a tight sqrt 2 approximation for linear 3 cut we investigate the approximability of the linear 3 cut problem in directed graphs the input here is a directed graph d v e with node weights and three specified terminal nodes s r t in v and the goal is to find a minimum weight subset of non terminal nodes whose removal ensures that s cannot reach r and t and r cannot reach t the precise approximability of linear 3 cut has been wide open until now the best known lower bound under the unique games conjecture ugc was 4 3 while the best known upper bound was 2 using a trivial algorithm in this work we completely close this gap we present a sqrt 2 approximation algorithm and show that this factor is tight under ugc our contributions are twofold 1 we analyze a natural two step deterministic rounding scheme through the lens of a single step randomized rounding scheme with non trivial distributions and 2 we construct integrality gap instances that meet the upper bound of sqrt 2 our gap instances can be viewed as a weighted graph sequence converging to a graph limit structure we complement our results by showing connections between the linear 3 cut problem and other fundamental cut problems in directed graphs

training free generic object detection using locally adaptive regression kernels we present a generic detection localization algorithm capable of searching for a visual object of interest without training the proposed method operates using a single example of an object of interest to find similar matches does not require prior knowledge learning about objects being sought and does not require any preprocessing step or segmentation of a target image our method is based on the computation of local regression kernels as descriptors from a query which measure the likeness of a pixel to its surroundings salient features are extracted from said descriptors and compared against analogous features from the target image this comparison is done using a matrix generalization of the cosine similarity measure we illustrate optimality properties of the algorithm using a naive bayes framework the algorithm yields a scalar resemblance map indicating the likelihood of similarity between the query and all patches in the target image by employing nonparametric significance tests and nonmaxima suppression we detect the presence and location of objects similar to the given query the approach is extended to account for large variations in scale and rotation high performance is demonstrated on several challenging data sets indicating successful detection of objects in diverse contexts and under different imaging conditions

fptass for trimming weighted trees given a tree with nonnegative edge cost and nonnegative vertex weight and a number k 0 we consider the following four cut problems cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized the minmstcut problem cut vertices of weight at mostk and minimize the total cost of the edges being deleted can be solved in linear time and space and the other three problems are np hard in this paper we design an o nl e time o l 2 e n space algorithm for maxmstcut and o nl 1 e logn time o l 2 e n space algorithms for the other two problems minlstcut and maxlstcut where n is the number of vertices in the tree l the number of leaves and e 0 the prescribed error bound

nonparametric multivariate density estimation a comparative study the paper algorithmically and empirically studies two major types of nonparametric multivariate density estimation techniques where no assumption is made about the data being drawn from any of known parametric families of distribution the first type is the popular kernel method and several of its variants which uses locally tuned radial basis e g gaussian functions to interpolate the multidimensional density the second type is based on an exploratory projection pursuit technique which interprets the multidimensional density through the construction of several 1d densities along highly interesting projections of multidimensional data performance evaluations using training data from mixture gaussian and mixture cauchy densities are presented the results show that the curse of dimensionality and the sensitivity of control parameters have a much more adverse impact on the kernel density estimators than on the projection pursuit density estimators

lower bounds for the happy coloring problems in this paper we study the maximum happy vertices and the maximum happy edges problems mhv and mhe for short very recently the problems attracted a lot of attention and were studied in agrawal 17 aravind et al 16 choudhari and reddy 18 misra and reddy 17 main focus of our work is lower bounds on the computational complexity of these problems established lower bounds can be divided into the following groups np hardness of the above guarantee parameterization kernelization lower bounds answering questions of misra and reddy 17 exponential lower bounds under the set cover conjecture and the exponential time hypothesis and inapproximability results moreover we present an mathcal o ell k randomized algorithm for mhv and an mathcal o 2 k algorithm for mhe where ell is the number of colors used and k is the number of required happy vertices or edges these algorithms cannot be improved to subexponential taking proved lower bounds into account

improved parameterized and exact algorithms for cut problems on trees we study the multicut on trees and the generalized multiway cut on trees problems for the multicut on trees problem we present a parameterized algorithm that runs in time o k where 2 1 1 554 is the positive root of the polynomial x 4 2 x 2 1 this improves the current best algorithm of chen et al that runs in time o 1 619 k for the generalized multiway cut on trees problem we show that this problem is solvable in polynomial time if the number of terminal sets is fixed this answers an open question posed in a recent paper by liu and zhang by reducing the generalized multiway cut on trees problem to the multicut on trees problem our results give a parameterized algorithm that solves the generalized multiway cut on trees problem in time o k

parameter transformations for improved approximate confidence regions in nonlinear least squares in a previous paper it was shown that parameter effects nonlinearities of a nonlinear regression model experimental design parameterization combination can be quantified by means of a parameter effects curvature array a based on second derivatives of the model function in this paper the individual terms of a are interpreted and local compensation methods are suggested a method of computing the parameter effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter effects nonlinearity

constrained coalition formation on valuation structures coalition structure generation is the problem of partitioning the agents of a given environment into disjoint and exhaustive coalitions so that the whole available worth is maximized while this problem has been classically studied in settings where all coalitions are allowed to form it has been recently reconsidered in the literature moving from the observation that environments often forbid the formation of certain coalitions by following this latter perspective a model for coalition structure generation is proposed where constraints of two different kinds can be expressed simultaneously indeed the model is based on the concept of valuation structure which consists of a set of pivotal agents that are pairwise incompatible plus an interaction graph prescribing that a coalition c can form only if the subgraph induced over the nodes agents in c is connected it is shown that valuation structures can be used to model a number of relevant problems arising in real world application domains then the complexity of coalition structure generation over valuation structures is studied by assuming that the functions associating each coalition with its worth are given as input according to some compact encodingrather than explicitly listing all exponentially many associations in particular islands of tractability are identified based on the topological properties of the underlying interaction graphs and on suitable algebraic properties of the given worth functions finally stability issues over valuation structures are studied too by considering the core as the prototypical solution concept

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

the multi multiway cut problem in this paper we define and study a natural generalization of the multicut and multiway cut problems the minimum multi multiway cut problem the input to the problem is a weighted undirected graph g v e and k sets s 1 s 2 s k of vertices the goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets s 1 s 2 s k i e disconnects every pair of vertices u and v such that u email protected s i for some i this problem generalizes both the multicut problem when s i 2 for email protected email protected k and the multiway cut problem when k 1 we present an approximation algorithm for the multi multiway cut problem with an approximation ratio which matches that obtained by garg vazirani and yannakakis on the standard multicut problem namely our algorithm has an o logk approximation ratio moreover we consider instances of the minimum multi multiway cut problem which are known to have an optimal solution of light weight we show that our algorithm has an approximation ratio substantially better than o logk when restricted to such light instances specifically we obtain an o loglp approximation algorithm for the problem when all edge weights are at least 1 here lp denotes the value of a natural linear programming relaxation of the problem the latter improves the o loglplogloglp approximation ratio for the minimum multicut problem implied by the work of seymour and even et al

multicuts and integral multiflows in rings we show how to solve in polynomial time the multicut and the maximum integral multiflow problems in rings moreover we give linear time procedures to solve both problems in rings with uniform capacities

a polynomial algorithm for the k cut problem for fixed k the k cut problem is to find a partition of an edge weighted graph into k nonempty components such that the total edge weight between components is minimum this problem is np complete for an arbitrary k and its version involving fixing a vertex in each component is np hard even for k 3 we present a polynomial algorithm for k fixed that runs in onki 2 3k 2 4tn m steps where tn m is the running time required to find the minimum s t cut on a graph with n vertices and m edges

minimal multicut and maximal integer multiflow a survey we present a survey about the maximum integral multiflow and minimum multicut problems and their subproblems such as the multiterminal cut and the unsplittable flow problems we consider neither continuous multiflow nor minimum cost multiflow most of the results are very recent and some are new we recall the dual relationship between both problems give complexity results and algorithms firstly in unrestricted graphs and secondly in several special graphs trees bipartite or planar graphs a table summarizes the most important results

approximate max flow min multi cut theorems and their applications consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed we prove the following approximate max flow min multicut theorem dst frac mbox rm min multicut o log k leq mbox rm max flow leq mbox rm min multicut noindent where k is the number of commodities our proof is constructive it enables us to find a multicut within o log k of the max flow and hence also the optimal multicut in addition the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of leighton and rao and klein et al and thereby obtain an improved bound for the latter problem

polynomial flow cut gaps and hardness of directed cut problems we study the multicut and the sparsest cut problems in directed graphs in the multicut problem we are a given an n vertex graph g along with k source sink pairs and the goal is to find the minimum cardinality subset of edges whose removal separates all source sink pairs the sparsest cut problem has the same input but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of source sink pairs that are separated by this deletion the natural linear programming relaxation for multicut corresponds by lp duality to the well studied maximum fractional multicommodity flow problem while the standard lp relaxation for sparsest cut corresponds to maximum concurrent flow therefore the integrality gap of the linear programming relaxation for multicut sparsest cut is also the flow cut gap the largest gap achievable for any graph between the maximum flow value and the minimum cost solution for the corresponding cut problem our first result is that the flow cut gap between maximum multicommodity flow and minimum multicut is n1 7 in directed graphs we show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs these results improve upon a long standing lower bound of log n for both types of flow cut gaps we notice that these polynomially large flow cut gaps are in a sharp contrast to the undirected setting where both these flow cut gaps are known to be log n our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2 log1 n for any constant 0 unless np zpp this improves upon the recent log n log log n hardness result for these problems we also show that existence of pcp s for np with perfect completeness polynomially small soundness and constant number of queries would imply a polynomial factor hardness of approximation for both these problems all our results hold for directed acyclic graphs

using branch and price and cut to solve origin destination integer multicommodity flow problems we present a column generation model and branch and price and cut algorithm for origin destination integer multicommodity flow problems the origin destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity defined in this case by an origin destination pair may use only one path from origin to destination branch and price and cut is a variant of branch and bound with bounds provided by solving linear programs using column and cut generation at nodes of the branch and bound tree because our model contains one variable for each origin destination path for every commodity the linear programming relaxations at nodes of the branch and bound tree are solved using column generation i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality we devise a new branching rule that allows columns to be generated efficiently at each node of the branch and bound tree then we describe cuts cover inequalities that can be generated at each node of the branch and bound tree these cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry we detail the implementation of our combined column and cut generation method and present computational results for a set of test problems arising from telecommunications applications we illustrate the value of our branching rule when used to find a heuristic solution and compare branch and price and branch and price and cut methods to find optimal solutions for highly capacitated problems

efficient particle filtering via sparse kernel density estimation particle filters pfs are bayesian filters capable of modeling nonlinear non gaussian and nonstationary dynamical systems recent research in pfs has investigated ways to appropriately sample from the posterior distribution maintain multiple hypotheses and alleviate computational costs while preserving tracking accuracy to address these issues a novel utilization of the support vector data description svdd density estimation method within the particle filtering framework is presented the svdd density estimate can be integrated into a wide range of pfs to realize several benefits it yields a sparse representation of the posterior density that reduces the computational complexity of the pf the proposed approach also provides an analytical expression for the posterior distribution that can be used to identify its modes for maintaining multiple hypotheses and computing the map estimate and to directly sample from the posterior we present several experiments that demonstrate the advantages of incorporating a sparse kernel density estimate in a particle filter

discrete convexity and polynomial solvability in minimum 0 extension problems a 0 0 extension of graph vargamma is a metric d d on a set v v containing the vertex set v vargamma v of vargamma such that d d extends the shortest path metric of vargamma and for all x in v x v there exists a vertex s s in vargamma with d x s 0 d x s 0 the minimum 0 0 extension problem 0 ext vargamma on vargamma is given a set v supseteq v vargamma v v and a nonnegative cost function c c defined on the set of all pairs of v v find a 0 0 extension d d of vargamma with sum xy c xy d x y xyc xy d x y minimum the 0 0 extension problem generalizes a number of basic combinatorial optimization problems such as minimum s t s t cut problem and multiway cut problem karzanov proved the polynomial solvability of 0 ext vargamma for a certain large class of modular graphs vargamma and raised the question what are the graphs vargamma for which 0 ext vargamma can be solved in polynomial time he also proved that 0 ext vargamma is np hard if vargamma is not modular or not orientable in a certain sense in this paper we prove the converse if vargamma is orientable and modular then 0 ext vargamma can be solved in polynomial time this completes the classification of graphs vargamma for which 0 ext vargamma is tractable to prove our main result we develop a theory of discrete convex functions on orientable modular graphs analogous to discrete convex analysis by murota and utilize a recent result of thapper and ivn on valued csp

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

rounding algorithms for a geometric embedding of minimum multiway cut given an undirected graph with edge costs and a subset ofk 3 nodes calledterminals a multiway ork way cut is a subset of the edges whose removal disconnects each terminal from the others the multiway cut problem is to find a minimum cost multiway cut this problem is max snp hard recently calinescu et al calinescu g h karloff y rabani 2000 an improved approximation algorithm for multiway cut j comput system sci 60 3 564 574 gave a novel geometric relaxation of the problem and a rounding scheme that produced a 3 2 1 k approximation algorithm in this paper we study their geometric relaxation in particular we study the worst case ratio between the value of the relaxation and the value of the minimum multicut the so called integrality gap of the relaxation fork 3 we show the integrality gap is 12 11 giving tight upper and lower bounds that is we exhibit a family of graphs with integrality gaps arbitrarily close to 12 11 and give an algorithm that finds a cut of value 12 11 times the relaxation value our lower bound shows that this is the best possible performance guarantee for any algorithm based purely on the value of the relaxation our upper bound meets the lower bound and improves the factor of 7 6 shown by calinescu et al for allk we show that there exists a rounding scheme with performance ratio equal to the integrality gap and we give explicit constructions of polynomial time rounding schemes that lead to improved upper bounds fork 4 and 5 our best upper bounds are based on computer constructed rounding schemes with computer proofs of correctness for generalk we give an algorithm with performance ratio 1 3438 e k our results were discovered with the help of computational experiments that we also describe here

the complexity of multicut and mixed multicut problems in di graphs the multicut problem with respect to the fixed pattern di graph q is the following given a di graph g an integer k and a 1 1 mapping p of the vertices of q to a subset x of v g determine whether there exists a set s of at most k edges arcs of g such that g s has no p q p q path for every edge arc qq of q we first prove a dichotomy result for the complexity they are all either polynomial or np complete for multicut problems in di graphs in terms of the pattern q then we consider a variant where the pattern is a mixed graph m the input is a digraph d an integer k and a 1 1 mapping p from v m to some subset of v d and the goal is to decide whether there exists a set s of at most k arcs of d that meets every p q p q path in d for every arc qq of m and every p p p p path in the underlying undirected graph of d for every edge pp of m again we prove a dichotomy result for the complexity of such mixed multicut problems in terms of the pattern m it turns out that the only polynomial cases are those where the pattern is either a polynomial graph pattern or a polynomial digraph pattern as soon as m contains both edges and arcs the m mixed cut problem becomes np complete finally we prove that the directed feedback arc set problem is np complete even in the class of digraphs which have a feedback vertex set of size 2

a forward constrained regression algorithm for sparse kernel density estimation using the classical parzen window pw estimate as the target function the sparse kernel density estimator is constructed in a forward constrained regression fcr manner the proposed algorithm selects significant kernels one at a time while the leave one out loo test score is minimized subject to a simple positivity constraint in each forward stage the model parameter estimation in each forward stage is simply the solution of jackknife parameter estimator for a single parameter subject to the same positivity constraint check for each selected kernels the associated kernel width is updated via the gauss newton method with the model parameter estimate fixed the proposed approach is simple to implement and the associated computational cost is very low numerical examples are employed to demonstrate the efficacy of the proposed approach

complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs multicut is a fundamental network communication and connectivity problem it is defined as given an undirected graph and a collection of pairs of terminal vertices find a minimum set of edges or vertices whose removal disconnects each pair we mainly focus on the case of removing vertices where we distinguish between allowing or disallowing the removal of terminal vertices complementing and refining previous results from the literature we provide several np completeness and fixedparameter tractability results for restricted classes of graphs such as trees interval graphs and graphs of bounded treewidth

complexity of the multicut problem in its vanilla partial and generalized versions in graphs of bounded treewidth abstract in the multicut problem the input consists of a graph and a set of pairs of terminal vertices that have to be disconnected by the removal of a set of edges with minimum total weight the partial multicut problem generalizes multicut by only requiring a given number of pairs of terminals to be disconnected by the removal of edges it has been shown that multicut remains apx hard if the treewidth t w g of its input graph g is bounded but that it is fpt w r t t w g h where h is the demand graph whose vertices are the terminals and whose edges are the pairs of terminals we prove that this also holds for partial multicut furthermore it has been proved that multicut also becomes polynomial time solvable if t w g is bounded and h is complete this is the multiterminal cut problem we extend this result in two directions by proving that multicut remains polynomial time solvable if t w g h is bounded and that this remains true for partial multicut finally we show that if we further generalize the problem to allow non unitary profits for pairs of terminals then the problem is weakly np hard and has an fptas if t w g h is bounded and becomes apx hard if t w g h is bounded

a new approach to the minimum cut problem this paper present a new approach to finding minimum cuts in undirected graphs the fundamental principle is simple the edges in a graph s minimum cut form an extremely small fraction of the graph s edges using this idea we give a randomized strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability the algorithm runs in o n 2 log 3 n time a significant improvement over the previous o mn time bounds based on maximum flows it is simple and intuitive and uses no complex data structures our algorithm can be parallelized to run in rnc with n 2 processors this gives the first proof that the minimum cut problem can be solved in rnc the algorithm does more than find a single minimum cut it finds all of them with minor modifications our algorithm solves two other problems of interest our algorithm finds all cuts with value within a multiplicative factor of of the minimum cut s in expected o n 2 time or in rnc with n 2 processors the problem of finding a minimum multiway cut of graph into r pieces is solved in expected o n 2 r 1 time or in rnc with n 2 r 1 processors the trace of the algorithm s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph this data structure can be efficiently transformed into the more standard cactus representing for minimum cuts

correlation clustering in general weighted graphs we consider the following general correlation clustering problem n bansal a blum s chawla correlation clustering in proc 43rd annu ieee symp on foundations of computer science vancouver canada november 2002 pp 238 250 given a graph with real nonnegative edge weights and a edge labelling partition the vertices into clusters to minimize the total weight of cut edges and uncut edges thus edges with large weights representing strong correlations between endpoints encourage those endpoints to belong to a common cluster while edges with large weights encourage the endpoints to belong to different clusters in contrast to most clustering problems correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering both of these parameters are effectively chosen to be best possible by the problem definition correlation clustering was introduced by bansal et al correlation clustering in proc 43rd annu ieee syrup on foundations of computer science vancouver canada november 2002 pp 238 250 motivated by both document clustering and agnostic learning they proved np hardness and gave constant factor approximation algorithms for the special case in which the graph is complete full information and every edge has the same weight we give an o log n approximation algorithm for the general case based on a linear programming rounding and the region growing technique we also prove that this linear program has a gap of log n and therefore our approximation is tight under this approach we also give an o r3 approximation algorithm for kr r minor free graphs on the other hand we show that the problem is equivalent to minimum multicut and therefore apx hard and difficult to approximate better than log n

greedy splitting algorithms for approximating multiway partition problems given a system v t f k where v is a finite set open image in new window is a submodular function and k 2 is an integer the general multiway partition problem mpp asks to find a k partition open image in new window v1 v2 vk of v that satisfiesopen image in new windowfor all i and minimizes f v1 f v2 f vk where open image in new window is a k partition of open image in new window hold mpp formulation captures a generalization in submodular systems of many np hard problems such as k way cut multiterminal cut target split and their generalizations in hypergraphs this paper presents a simple and unified framework for developing and analyzing approximation algorithms for various mpps

on weighted multiway cuts in trees a min max theorem is developed for the multiway cut problem of edge weighted trees we present a polynomial time algorithm to construct an optimal dual solution if edge weights come in unary representation applications to biology also require some more complex edge weights we describe a dynamic programming type algorithm for this more general problem from biology and show that our min max theorem does not apply to it

packing and covering with linear programming a survey this paper considers the polyhedral results and the min max results on packing and covering problems of the decade since the strong perfect graph theorem published in 2006 the main such results are available for the packing problem however there are still important polyhedral questions that remain open for the covering problem the main questions are still open although there has been important progress we survey some of the main results with emphasis on those where linear programming and graph theory come together they mainly concern the covering of cycles or dicycles in graphs or signed graphs either with vertices or edges this includes the multicut and integral multiflow problems

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

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

parameterized graph separation problems we consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible the following five problems are studied delete k vertices such that a each of the given l terminals is separated from the others b each of the given l pairs of terminals is separated c exactly l vertices are cut away from the graph d exactly l connected vertices are cut away from the graph e the graph is separated into at least l components we show that if both k and l are parameters then a b and d are fixed parameter tractable while c and e are w 1 hard

a faster strongly polynomial minimum cost flow algorithm in this paper we present a new strongly polynomial time algorithm for the minimum cost flow problem based on a refinement of the edmonds karp scaling technique our algorithm solves the uncapacitated minimum cost flow problem as a sequence of on log n shortest path problems on networks with n nodes and m arcs and runs in on log nm n log n time using a standard transformation this approach yields an om log nm n log n algorithm for the capacitated minimum cost flow problem this algorithm improves the best previous strongly polynomial time algorithm due to z galil and e tardos by a factor of n2 m our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds say m is much less than m in this case the running time of the algorithm is om n log nm n log n

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

on the complexity of the selective graph coloring problem in some special classes of graphs in this paper we consider the selective graph coloring problem given an integer k 1k 1 and a graph g v e g v e with a partition v1 vpv1 vp of vv it consists in deciding whether there exists a set v v in gg such that v vi 1 v vi 1 for all i 1 p i 1 p and such that the graph induced by v v is kk colorable we investigate the complexity status of this problem in various classes of graphs

large scale integer programs in image analysis an important problem in image analysis is to segment an image into regions with different class labels this is relevant in applications in medicine and cartography in a proper statistical framework this problem may be viewed as a discrete optimization problem we present two integer linear programming formulations of the problem and study some properties of these models and associated polytopes different algorithms for solving these problems are suggested and compared on some realistic data in particular a lagrangian algorithm is shown to have a very promising performance the algorithm is based on the technique of cost splitting and uses the fact that certain relaxed problems may be solved as shortest path problems

efficient kernel density estimation using the fast gauss transform with applications to color modeling and tracking many vision algorithms depend on the estimation of a probability density function from observations kernel density estimation techniques are quite general and powerful methods for this problem but have a significant disadvantage in that they are computationally intensive in this paper we explore the use of kernel density estimation with the fast gauss transform fgt for problems in vision the fgt allows the summation of a mixture of ill gaussians at n evaluation points in o m n time as opposed to o mn time for a naive evaluation and can be used to considerably speed up kernel density estimation we present applications of the technique to problems from image segmentation and tracking and show that the algorithm allows application of advanced statistical techniques to solve practical vision problems in real time with today s computers

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

the complexity of soft constraint satisfaction over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types one very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations known as polymorphisms in this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints we launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation which we call a multimorphism we show that many tractable sets of soft constraints both established and novel can be characterised by the presence of particular multimorphisms we also show that a simple set of np hard constraints has very restricted multimorphisms finally we use the notion of multimorphism to give a complete classification of complexity for the boolean case which extends several earlier classification results for particular special cases

on the parameterized complexity of separating certain sources from the target abstract multicut is proven to be fpt parameterized by the size of the cutset p we focus on its generalization partial multicut setting the number of pairs of terminals to be separated as a parameter r it asks for the minimum edge vertex cut whose removal would separate r sources s i from their target t i our main contributions concern its single target subproblem partial one target cut potc we show that potc when edges are to be cut is fpt parameterized by the cutset size p but is w 1 hard when parameterized only with r the vertex cutting version of potc is w 1 hard regardless of parameter configurations involving r and p

beating the 2 approximation factor for global bicut in the fixed terminal bicut problem the input is a directed graph with two specified nodes s and t and the goal is to find a smallest subset of edges whose removal ensures that s cannot reach tandt cannot reach s in the global bicut problem the input is a directed graph and the goal is to find a smallest subset of edges whose removal ensures that there exist two nodes s and t such that s cannot reach tandt cannot reach s fixed terminal bicut and global bicut are natural extensions of s t min cut and global min cut respectively from undirected graphs to directed graphs fixed terminal bicut is np hard admits a simple 2 approximation and does not admit a 2 epsilon approximation for any constant epsilon 0 assuming the unique games conjecture in this work we show that global bicut admits a 2 1 448 approximation thus improving on the approximability of the global variant in comparison to the fixed terminal variant

improving multicut in directed trees by upgrading nodes abstract in this paper we consider the network improvement problem for multicut by upgrading nodes in a directed tree t v e with multiple sources and multiple terminals in a node based upgrading model a node v can be upgraded at the expense of c v and such an upgrade reduces weights on all edges incident to v the objective is to upgrade a minimum cost subset s v of nodes such that the resulting network has a multicut in which no edge has weight larger than a given value d we first obtain a minimum cardinality node multicut v c for tree t then find the minimum cost upgrading set based on the upgrading sets for the subtrees rooted at the nodes in v c we show that our algorithm is polynomial when the number of source terminal pairs is upper bounded by a given value

fixed parameter tractability for subset feedback set problems with parity constraints the subset feedback set problem which is a generalization of the well known feedback vertex set problem is that we are given an undirected graph g with a vertex subset s and a positive integer k and the goal is to find a vertex set x of size at most k such that g x has no s cycle where an s cycle is a cycle having at least one vertex of s it was recently shown that this problem is fixed parameter tractable where k is the parameter in this paper we further generalize this problem to one with the parity constraints and show the fixed parameter tractability 1 for a parameter k there exists a fixed parameter algorithm that either finds a vertex set x of size k such that g x has no s cycle of even length or concludes that such a vertex set does not exist 2 for a parameter k there exists a fixed parameter algorithm that either finds a vertex set x of size k such that g x has no s cycle of odd length or concludes that such a vertex set does not exist

the multi terminal maximum flow network interdiction problem abstract this paper defines and studies the multi terminal maximum flow network interdiction problem mtnip in which a network user attempts to maximize flow in a network among k 3 pre specified node groups while an interdictor uses limited resources to interdict network arcs to minimize this maximum flow the paper proposes an exact mtnip e and an approximating model mpnim to solve this np hard problem and presents computational results to compare the models mtnip e is obtained by first formulating mtnip as bi level min max program and then converting it into a mixed integer program where the flow is explicitly minimized mpnim is binary integer program that does not minimize the flow directly it partitions the node set into disjoint subsets such that each node group is in a different subset and minimizes the sum of the arc capacities crossing between different subsets computational results show that mpnim can solve all instances in a few seconds while mtnip e cannot solve about one third of the problems in 24 hour the optimal objective function values of both models are equal to each other for some problems while they differ from each other as much as 46 2 in the worst case however when the post interdiction flow capacity incurred by the solution of mpnim is computed and compared to the objective value of mtnip e the largest difference is only 7 90 implying that mpnim may be a very good approximation to mtnip e

a histogram transform for probabilitydensity function estimation the estimation of multivariate probability density functions has traditionally been carried out by mixtures of parametric densities or by kernel density estimators here we present a new nonparametric approach to this problem which is based on the integration of several multivariate histograms computed over affine transformations of the training data our proposal belongs to the class of averaged histogram density estimators the inherent discontinuities of the histograms are smoothed while their low computational complexity is retained we provide a formal proof of the convergence to the real probability density function as the number of training samples grows and we demonstrate the performance of our approach when compared with a set of standard probability density estimators

optimal 3 terminal cuts and linear programming given an undirected graph g v e and three specified terminal nodes t1 t2 t3 a 3 cut is a subset a of e such that no two terminals are in the same component of g a if a non negative edge weight ce is specified for each e e the optimal 3 cut problem is to find a 3 cut of minimum total weight this problem is hard and in fact is max hard an approximation algorithm having performance guarantee has recently been given by c linescu karloff and rabani it is based on a certain linear programming relaxation for which it is shown that the optimal 3 cut has weight at most times the optimal lp value it is proved here that can be improved to and that this is best possible as a consequence we obtain an approximation algorithm for the optimal 3 cut problem having performance guarantee in addition we show that is best possible for this algorithm

locally defined principal curves and surfaces principal curves are defined as self consistent smooth curves passing through the middle of the data and they have been used in many applications of machine learning as a generalization dimensionality reduction and a feature extraction tool we redefine principal curves and surfaces in terms of the gradient and the hessian of the probability density estimate this provides a geometric understanding of the principal curves and surfaces as well as a unifying view for clustering principal curve fitting and manifold learning by regarding those as principal manifolds of different intrinsic dimensionalities the theory does not impose any particular density estimation method can be used with any density estimator that gives continuous first and second derivatives therefore we first present our principal curve surface definition without assuming any particular density estimation method afterwards we develop practical algorithms for the commonly used kernel density estimation kde and gaussian mixture models gmm results of these algorithms are presented in notional data sets as well as real applications with comparisons to other approaches in the principal curve literature all in all we present a novel theoretical understanding of principal curves and surfaces practical algorithms as general purpose machine learning tools and applications of these algorithms to several practical problems

a bayesian analysis of simultaneous equation models by combining recursive analytical and numerical approaches abstract inference on simultaneous equation models using a bayesian approach is still far from a routine procedure serious numerical problems may occur in most practical situations in this paper a unifying framework is proposed using recent analytical see steel 1988 and well documented poly t based methods in order to reduce the dimension of the final monte carlo integration different integration sequences are examined and the implications for various model and prior assumptions are tabulated an application to klein s model i clearly illustrates the computational feasibility and numerical stability of the hybrid method proposed

bayesian estimation of common parameters from multiresponse data with missing observations new relations are derived for bayesian estimation of parameters in nonlinear multiresponse models modal and interval estimates are provided for the parameter vector of a multiresponse expectation model and for the covariance matrix of a multivariate normal error distribution incomplete observation sets are handled directly two chemical examples are analyzed

color texture segmentation using unsupervised graph cuts this paper proposes a novel approach to color texture segmentation based on graph cut techniques which finds an optimal color texture segmentation of a color textured image by regarding it as a minimum cut problem in a weighted graph a new texture descriptor based on the texton theory is introduced to efficiently represent texture attributes of the given image then the segmentation is formulated in terms of energy minimization with graph cuts where color and texton features are modelled with a multivariate finite mixture model with an unknown number of components contrary to previous supervised graph cut approaches our method finds minimum cuts using split moves in an unsupervised way the segmentation result including the number of segments is determined during the split moves without user interaction thus our method is called unsupervised graph cuts experimental results of color texture segmentation using various images including the mit vistex datasets and the berkeley datasets are presented and analyzed in terms of precision and recall to verify its effectiveness

correlation clustering with constrained cluster sizes and extended weights bounds we consider the problem of correlation clustering on graphs with constraints on both the cluster sizes and the positive and negative weights of edges our contributions are twofold first we introduce the problem of correlation clustering with bounded cluster sizes second we extend the regime of weight values for which the clustering may be performed with constant approximation guarantees in polynomial time and apply the results to the bounded cluster size problem

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

partial multicuts in trees let t v e be an undirected tree in which each edge is associated with a non negative cost and let s1 t1 sk tk be a collection of k distinct pairs of vertices given a requirement parameter t k the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from t disconnects at least t out of these k pairs this problem generalizes the well known multicut on a tree problem in which we are required to disconnect all given pairs the main contribution of this paper is an 8 3 e approximation algorithm for partial multicut on a tree whose run time is strongly polynomial for any fixed e 0 this result is achieved by introducing problem specific insight to the general framework of using the lagrangian relaxation technique in approximation algorithms our algorithm utilizes a heuristic for the closely related prize collecting variant in which we are not required to disconnect all pairs but rather incur penalties for failing to do so we provide a lagrangian multiplier preserving algorithm for the latter problem with an approximation factor of 2 finally we present a new 2 approximation algorithm for multicut on a tree based on lp rounding

algorithmic aspects of homophyly of networks abstract we investigate the algorithmic problems of the homophyly phenomenon in networks given an undirected graph g v e and a vertex coloring c v 1 2 k of g we say that a vertex v v is happy if v shares the same color with all its neighbors and unhappy otherwise and that an edge e e is happy if its two endpoints have the same color and unhappy otherwise supposing c is a partial vertex coloring of g we define the maximum happy vertices problem mhv for short as to color all the remaining vertices such that the number of happy vertices is maximized and the maximum happy edges problem mhe for short as to color all the remaining vertices such that the number of happy edges is maximized let k be the number of colors allowed in the problems we show that both mhv and mhe can be solved in polynomial time if k 2 and that both mhv and mhe are np hard if k 3 we devise a max 1 k 3 approximation algorithm for the mhv problem where is the maximum degree of vertices in the input graph and a 1 2 approximation algorithm for the mhe problem this is the first theoretical progress of these two natural and fundamental new problems

discrete and continuous models for partitioning problems recently variational relaxation techniques for approximating solutions of partitioning problems on continuous image domains have received considerable attention since they introduce significantly less artifacts than established graph cut based techniques this work is concerned with the sources of such artifacts we discuss the importance of differentiating between artifacts caused by discretization and those caused by relaxation and provide supporting numerical examples moreover we consider in depth the consequences of a recent theoretical result concerning the optimality of solutions obtained using a particular relaxation method since the employed regularizer is quite tight the considered relaxation generally involves a large computational cost we propose a method to significantly reduce these costs in a fully automatic way for a large class of metrics including tree metrics thus generalizing a method recently proposed by strekalovskiy and cremers ieee conference on computer vision and pattern recognition pp 1905 1911 2011

the complexity of multiterminal cuts in the multiterminal cut problem one is given an edge weighted graph and a subset of the vertices called terminals and is asked for a minimum weight set of edges that separates each terminal from all the others when the number k of terminals is two this is simply the mincut max flow problem and can be solved in polynomial time it is shown that the problem becomes np hard as soon as k 3 but can be solved in polynomial time for planar graphs for any fixed k the planar problem is np hard however if k is not fixed a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2 2 k of the optimal cut weight is also described

multiresponse estimation with special application to linear systems of differential equations we present a multiresponse estimation procedure for parameters of systems described by as the procedure features a generalized gauss newton algorithm for optimizing the determinant criterion efficient evaluation of the expectation function and its derivatives directly from the reaction network or compartment diagram automatic determination of starting values and eficient computational procedures for handling linear constraints on the responses

nonlinear filtering using a new proposal distribution and the improved fast gauss transform with tighter performance bounds we propose a new approach to approximating the chapman kolmogorov equation cke for particle based nonlinear filtering algorithms using a new proposal distribution and the improved fast gauss transform ifgt with tighter accuracy bounds the new proposal distribution used to obtain a monte carlo mc approximation of the cke is based on the proposal distribution found in the auxiliary marginal particle filter ampf by using mc integration to approximate the integrals of the ampf proposal distribution as well as the cke we demonstrate significant improvement in terms of both error and computation time we consider the additive state noise case where the evaluation of the cke is equivalent to performing kernel density estimation kde thus fast methods such as the ifgt can be used in practice the ifgt demonstrates performance far better than that which is predicted by current error analysis therefore the existing bounds are not useful for determining the ifgt parameters which in practice have to be chosen experimentally in order to obtain satisfactory compromise between accuracy and speed of the filtering algorithm we provide in this paper much improved performance bounds for the ifgt and which unlike the previous bound are consistent with the expectation that the error decreases as the truncation order of the ifgt increases the new bounds lead to a new definition of the ifgt parameters and give important insight into the effect that these parameters have on the error therefore facilitating the choice of parameters in practice the experimental results show that we can obtain similar error to the sequential importance sampling sis particle filter while using fewer particles furthermore the choice of the new ifgt parameters remains roughly the same for all the examples that we give

multivariate online kernel density estimation with gaussian kernels we propose a novel approach to online estimation of probability density functions which is based on kernel density estimation kde the method maintains and updates a non parametric model of the observed data from which the kde can be calculated we propose an online bandwidth estimation approach and a compression revitalization scheme which maintains the kde s complexity low we compare the proposed online kde to the state of the art approaches on examples of estimating stationary and non stationary distributions and on examples of classification the results show that the online kde outperforms or achieves a comparable performance to the state of the art and produces models with a significantly lower complexity while allowing online adaptation

approximation algorithms for disjoint paths and related routing and packing problems given a network and a set of connection requests on it we consider the maximum edge disjoint paths and related generalizations and routing problems that arise in assigning paths for these requests we present improved approximation algorithms and or integrality gaps for all problems considered the central theme of this work is the underlying multicommodity flow relaxation applications of these techniques to approximating families of packing integer programs are also presented

probability density function estimation with the frequency polygon transform a probability density function estimator is proposed which is based on frequency polygons its convergence to the true density is formally proved a mode finding algorithm is also proposed as an alternative to mean shift our approach outperforms histogram and kernel based estimators in synthetic and real datasets our proposal is shown to be suitable to object tracking in video sequences most current nonparametric approaches to probability density function estimation are based on the kernel density estimator also known as the parzen window estimator a usual alternative is the multivariate histogram which features a low computational complexity multivariate frequency polygons have often been neglected even though they share many of the advantages of the histograms while they are continuous unlike the histograms here we build on our previous work on histograms in order to propose a new probability density estimator which is based on averaging multivariate frequency polygons the convergence of the estimator is formally proved experiments are carried out with synthetic and real machine learning datasets finally image denoising and object tracking applications are also considered

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

approximation algorithms for classification problems with pairwise relationships metric labeling and markov random fields in a traditional classification problem we wish to assign one of k labels or classes to each of n objects in a way that is consistent with some observed data that we have about the problem an active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified this issue is one of the principal motivations for the framework of markov random fields and it arises in areas such as image processing biometry and document analysis in its most basic form this style of analysis seeks to find a classification that optimizes a combinatorial function consisting of assignment costs based on the individual choice of label we make for each object and separation costs based on the pair of choices we make for two related objects we formulate a general classification problem of this type the metric labeling problem we show that it contains as special cases a number of standard classification frameworks including several arising from the theory of markov random fields from the perspective of combinatorial optimization our problem can be viewed as a substantial generalization of the multiway cut problem and equivalent to a type of uncapacitated quadratic assignment problem we provide the first nontrivial polynomial time approximation algorithms for a general family of classification problems of this type our main result is an o log k log log k approximation algorithm for the metric labeling problem with respect to an arbitrary metric on a set of k labels and an arbitrary weighted graph of relationships on a set of objects for the special case in which the labels are endowed with the uniform metric all distances are the same our methods provide a 2 approximation algorithm

fast algorithms for shortest paths in planar graphs with applications graph decomposition and data structures techniques are presented that exploit the structure of planar graphs to yield faster algorithms for a number of shortest path and related problems improved

on the hardness of finding near optimal multicuts in directed acyclic graphs abstract given an edge weighted di graph and a list of source sink pairs of vertices of this graph the minimum multicut problem consists in selecting a minimum weight set of edges or arcs whose removal leaves no path from each source to the corresponding sink this is a well known np hard problem and improving several previous results we show that it remains apx hard in unweighted directed acyclic graphs dag even with only two source sink pairs this is also true if we remove vertices instead of arcs

multiresponse parameter estimation with a new and noninformative prior summary a noninformative prior distribution for the covariance matrix of a normal distribution is derived from a data relocated likelihood function this new prior distribution also follows from jeffreys s rule resulting modifications of current formulae for multiresponse parameter estimation are summarized