linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

the hirsch conjecture for dual transportation polyhedra an algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most m 1 n 1 extreme edges

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

preservation under extensions on well behaved finite structures a class of relational structures is said to have the extension preservation property if every first order sentence that is preserved under extensions on the class is equivalent to an existential sentence the class of all finite structures does not have the extension preservation property we study the property on classes of finite structures that are better behaved we show that the property holds for classes of acyclic structures structures of bounded degree and more generally structures that are wide in a sense that we will make precise we also show that the preservation property holds for the class of structures of treewidth at most k for any k in contrast we show that the property fails for the class of planar graphs

subexponential parameterized algorithms on bounded genus graphs and h minor free graphs we introduce a new framework for designing fixed parameter algorithms with subexponential running time 2o k no 1 our results apply to a broad family of graph problems called bidimensional problems which includes many domination and problems such as vertex cover feedback vertex set minimum maximal matching dominating set edge dominating set disk dimension and many others restricted to bounded genus graphs phrased as bipartite graph problem furthermore it is fairly straightforward to prove that a problem is bidimensional in particular our framework includes as special cases all previously known problems to have such subexponential algorithms previously these algorithms applied to planar graphs single crossing minor free graphs and or map graphs we extend these results to apply to bounded genus graphs as well in a parallel development of combinatorial results we establish an upper bound on the treewidth or branchwidth of a bounded genus graph that excludes some planar graph h as a minor this bound depends linearly on the size v h of the excluded graph h and the genus g g of the graph g and applies and extends the graph minors work of robertson and seymour building on these results we develop subexponential fixed parameter algorithms for dominating set vertex cover and set cover in any class of graphs excluding a fixed graph h as a minor in particular this general category of graphs includes planar graphs bounded genus graphs single crossing minor free graphs and any class of graphs that is closed under taking minors specifically the running time is 2o k nh where h is a constant depending only on h which is polynomial for k o log2 n we introduce a general approach for developing algorithms on h minor free graphs based on structural results about h minor free graphs at the heart of robertson and seymour s graph minors work we believe this approach opens the way to further development on problems in h minor free graphs

on the length of simplex paths the assignment case this paper reports on an experimental study of the distribution of the length of simplex paths for the optimal assignment problem we study the distribution of the pivot counts for a version of the simplex method that with essentially equal probabilities introduces any variable with negative reduced cost into the basis in this situation the distribution of the pivot counts turns out to be normally distributed and independent of the actual cost coefficients provided these are sufficiently spread out further the mean and standard deviation grow only moderately with the size of the problem namely asd1 8 andd1 5 respectively for ad d problem implying in particular that the pivot counts concentrate around the mean with growingd the usual simplex method on the other hand gives a growth ofd1 6 hence a large part of the favourable polynomial growth experienced on practical problems may be attributed to the fact that the simplex paths are rather short on the average at least for assignment problems

highly acyclic groups hypergraph covers and the guarded fragment we construct finite groups whose cayley graphs have large girth even with respect to a discounted distance measure that contracts arbitrarily long sequences of edges from the same color class subgroup and only counts transitions between color classes cosets these groups are shown to be useful in the construction of finite bisimilar hypergraph covers that avoid any small cyclic configurations we present two applications to the finite model theory of the guarded fragment a strengthening of the known finite model property for gf and the characterization of gf as the guarded bisimulation invariant fragment of first order logic in the sense of finite model theory

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

on the complexity of connection games in this paper we study three connection games among the most widely played havannah twixt and slither we show that determining the outcome of an arbitrary input position is pspace complete in all three cases our reductions are based on the popular graph problem generalized geography and on hex itself we also consider the complexity of generalizations of hex parameterized by the length of the solution and establish that while short generalized hex is w1 hard short hex is fpt finally we prove that the ultra weak solution to the empty starting position in hex cannot be fully adapted to any of these three games the games of havannah twixt and slither are pspace complete the game of slither does not admit draws on finite planar grid boards short generalized hex is w1 hard and short hex is fixed parameter tractable

improved approximation algorithms for minimum weight vertex separators we develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces unlike in the edge case we show that embeddings into l 1 and even euclidean embeddings are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes we obtain an o sqrt log n approximation for minimum ratio vertex cuts in general graphs based on a new semidefinite relaxation of the problem and a tight analysis of the integrality gap which is shown to be theta sqrt log n we also prove an optimal o log k approximate max flow min vertex cut theorem for arbitrary vertex capacitated multicommodity flow instances on k terminals for uniform instances on any excluded minor family of graphs we improve this to o 1 and this yields a constant factor approximation for minimum ratio vertex cuts in such graphs previously this was known only for planar graphs and for general excluded minor families the best known ratio was o log n these results have a number of applications we exhibit an o sqrt log n pseudoapproximation for finding balanced vertex separators in general graphs in fact we achieve an approximation ratio of o sqrt log opt where opt is the size of an optimal separator improving over the previous best bound of o log opt likewise we obtain improved approximation ratios for treewidth in any graph of treewidth k we show how to find a tree decomposition of width at most o k sqrt log k whereas previous algorithms yielded o k log k for graphs excluding a fixed graph as a minor which includes e g bounded genus graphs we give a constant factor approximation for the treewidth this in turn can be used to obtain polynomial time approximation schemes for several problems in such graphs

dominating set is fixed parameter tractable in claw free graphs we show that the dominating set problem parameterized by solution size is fixed parameter tractable fpt in graphs that do not contain the claw k 1 3 the complete bipartite graph on four vertices where the two parts have one and three vertices respectively as an induced subgraph we present an algorithm that uses 2 o k 2 n o 1 time and polynomial space to decide whether a claw free graph on n vertices has a dominating set of size at most k note that this parameterization of dominating set is w 2 hard on the set of all graphs and thus is unlikely to have an fpt algorithm for graphs in general the most general class of graphs for which an fpt algorithm was previously known for this parameterization of dominating set is the class of k i j free graphs which exclude for some fixed i j n the complete bipartite graph k i j as a subgraph for i j 2 the class of claw free graphs and any class of k i j free graphs are not comparable with respect to set inclusion we thus extend the range of graphs over which this parameterization of dominating set is known to be fixed parameter tractable we also show that in some sense it is the presence of the claw that makes this parameterization of the dominating set problem hard more precisely we show that for any t 4 the dominating set problem parameterized by the solution size is w 2 hard in graphs that exclude the t claw k 1 t as an induced subgraph our arguments also imply that the related connected dominating set and dominating clique problems are w 2 hard in these graph classes finally we show that for any t n the clique problem parameterized by solution size which is w 1 hard on general graphs is fpt in t claw free graphs our results add to the small and growing collection of fpt results for graph classes defined by excluded subgraphs rather than by excluded minors

graph editing problems with extended regularity constraints graph editing problems offer an interesting perspective on sub and supergraph identification problems for a large variety of target properties they have also attracted significant attention in recent years particularly in the area of parameterized complexity as the problems have rich parameter ecologies in this paper we examine generalisations of the notion of editing a graph to obtain a regular subgraph in particular we extend the notion of regularity to include two variants of edge regularity along with the unifying constraint of strong regularity we present a number of results with the central observation that these problems retain the general complexity profile of their regularity based inspiration when the number of edits k and the maximum degree r are taken together as a combined parameter the problems are tractable i e in fpt but are otherwise intractable we also examine variants of the basic editing to obtain a regular subgraph problem from the perspective of parameterizing by the treewidth of the input graph in this case the treewidth of the input graph essentially becomes a limiting parameter on the natural k r parameterization

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

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

on locally constructible spheres and balls durhuus and jonsson 1995 introduced the class of locally constructible lc 3 spheres and showed that there are only exponentially many combinatorial types of simplicial lc 3 spheres such upper bounds are crucial for the convergence of models for 3d quantum gravity

new results on the average behavior of simplex algorithms it has been a challenge for mathematicians to theoreti cally confirm the extremely good performance of simplex algorithms for linear programming we have confirmed that a certain variant of the simplex method solves problems of order m x n in an expected num ber of steps which is bounded between two quadratic functions of the smaller dimension of the problem our probabilistic assumptions are rather weak 1 introduction we consider the linear programming problem of order m x n in the form maximize ctx over all x 0 in rn such that ax 5 6 where a e rmxn it has been observed that the simplex algorithms for linear programming developed by george dantzig dl work extremely well a central question in the field of analysis of algorithms is to estimate the expected number of steps that these algorithms perform relative to different probability distributions of inputs some background is given in 52 in am1 and t we consider a variant of the simplex method called the lexicographic self dual method we find that the expected number of steps of this variant relative to a rather weak probabilistic model is only o min m n 2 this is the first polynomial average case upper bound for a simplex algorithm which is capable of solving any linear programming problem the first two authors am21 have also determined a quadratic lower bound so the behavior of this variant is indeed quadratic whereas it has been conjectured that other variants take a linear expected number of steps with various choices of the starting point this variant can simulate different constraint by constraint and variable dimension algorithms as well as combinations thereof me2 adler karp and shamir aks provide a different proof of similar upper bound for a certain class of constraint by constraint algorithms it follows from me21 that their variant can also be simulated by the self dual algorithm with an appropriate starting point

on preservation under homomorphisms and unions of conjunctive queries unions of conjunctive queries also known as select project join union queries are the most frequently asked queries in relational database systems these queries are definable by existential positive first order formulas and are preserved under homomorphisms a classical result of mathematical logic asserts that the existential positive formulas are the only first order formulas up to logical equivalence that are preserved under homomorphisms on all structures finite and infinite the question of whether the homomorphism preservation theorem holds for the class of all finite structures resisted solution for a long time it was eventually shown that unlike other classical preservation theorems the homomorphism preservation theorem does hold in the finite in this article we show that the homomorphism preservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory specifically we show that this result holds for all classes of finite structures of bounded degree all classes of finite structures of bounded treewidth and more generally all classes of finite structures whose cores exclude at least one minor

large induced subgraphs via triangulations and cmso we obtain an algorithmic metatheorem for the following optimization problem let varphi be a counting monadic second order logic cmso formula and t geq 0 be an integer for a given graph g v e the task is to maximize x subject to the following there is a set f subseteq v such that x subseteq f the subgraph g f induced by f is of treewidth at most t and the structure g f x models varphi i e g f x models varphi we give an algorithm solving this optimization problem on any n vertex graph g in time cal o pi g cdot n t 4 cdot f t varphi where pi g is the set of all potential maximal cliques in g and f is a function of t and varphi only pipelined with the known bounds on the number of potential maximal cliques in different graph classes there are a plethora of algorithmic consequences extending and subsuming many known results on polynomial time algorithms for graph classes we also show that all potential maximal cliques of g can be enumerated in time cal o 1 7347 n this implies the existence of an exact exponential algorithm of running time cal o 1 7347 n for many np hard problems related to finding maximum induced subgraphs with different properties

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

meta kernelization in a parameterized problem every instance i comes with a positive integer k the problem is said to admit a polynomial kernel if in polynomial time one can reduce the size of the instance i to a polynomial in k while preserving the answer in this work we give two meta theorems on kernelization the first theorem says that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus these theorems unify and extend all previously known kernelization results for planar graph problems

unions of oriented matroids union or sum is an operation on ordinary matroids that yields great insight into matroid structure it has a wealth of applications and as pointed out by mirsky and others allows the treatment without difficulty of problems whose solution has previously required extremely complex arguments for all its power the operation is expressed simply in terms of independent sets in this paper the corresponding operation for oriented matroids is formulated in a constructive way by making use of the multiply ordered sets introduced by gutierrez novoa and algorithms are given for generating the union matroid for simple oriented matroids that is oriented matroids whose underlying matroids are uniform it is also shown that union yields another simple matroid and that the union operation may be carried out by concatenation of two set systems this construction is also useful in obtaining combinatorial types of some polytopes and convex polyhedral sets in particular concatenation is used to determine the polyhedral set found by klee and walkup as a counterexample to the polyhedral hirsch and d step conjectures

finding large degree anonymous subgraphs is hard a graph is said to be k anonymous for an integer k if for every vertex in the graph there are at least k 1 other vertices with the same degree we examine the computational complexity of making a given undirected graph k anonymous either through at most s vertex deletions or through at most s edge deletions the corresponding problem variants are denoted by anonym v del and anonym e del we present a variety of hardness results most of them hold for both problems the two variants are intractable from the parameterized as well as from the approximation point of view in particular we show that both variants remain np hard on very restricted graph classes like trees even if k 2 we further prove that both variants are w1 hard with respect to the combined parameter solutions size s and anonymity level k with respect to approximability we obtain hardness results showing that neither variant can be approximated in polynomial time within a factor better than n 1 2 unless p np furthermore for the optimization variants where the solution size s is given and the task is to maximize the anonymity level k this inapproximability result even holds if we allow a running time of f s n o 1 for any computable function f on the positive side we classify both problem variants as fixed parameter tractable with respect to the combined parameter solution size s and maximum degree

the parameterized complexity of k edge induced subgraphs we prove that finding a k edge induced subgraph is fixed parameter tractable thereby answering an open problem of leizhen cai our algorithm is based on several combinatorial observations gauss famous eureka theorem and a generalization of the well known fpt algorithm for the model checking problem for first order logic on graphs with locally bounded tree width due to frick and grohe on the other hand we show that two natural counting versions of the problem are hard hence the k edge induced subgraph problem is one of the very few known examples in parameterized complexity that are easy for decision while hard for counting

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

on the treewidth of dynamic graphs dynamic graph theory is a novel growing area that deals with graphs that change over time and is of great utility in modeling modern wireless mobile and dynamic environments as a graph evolves possibly arbitrarily it is challenging to identify the graph properties that can be preserved over time and understand their respective computability in this paper we are concerned with the treewidth of dynamic graphs we focus on metatheorems which allow the generation of a series of results based on general properties of classes of structures in graph theory two major metatheorems on treewidth provide complexity classifications by employing structural graph measures and finite model theory courcelle s theorem gives a general tractability result for problems expressible in monadic second order logic on graphs of bounded treewidth and frick and grohe demonstrate a similar result for first order logic and graphs of bounded local treewidth we extend these theorems by showing that dynamic graphs of bounded local treewidth where the length of time over which the graph evolves and is observed is finite and bounded can be modeled in such a way that the local treewidth of the underlying graph is maintained we show the application of these results to problems in dynamic graph theory and dynamic extensions to static problems in addition we demonstrate that certain widely used dynamic graph classes have bounded local treewidth

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

discrete morse theory for manifolds with boundary we introduce a version of discrete morse theory specific for manifolds with boundary the idea is to consider morse functions for which all boundary cells are critical we obtain relative morse inequalities relating the homology of the manifold to the number of interior critical cells we also derive a ball theorem in analogy to forman s sphere theorem the main corollaries of our work are 1 for each d 3 and for each k 0 there is a pl d sphere on which any discrete morse function has more than k critical d 1 cells this solves a problem by chari 2 for fixed d and k there are exponentially many combinatorial types of simplicial d manifolds counted with respect to the number of facets that admit discrete morse functions with at most k critical interior d 1 cells this connects discrete morse theory to enumerative combinatorics discrete quantum gravity 3 the barycentric subdivision of any simplicial constructible d ball is collapsible this almost solves a problem by hachimori 4 every constructible ball collapses onto its boundary minus a facet this improves a result by the author and ziegler 5 any 3 ball with a knotted spanning edge cannot collapse onto its boundary minus a facet this strengthens a classical result by bing and a recent result by the author and ziegler

deciding first order properties of locally tree decomposable structures we introduce the concept of a class of graphs or more generally relational structures being locally tree decomposable there are numerous examples of locally tree decomposable classes among them the class of planar graphs and all classes of bounded valence or of bounded tree width we also consider a slightly more general concept of a class of structures having bounded local tree width we show that for each property of structures that is definable in first order logic and for each locally tree decomposable class c of structures there is a linear time algorithm deciding whether a given structure a c has property for classes c of bounded local tree width we show that for every k 1 there is an algorithm solving the same problem in time o n1 1 k where n is the cardinality of the input structure

bivariate complexity analysis of almost forest deletion abstract in this paper we study a generalization of classic feedback vertex set problem in the realm of multivariate complexity analysis we say that a graph f is an l forest if we can delete at most l edges from f to get a forest that is f is at most l edges away from being a forest in this paper we introduce the almost forest deletion problem where given a graph g and integers k and l the question is whether there exists a subset of at most k vertices such that its deletion leaves us an l forest we show that this problem admits an algorithm with running time 2 o k l n o 1 and a kernel of size o k l k l we also show that the problem admits a 2 o tw n o 1 algorithm on bounded treewidth graphs using which we design a subexponential algorithm for the problem on planar graphs

faster approximation schemes and parameterized algorithms on odd h minor free graphs we improve the running time of the general algorithmic technique known as baker s approach 1994 1 on h minor free graphs from o n f h to o f h n o 1 the numerous applications include e g a 2 approximation for coloring and ptases for various problems such as dominating set and max cut where we obtain similar improvements on classes of odd minor free graphs which have gained significant attention in recent time we obtain a similar acceleration for a variant of the structural decomposition theorem proved by demaine et al 2010 20 we use these algorithms to derive faster 2 approximations furthermore we present the first ptases and subexponential fpt algorithms for independent set and vertex cover on these graph classes using a novel dynamic programming technique we also introduce a technique to derive nearly subexponential parameterized algorithms on h minor free graphs our technique applies in particular to problems such as steiner tree directed subgraph with a property directed longest path and connected independent dominating set on some or all proper minor closed graph classes we obtain as a corollary that all problems with a minor monotone subexponential kernel and amenable to our technique can be solved in subexponential fpt time onh minor free graphs this results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality

decompositions of simplicial balls and spheres with knots consisting of few edges constructibility is a condition on pure simplicial complexes that is weaker than shellability in this paper we show that non constructible triangulations of the d dimensional sphere exist for every d geq 3 this answers a question of danaraj and klee 10 it also strengthens a result of lickorish 16 about non shellable spheres furthermore we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non trivial knot with few edges in a 3 sphere or 3 ball and a similar hierarchy for 3 balls with a knotted spanning arc that consists of few edges

the hirsch conjecture holds for normal flag complexes using an intuition from metric geometry we prove that any flag normal simplicial complex satisfies the nonrevisiting path conjecture as a consequence the diameter of its facet ridge graph is smaller than the number of vertices minus the dimension as in the hirsch conjecture this proves the hirsch conjecture for all flag polytopes and more generally for all connected flag homology manifolds

the parameterized complexity of editing graphs for bounded degeneracy we examine the parameterized complexity of the problem of editing a graph to obtain an r degenerate graph we show that for the editing operations vertex deletion and edge deletion both separately and combined the problem is w p complete and remains w p complete even if the input graph is already r 1 degenerate or has maximum degree 2r 1 for all r 2 we also demonstrate fixed parameter tractability for several clique based problems when the input graph has bounded degeneracy

on finding short resolution refutations and small unsatisfiable subsets we consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps and whether a given set of clauses contains an unsatisfiable subset of size at most k we show that both problems are complete for the class w 1 the first level of the w hierarchy of fixed parameter intractable problems our results remain true if restricted to 3 sat instances and or to various restricted versions of resolution including tree like resolution input resolution and read once resolution applying a metatheorem of frick and grohe we show that restricted to classes of sets of clauses of locally bounded treewidth the considered problems are fixed parameter tractable for example the problems are fixed parameter tractable for planar cnf formulas

deciding first order properties of nowhere dense graphs nowhere dense graph classes introduced by neset il and ossona de mendez 2010 2011 form a large variety of classes of sparse graphs including the class of planar graphs actually all classes with excluded minors and also bounded degree graphs and graph classes of bounded expansion we show that deciding properties of graphs definable in first order logic is fixed parameter tractable on nowhere dense graph classes parameterized by the length of the input formula at least for graph classes closed under taking subgraphs this result is optimal it was known before that for all classes c of graphs closed under taking subgraphs if deciding first order properties of graphs in c is fixed parameter tractable then c must be nowhere dense under a reasonable complexity theoretic assumption as a by product we give an algorithmic construction of sparse neighborhood covers for nowhere dense graphs this extends and improves previous constructions of neighborhood covers for graph classes with excluded minors at the same time our construction is considerably simpler than those our proofs are based on a new game theoretic characterization of nowhere dense graphs that allows for a recursive version of locality based algorithms on these classes on the logical side we prove a rank preserving version of gaifman s locality theorem

testing first order properties for subclasses of sparse graphs we present a linear time algorithm for deciding first order fo properties in classes of graphs with bounded expansion a notion recently introduced by neset il and ossona de mendez this generalizes several results from the literature because many natural classes of graphs have bounded expansion graphs of bounded tree width all proper minor closed classes of graphs graphs of bounded degree graphs with no subgraph isomorphic to a subdivision of a fixed graph and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges we deduce that there is an almost linear time algorithm for deciding fo properties in classes of graphs with locally bounded expansion more generally we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion after a linear time initialization the data structure allows us to test an fo property in constant time and the data structure can be updated in constant time after addition deletion of an edge provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class all our results also hold for relational structures and are based on the seminal result of neset il and ossona de mendez on the existence of low tree depth colorings