equivalence between the model checking for existential first order logic the homomorphism problem for relational structures and the substructure isomorphism problem our main tractability result shows that model checking for first order formulas is fixed parameter tractable when restricted to a class of input structures with an excluded minor on the intractability side for every t ge 0 we prove an equivalence between model checking for first order formulas with t quantifier alternations and the parameterized halting problem for alternating turing machines with t alternations we discuss the close connection between this alternation hierarchy and downey and fellows w hierarchy on a more abstract level we consider two forms of definability called fagin definability and slicewise definability that are appropriate for describing parameterized problems we give a characterization of the class fpt of all fixed parameter tractable problems in terms of slicewise definability in finite variable least fixed point logic which is reminiscent of the immerman vardi theorem characterizing the class ptime in terms of definability in least fixed point logic

finding a path of superlogarithmic length we consider the problem of finding a long simple path in an undirected graph we present a polynomial time algorithm that finds a path of length omega bigl log l log log l 2 bigr where l denotes the length of the longest simple path in the graph this establishes the performance ratio o n log log n log n 2 for the longest path problem where n denotes the number of vertices in the graph

on the compressibility of mathcal np instances and cryptographic applications we study compression that preserves the solution to an instance of a problem rather than preserving the instance itself our focus is on the compressibility of mathcal np decision problems we consider mathcal np problems that have long instances but relatively short witnesses the question is whether one can efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not we want the length of the compressed instance to be polynomial in the length of the witness and polylog in the length of original input such compression enables succinctly storing instances until a future setting will allow solving them either via a technological or algorithmic breakthrough or simply until enough time has elapsed in this paper we first develop the basic complexity theory of compression including reducibility completeness and a stratification of mathcal np with respect to compression we then show that compressibility say of sat would have vast implications for cryptography including constructions of one way functions and collision resistant hash functions from any hard on average problem in mathcal np and cryptanalysis of key agreement protocols in the bounded storage model when mixed with time complexity based cryptography

boundary stabilization for a coupled system abstract we investigate in this work the global existence of weak solutions for a nonlinear coupled system with mixed type boundary conditions more precisely dirichlet and feedback boundary conditions further we also prove the exponential decay of the energy associated with these solutions

existence and boundary stabilization of solutions for the coupled semilinear system abstract we investigate the global existence of both strong and weak solutions for a semilinear coupled system with homogeneous feedback boundary conditions in bounded open domain in r n with n n we also prove the exponential decay of total energy associated with weak solutions