searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

heat flow on alexandrov spaces we prove that on compact alexandrov spaces with curvature bounded below the gradient flow of the dirichlet energy in the space produces the same evolution as the gradient flow of the relative entropy in the wasserstein space this means that the heat flow is well defined by either one of the two gradient flows combining properties of these flows we are able to deduce the lipschitz continuity of the heat kernel as well as bakry emery gradient estimates and the condition our identification is established by purely metric means unlike preceding results relying on pde techniques our approach generalizes to the case of heat flow with drift 2012 wiley periodicals inc

the complexity of pursuit on a graph abstract a robber and k cops choose starting vertices in a graph and move in alternation from vertex to vertex along the edges of the graph capture occurs if a cop ever shares a vertex with the robber if k is not fixed and if either the graph is directed or initial positions are given we show that the problem is exptime complete similar techniques lead to the pspace and exptime completeness of some other combinatorial games that were previously only known to be np and pspace hard

non contraction of heat flow on minkowski spaces we study contractivity properties of gradient flows for functions on normed spaces or more generally on finsler manifolds contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions this is different from the usual convexity along geodesics in non riemannian finsler manifolds as an application we show that the heat flow on minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic wasserstein distance

on the computational complexity of a game of cops and robbers we study the computational complexity of a perfect information two player game proposed by aigner and fromme 1984 1 the game takes place on an undirected graph where n simultaneously moving cops attempt to capture a single robber all moving at the same speed the players are allowed to pick their starting positions at the first move the question of the computational complexity of deciding this game was raised by goldstein and reingold 1995 9 we prove that the game is hard for pspace

on the desirability of acyclic database schemes a class of database schemes called acychc was recently introduced it is shown that this class has a number of desirable properties in particular several desirable properties that have been studied by other researchers m very different terms are all shown to be eqmvalent to acydicity in addition several equivalent charactenzauons of the class m terms of graphs and hypergraphs are given and a smaple algorithm for determining acychclty is presented also given are several eqmvalent characterizations of those sets m of multivalued dependencies such that m is the set of murlvalued dependencies that are the consequences of a given join dependency several characterizations for a conflict free in the sense of lien set of muluvalued dependencies are provided

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

uniform convexity and smoothness and their applications in finsler geometry we generalize the alexandrov toponogov comparison theorems to finsler manifolds under suitable upper lower resp bounds on the flag and tangent curvatures together with the 2 uniform convexity smoothness resp of tangent spaces we show the 2 uniform convexity smoothness resp of finsler manifolds as applications we prove the almost everywhere existence of the second order differentials of semi convex functions and of c concave functions with the quadratic cost function

recursive markov chains stochastic grammars and monotone systems of nonlinear equations we define recursive markov chains rmcs a class of finitely presented denumerable markov chains and we study algorithms for their analysis informally an rmc consists of a collection of finite state markov chains with the ability to invoke each other in a potentially recursive manner rmcs offer a natural abstract model for probabilistic programs with procedures they generalize in a precise sense a number of well studied stochastic models including stochastic context free grammars scfg and multi type branching processes mt bp we focus on algorithms for reachability and termination analysis for rmcs what is the probability that an rmc started from a given state reaches another target state or that it terminatesq these probabilities are in general irrational and they arise as least fixed point solutions to certain monotone systems of nonlinear equations associated with rmcs we address both the qualitative problem of determining whether the probabilities are 0 1 or in between and the quantitative problems of comparing the probabilities with a given bound or approximating them to desired precision we show that all these problems can be solved in pspace using a decision procedure for the existential theory of reals we provide a more practical algorithm based on a decomposed version of multi variate newton s method and prove that it always converges monotonically to the desired probabilities we show this method applies more generally to any monotone polynomial system we obtain polynomial time algorithms for various special subclasses of rmcs among these for scfgs and mt bps equivalently for 1 exit rmcs the qualitative problem can be solved in p time for linearly recursive rmcs the probabilities are rational and can be computed exactly in p time we show that our pspace upper bounds cannot be substantially improved without a breakthrough on long standing open problems the square root sum problem and an arithmetic circuit decision problem that captures p time on the unit cost rational arithmetic ram model we show that these problems reduce to the qualitative problem and to the approximation problem to within any nontrivial error for termination probabilities of general rmcs and to the quantitative decision problem for termination extinction of scfgs mt bps

the complexity of mixed multi unit combinatorial auctions tractability under structural and qualitative restrictions mixed multi unit combinatorial auctions mmucas are extensions of classical combinatorial auctions cas where bidders trade transformations of goods rather than just sets of goods solving mmucas i e determining the sequences of bids to be accepted by the auctioneer is computationally intractable in general however differently from classical combinatorial auctions little was known about whether polynomial time solvable classes of mmucas can be singled out on the basis of their characteristics the paper fills this gap by studying the computational complexity of mmuca instances under structural and qualitative restrictions which characterize interactions among bidders and types of bids involved in the various transformations respectively

nilpotency almost nonnegative curvature and the gradient flow on alexandrov spaces we show that almost nonnegatively curved m manifolds are up to finite cover nilpotent spaces in the sense of homotopy theory and have c m nilpotent fundamental groups we also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

geometric properties of cones with applications on the hellinger kantorovich space and a new distance on the space of probability measures abstract we study general geometric properties of cone spaces and we apply them on the hellinger kantorovich space m x hk we exploit a two parameter scaling property of the hellinger kantorovich metric hk and we prove the existence of a distance shk on the space of probability measures that turns the hellinger kantorovich space m x hk into a cone space over the space of probabilities measures p x shk we provide a two parameter rescaling of geodesics in m x hk and for p x shk we obtain a full characterization of the geodesics we finally prove finer geometric properties including local angle condition and partial k semiconcavity of the squared distances that will be used in a future paper to prove existence of gradient flows on both spaces

pursuing a fast robber on a graph the cops and robbers game as originally defined independently by quilliot and by nowakowski and winkler in the 1980s has been much studied but very few results pertain to the algorithmic and complexity aspects of it in this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is np hard and that the parameterized version of the problem is w 2 hard the proof extends to the case where the robber moves s time faster than the cops we show that on split graphs the problem is polynomially solvable if s 1 but is np hard if s 2 we further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for email protected 2 finally we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops

hybrid tractability of valued constraint problems the constraint satisfaction problem csp is a central generic problem in computer science and artificial intelligence it provides a common framework for many theoretical problems as well as for many real life applications valued constraint problems are a generalisation of the csp which allow the user to model optimisation problems considerable effort has been made in identifying properties which ensure tractability in such problems in this work we initiate the study of hybrid tractability of valued constraint problems that is properties which guarantee tractability of the given valued constraint problem but which do not depend only on the underlying structure of the instance such as being tree structured or only on the types of valued constraints in the instance such as submodularity we present several novel hybrid classes of valued constraint problems in which all unary constraints are allowed which include a machine scheduling problem constraint problems of arbitrary arities with no overlapping nogoods and the softalldiff constraint with arbitrary unary valued constraints an important tool in our investigation will be the notion of forbidden substructures

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

a new characterization of talagrand s transport entropy inequalities and applications we show that talagrand s transport inequality is equivalent to a restricted logarithmic sobolev inequality this result clarifies the links between these two important functional inequalities as an application we give the first proof of the fact that talagrand s inequality is stable under bounded perturbations

greedy strategies and larger islands of tractability for conjunctive queries and constraint satisfaction problems most structural decomposition methods can be characterized through hypergraph games that are variations of the robber and cops graph game that characterizes the notion of treewidth decomposition trees correspond to monotone winning strategies of the cops where the escape space of the robber on the hypergraph is shrunk monotonically cops using non monotonic strategies are more powerful but such strategies do not correspond to valid decompositions in general the paper provides a general way to exploit the power of non monotonic strategies by allowing a greedy form of non monotonicity it is shown that deciding the existence of a non monotone greedy winning strategy and compute one if any is tractable moreover from greedy strategies we can compute valid decomposition trees efficiently as a consequence we are able to add power to structural methods and to obtain larger islands of tractability such as the one based on the new notion of greedy hypertree decomposition

on the power of structural decompositions of graph based representations of constraint problems the constraint satisfaction problem csp is a central issue of research in artificial intelligence due to its intractability many efforts have been made in order to identify tractable classes of csp instances and in fact deep and useful results have already been achieved in particular this paper focuses on structural decomposition methods which are essentially meant to look for near acyclicity properties of the graphs or hypergraphs that encode the structure of the constraints interactions in general constraint scopes comprise an arbitrary number of variables and thus this structure may be naturally encoded via hypergraphs however in many practical applications decomposition methods are applied over suitable graph representations of the possibly non binary csp instances at hand despite the great interest in such binary approaches a formal analysis of their power in terms of their ability of identifying islands of tractability was missing in the literature the aim of this paper is precisely to fill this gap by studying the relationships among binary structural methods and by providing a clear picture of the tractable fragments of csp that can be specified with respect to each of these decomposition approaches when they are applied to binary representations of non binary csp instances in particular various long standing questions about primal dual and incidence graph encodings are answered the picture is then completed by comparing methods on binary encodings with methods specifically conceived for non binary constraints

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

gradient flows and evolution variational inequalities in metric spaces i structural properties abstract this is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space x d that can be characterized by evolution variational inequalities evi we present new results concerning the structural properties of solutions to the evi formulation such as contraction regularity asymptotic expansion precise energy identity stability asymptotic behavior and their link with the geodesic convexity of the driving functional under the crucial assumption of the existence of an evi gradient flow we will also prove two main results the equivalence with the de giorgi variational characterization of curves of maximal slope the convergence of the minimizing movement jko scheme to the evi gradient flow with an explicit and uniform error estimate of order 1 2 with respect to the step size independent of any geometric hypothesis as upper or lower curvature bounds on d in order to avoid any compactness assumption we will also introduce a suitable relaxation of the minimizing movement algorithm obtained by the ekeland variational principle and we will prove its uniform convergence as well

the vertex separation and search number of a graph abstract we relate two concepts in graph theory and algorithmic complexity namely the search number and the vertex separation of a graph let s g denote the search number and vs g denote the vertex separation of a connected undirected graph g we show that vs g s g vs g 2 and we give a simple transformation from g to g such that vs g s g we characterize those trees having a given vertex separation and describe the smallest such trees we also note that there exist trees for which the difference between search number and vertex separation is indeed 2 we give algorithms that for any tree t compute vs t in linear time and compute an optimal layout with respect to vertex separation in time o n log n vertex separation has previously been related to progressive black white pebble demand and has been shown to be identical to a variant of search number node search number and to path width which has been related to gate matrix layout cost all these properties are known to be computationally intractable for fixed k an o n log 2 n algorithm is known which decides whether a graph has path width at most k

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

tree projections and structural decomposition methods minimality and game theoretic characterization tree projections provide a mathematical framework that encompasses all the various purely structural decomposition methods that have been proposed in the literature to single out classes of nearly acyclic hyper graphs such as the tree decomposition method which is the most powerful decomposition method on graphs and the generalized hypertree decomposition method which is its natural counterpart on arbitrary hypergraphs the paper analyzes this framework by focusing in particular on minimal tree projections that is on tree projections without useless redundancies first it is shown that minimal tree projections enjoy a number of properties that are usually required for normal form decompositions in various structural decomposition methods in particular they enjoy the same kind of connection properties as minimal tree decompositions of graphs with the result being tight in the light of the negative answer that is provided to the open question about whether they enjoy a slightly stronger notion of connection property defined to speed up the computation of hypertree decompositions second it is shown that tree projections admit a natural game theoretic characterization in terms of the robber and captain game in this game as for the robber and cops game characterizing tree decompositions the existence of winning strategies implies the existence of monotone ones as a special case the robber and captain game can be used to characterize the generalized hypertree decomposition method where such a game theoretic characterization was missing and asked for besides their theoretical interest these results have immediate algorithmic applications both for the general setting and for structural decomposition methods that can be recast in terms of tree projections

gradient flows on wasserstein spaces over compact alexandrov spaces we establish the existence of euclidean tangent cones on wasserstein spaces over compact alexandrov spaces of curvature bounded below by using this riemannian structure we formulate and construct gradient flows of functions on such spaces if the underlying space is a riemannian manifold of nonnegative sectional curvature then our gradient flow of the free energy produces a solution of the linear fokker planck equation

decomposing combinatorial auctions and set packing problems combinatorial auctions allow bidders to bid on bundles of items rather than just on single items the winner determination problem in combinatorial auctions is the problem of determining the allocation of items to bidders such that the sum of the accepted bid prices is maximized this problem is equivalent to the well known maximum weight set packing problem even though these problems are np hard in general they can be solved in polynomial time on instances whose associated item graphs have bounded treewidth called structured item graphs however the tractability of determining whether for a given problem instance a structured item graph of fixed treewidth exists and if so computing one efficiently was an open problem in this article we solve this problem by proving that deciding the existence of structured item graphs is computationally intractable even for treewidth 3 motivated by this unfavorable complexity result we investigate other structural restrictions and we show that the notion of hypertree decomposition a well studied measure of hypergraph cyclicity turns out to be most useful here indeed we show that the winner determination problem is solvable in polynomial time on instances whose dual auction hypergraphs have bounded hypertree width our solution method is based on encoding winner determination via a constraint satisfaction optimization problem and on exhibiting an algorithm to solve this latter problem efficiently for such structurally restricted instances the class of tractable instances identified by our approach while being efficiently recognizable properly contains the class of instances having a structured item graph moreover on the larger class our method solves winner determination with the same asymptotic complexity as the best algorithm proposed in the literature for the subclass of structured item graphs hypertree decompositions can equally profitably be applied to the maximum weight independent set problem which is the dual problem of maximum weight set packing

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

hopfield neural network flow a geometric viewpoint we provide gradient flow interpretations for the continuous time continuous state hopfield neural network hnn the ordinary and stochastic differential equations associated with the hnn were introduced in the literature as analog optimizers and were reported to exhibit good performance in numerical experiments in this work we point out that the deterministic hnn can be transcribed into amari s natural gradient descent and thereby uncover the explicit relation between the underlying riemannian metric and the activation functions by exploiting an equivalence between the natural gradient descent and the mirror descent we show how the choice of activation function governs the geometry of the hnn dynamics for the stochastic hnn we show that the so called diffusion machine while not a gradient flow itself induces a gradient flow when lifted in the space of probability measures we characterize this infinite dimensional flow as the gradient descent of certain free energy with respect to a wasserstein metric that depends on the geodesic distance on the ground manifold furthermore we demonstrate how this gradient flow interpretation can be used for fast computation via recently developed proximal algorithms

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

the complexity of zero visibility cops and robber in this work we deal with the computational complexity aspects of the zero visibility cops and robber game we provide an algorithm that computes the zero visibility copnumber of a tree in linear time and show that the corresponding decision problem is np complete even for the class of starlike graphs

degrees of acyclicity for hypergraphs and relational database schemes database schemes winch intuitively are collecuons of table skeletons can be wewed as hypergraphs a hypergraph is a generalization of an ordinary undirected graph such that an edge need not contain exactly two nodes but can instead contain an arbitrary nonzero number of nodes a class of acychc database schemes was recently introduced a number of basic desirable propemes of database schemes have been shown to be equivalent to acyclicity this shows the naturalness of the concept however unlike the situation for ordinary undirected graphs there are several natural noneqmvalent notions of acyclicity for hypergraphs and hence for database schemes various desirable properties of database schemes are constdered and it is shown that they fall into several equivalence classes each completely characterized by the degree of acycliclty of the scheme the results are also of interest from a purely graph theoretic viewpomt the original notion of aeyclicity has the countermtmtive property that a subhypergraph of an acychc hypergraph can be cyclic this strange behavior does not occur for the new degrees of acyelicity that are considered

some geometric calculations on wasserstein space we compute the riemannian connection and curvature for the wasserstein space of a smooth compact riemannian manifold

on clique separators nearly chordal graphs and the maximum weight stable set problem clique separators in graphs are a helpful tool used by tarjan as a divide and conquer approach for solving various graph problems such as the maximum weight stable set mws problem maximum clique graph coloring and minimum fill in but few examples of graph classes having clique separators are known we use this method to solve mws in polynomial time for two classes where the unweighted maximum stable set ms problem is solvable in polynomial time by augmenting techniques but the complexity of the mws problem was open another example namely a result by alekseev for the mws problem on a subclass of p 5 free graphs obtained by clique separators can be improved by our techniques we also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion a graph is email protected if for each of its vertices the subgraph induced by the set of its nonneighbors has property p we deal with the cases email protected chordal perfect this also simplifies a result obtained by a method called struction

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

displacement convexity of generalized relative entropies abstract we investigate the m relative entropy which stems from the bregman divergence on weighted riemannian and finsler manifolds we prove that the displacement k convexity of the m relative entropy is equivalent to the combination of the nonnegativity of the weighted ricci curvature and the k convexity of the weight function we use this to show appropriate variants of the talagrand hwi and the logarithmic sobolev inequalities as well as the concentration of measures we also prove that the gradient flow of the m relative entropy produces a solution to the porous medium equation or the fast diffusion equation

duality on gradient estimates and wasserstein controls abstract we establish a duality between l p wasserstein control and l q gradient estimate in a general framework our result extends a known result for a heat flow on a riemannian manifold especially we can derive a wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound by applying our result to a subelliptic heat flow on a lie group we obtain a coupling of heat distributions which carries a good control of their relative distance

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

on atomic structure of p5 free subclasses and maximum weight independent set problem graph decompositions play a crucial role in structural graph theory and in designing efficient graph algorithms among them clique separator decomposition a decomposition tree of the graph whose leaves have no clique separator so called atoms used by tarjan for solving various optimization problems recently received much attention in this note we first derive the atomic structure of two subclasses of p 5 free graphs where p 5 is a chordless path on five vertices these results enable us to provide efficient solutions for the maximum weight independent set problem in these classes of graphs

more results on weighted independent domination weighted independent domination is an np hard graph problem which remains computationally intractable in many restricted graph classes only few examples of classes are available where the problem admits polynomial time solutions in the present paper we extend the short list of such classes with two new examples

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

tractable hypergraph properties for constraint satisfaction and conjunctive queries an important question in the study of constraint satisfaction problems csp is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem for binary csp instances that is where each constraint involves only two variables the situation is well understood the complexity of the problem essentially depends on the treewidth of the graph of the constraints grohe 2007 marx 2010b however this is not the correct answer if constraints with unbounded number of variables are allowed and in particular for csp instances arising from query evaluation problems in database theory formally if h is a class of hypergraphs then let csp h be csp restricted to instances whose hypergraph is in h our goal is to characterize those classes of hypergraphs for which csp h is polynomial time solvable or fixed parameter tractable parameterized by the number of variables note that in the applications related to database query evaluation we usually assume that the number of variables is much smaller than the size of the instance thus parameterization by the number of variables is a meaningful question the most general known property of h that makes csp h polynomial time solvable is bounded fractional hypertree width here we introduce a new hypergraph measure called submodular width and show that bounded submodular width of h which is a strictly more general property than bounded fractional hypertree width implies that csp h is fixed parameter tractable in a matching hardness result we show that if h has unbounded submodular width then csp h is not fixed parameter tractable and hence not polynomial time solvable unless the exponential time hypothesis eth fails the algorithmic result uses tree decompositions in a novel way instead of using a single decomposition depending on the hypergraph the instance is split into a set of instances all on the same set of variables as the original instance and then the new instances are solved by choosing a different tree decomposition for each of them the reason why this strategy works is that the splitting can be done in such a way that the new instances are uniform with respect to the number extensions of partial solutions and therefore the number of partial solutions can be described by a submodular function for the hardness result we prove via a series of combinatorial results that if a hypergraph h has large submodular width then a 3sat instance can be efficiently simulated by a csp instance whose hypergraph is h to prove these combinatorial results we need to develop a theory of multicommodity flows on hypergraphs and vertex separators in the case when the function b s defining the cost of separator s is submodular which can be of independent interest

the complexity of acyclic conjunctive queries this paper deals with the evaluation of acyclic booleanconjunctive queries in relational databases by well known resultsof yannakakis 1981 this problem is solvable in polynomial time its precise complexity however has not been pinpointed so far weshow that the problem of evaluating acyclic boolean conjunctivequeries is complete for logcfl the class of decision problems thatare logspace reducible to a context free language since logcfl iscontained in ac1 and nc2 the evaluation problem of acyclic booleanconjunctive queries is highly parallelizable we present a paralleldatabase algorithm solving this problem with alogarithmic number ofparallel join operations the algorithm is generalized to computingthe output of relevant classes of non boolean queries we also showthat the acyclic versions of the following well known database andai problems are all logcfl complete the query output tuple problemfor conjunctive queries conjunctive query containment clausesubsumption and constraint satisfaction the logcfl completenessresult is extended to the class of queries of bounded tree widthand to other relevant query classes which are more general than theacyclic queries

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below i we investigate the finiteness structure of a complete non compact n dimensional riemannian manifold m whose radial curvature at a base point of m is bounded from below by that of a non compact von mangoldt surface of revolution with its total curvature greater than we show as our main theorem that all busemann functions on m are exhaustions and that there exists a compact subset of m such that the compact set contains all critical points for any busemann function on m as corollaries by the main theorem m has finite topological type and the isometry group of m is compact

generalized hypertree decompositions np hardness and tractable variants the generalized hypertree width ghw h of a hypergraph h is a measure of its cyclicity classes of conjunctive queries or constraint satisfaction problems whose associated hypergraphs have bounded ghw are known to be solvable in polynomial time however it has been an open problem for several years if for a fixed constant k and input hypergraph h it can be determined in polynomial time whether ghw h k here this problem is settled by proving that even for k e 3 the problem is already np hard on the way to this result another long standing open problem originally raised by goodman and shmueli 1984 in the context of join optimization is solved it is proven that determining whether a hypergraph h admits a tree projection with respect to a hypergraph g is np complete our intractability results on generalized hypertree width motivate further research on more restrictive tractable hypergraph decomposition methods that approximate generalized hypertree decomposition ghd we show that each such method is dominated by a tractable decomposition method definable through a function that associates a set of partial edges to a hypergraph by using one particular such function we define the new component hypertree decomposition method which is tractable and strictly more general than other approximations to ghd published so far

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

new applications of clique separator decomposition for the maximum weight stable set problem graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms clique separators in graphs were used by tarjan as a divide and conquer approach for solving various problems such as the maximum weight stable set mws problem colouring and minimum fill in the basic tool is a decomposition tree of the graph whose leaves have no clique separator so called atoms and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms we give new examples where the clique separator decomposition works well for the mws problem our results improve and extend various recently published results in particular we describe the atom structure for some new classes of graphs whose atoms are p 5 free the p 5 is the induced path with five vertices and obtain new polynomial time results for the mws problem the complexity of this problem on the class of p 5 free graphs is still unknown

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

on algorithms for p 5 gem free graphs a graph is p5 gem free when it does not contain p5 an induced path with five vertices or a gem a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path p4 as an induced subgraph we present o n2 time recognition algorithms for chordal gem free graphs and for p5 gem free graphs using a characterization of p5 gem free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees a brandstadt d kratsch on the structure of p5 gem free graphs discrete appl math 145 2005 155 166 we give linear time algorithms for the following np complete problems on p5 gem free graphs minimum coloring maximum weight stable set maximum weight clique and minimum clique cover