models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

a randomized algorithm for closest point queries an algorithm for closest point queries is given the problem is this given a set s of n points in d dimensional space build a data structure so that given an arbitrary query point p a closest point in s to p can be found quickly the measure of distance is the euclidean norm this is sometimes called the post office problem the new data structure will be termed an rpo tree from randomized post office the expected time required to build an rpo tree is o n lceil d 2 rceil 1 epsilon for any fixed epsilon 0 and a query can be answered in o log n worst case time an rpo tree requires o n lceil d 2 rceil 1 epsilon space in the worst case the constant factors in these bounds depend on d and epsilon the bounds are average case due to the randomization employed by the algorithm and hold for any set of input points this result approaches the omega n lceil d 2 rceil worst case time required for any algorithm that constructs the voronoi

controllability and maximum matchings of complex networks previously the controllability problem of a linear time invariant dynamical system was mapped to the maximum matching mm problem on the bipartite representation of the underlying directed graph and the sizes of mms on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit here we present an alternative theory to estimate mm sizes based on the core percolation theory and the perfect matching of cores our theory is much more simplified and easily interpreted and can estimate mm sizes on random graphs with or without symmetry between out and in degree distributions our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems

statistical mechanics of the vertex cover problem we review recent progress in the study of the vertex cover problem vc the vc belongs to the class of np complete graph theoretical problems which plays a central role in theoretical computer science on ensembles of random graphs vc exhibits a coverable uncoverable phase transition very close to this transition depending on the solution algorithm easy hard transitions in the typical running time of the algorithms occur we explain a statistical mechanics approach which works by mapping the vc to a hard core lattice gas and then applying techniques such as the replica trick or the cavity approach using these methods the phase diagram of the vc could be obtained exactly for connectivities c e the solution of the vc exhibits full replica symmetry breaking the statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for the vc finally we describe recent results for the vc when studied on other ensembles of finite and infinite dimensional graphs

on approximations and incidence in cylindrical algebraic decompositions let p subset mathbb z x 1 cdots x r be a finite set this paper describes and analyzes a variant of the algorithm of collins and others for decomposing mathbb r r into semi algebraic cells so that the value of each f in p has constant sign positive negative or zero on the points of each cell the version here has several advantages 1 the boundary of each cell is a disjoint union of lower dimensional cells for each bounded cell alpha the pair bar alpha alpha is homeomorphic to a closed ball and its interior 2 an algorithm is presented which for fixed r computes incidence of cells in polynomial time 3 a priori estimates of the accuracy of approximations of roots of polynomials required in order to determine the combinatorial structure of the cell complex are given this avoids computation in algebraic number fields

near optimal configurations in mean field disordered systems we present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems our approach is based on a generalization of the cavity method to a system interacting with its ground state with this technique we study the random matching problem as well as the mean field diluted spin glass as a by product of this approach we calculate the de almeida thouless transition line of the spin glass on a fixed connectivity random graph

almost tight upper bounds for vertical decompositions in four dimensions we show that the complexity of the vertical decomposition of an arrangement of n fixed degree algebraic surfaces or surface patches in four dimensions is o n4 e for any e 0 this improves the best previously known upper bound for this problem by a near linear factor and settles a major problem in the theory of arrangements of surfaces open since 1989 the new bound can be extended to higher dimensions yielding the bound o n2d 4 e for any e 0 on the complexity of vertical decompositions in dimensions d 4 we also describe the immediate algorithmic applications of these results which include improved algorithms for point location range searching ray shooting robot motion planning and some geometric optimization problems

an optimal algorithm for constructing the weighted voronoi diagram in the plane let s denote a set of n points in the plane such that each point p has assigned a positive weight w p which expresses its capability to influence its neighbourhood in this sense the weighted distance of an arbitrary point x from p is given by de x p w p where de denotes the euclidean distance function the weighted voronoi diagram for s is a subdivision of the plane such that each point p in s is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of s an algorithm which constructs the weighted voronoi diagram for s in o n2 time is outlined in this paper the method is optimal as the diagram can consist of n2 faces edges and vertices

typical approximation performance for maximum coverage problem this study investigated the typical performance of approximation algorithms known as belief propagation the greedy algorithm and linear programming relaxation for maximum coverage problems in sparse biregular random graphs after we used the cavity method for a corresponding hard core lattice gas model results showed that two distinct thresholds of replica symmetry and its breaking exist in the typical performance threshold of belief propagation in the low density region the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses although the greedy algorithm and linear programming relaxation have the same approximation ratio in worst case performance their typical performance thresholds are mutually different indicating the importance of typical performance results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms

statistical mechanics of maximal independent sets the graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory a maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints it is known that finding minimum and maximum density maximal independent sets are hard optimization problems in this paper we use cavity method of statistical physics and monte carlo simulations to study the corresponding constraint satisfaction problem on random graphs we obtain the entropy of maximal independent sets within the replica symmetric and one step replica symmetry breaking frameworks shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms this is of particular relevance for the application to the study of strategic interactions in social and economic networks where maximal independent sets correspond to pure nash equilibria of a graphical game of public goods allocation

statistical mechanics of the minimum vertex cover problem in stochastic block models the minimum vertex cover min vc problem is a well known np hard problem earlier studies illustrate that the problem defined over the erdos renyi random graph with a mean degree c exhibits computational difficulty in searching the min vc set above a critical point c e 2 718 here we address how this difficulty is influenced by the mesoscopic structures of graphs for this we evaluate the critical condition of difficulty for the stochastic block model we perform a detailed examination of the specific cases of two equal size communities characterized by in and out degrees which are denoted by c in and c out respectively our analysis based on the cavity method indicates that the solution search once becomes difficult when c in c out exceeds e from below but becomes easy again when c out is sufficiently larger than c in in the region c out e experiments based on various search algorithms support the theoretical prediction

fault tolerance of random graphs with respect to connectivity mean field approximation for semidense random graphs the fault tolerance of random graphs with unbounded degrees with respect to connectivity is investigated which relates to the reliability of wireless sensor networks with unreliable relay nodes the model evaluates the network breakdown probability that a graph is disconnected after stochastic node removal to establish a mean field approximation for the model we propose the cavity method for finite systems the analysis enables us to obtain an approximation formula for random graphs with any number of nodes and an arbitrary degree distribution in addition its asymptotic analysis reveals that the phase transition occurs in semidense random graphs whose average degree grows logarithmically these results which are supported by numerical simulations coincide with the mathematical results indicating successful predictions by the mean field approximation for unbounded but not dense random graphs

statistical mechanics of the hitting set problem in this paper we present a detailed study of the hitting set hs problem this problem is a generalization of the standard vertex cover to hypergraphs one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle it can also be used in important practical tasks such as the group testing procedures where one wants to detect defective items in a large group by pool testing using a statistical mechanics approach based on the cavity method we study the phase diagram of the hs problem in the case of random regular hypergraphs depending on the values of the variables and tests degrees different situations can occur the hs problem can be either in a replica symmetric phase or in a one step replica symmetry breaking one in these two cases we give explicit results on the minimal density of particles and the structure of the phase space these problems are thus in some sense simpler than the original vertex cover problem where the need for a full replica symmetry breaking has prevented the derivation of exact results so far finally we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hitting set problem

maximum independent set on diluted triangular lattices core percolation and maximum independent set on random graphs have recently been characterized using the methods of statistical physics here we present a statistical physics study of these problems on bond diluted triangular lattices core percolation critical behavior is found to be consistent with the standard percolation values though there are strong finite size effects a transfer matrix method is developed and applied to find accurate values of the density and degeneracy of the maximum independent set on lattices of limited width but large length an extrapolation of these results to the infinite lattice limit yields high precision results which are tabulated these results are compared to results found using both vertex based and edge based local probability recursion algorithms which have proven useful in the analysis of hard computational problems such as the satisfiability problem

minimal vertex covers on finite connectivity random graphs a hard sphere lattice gas picture the minimal vertex cover or maximal independent set problem is studied on random graphs of finite connectivity analytical results are obtained by a mapping to a lattice gas of hard spheres of chemical radius 1 and they are found to be in excellent agreement with numerical simulations we give a detailed description of the replica symmetric phase including the size and entropy of the minimal vertex covers and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1 43 the replica symmetric solution breaks down at c e approximately 2 72 we give a simple one step replica symmetry broken solution and discuss the problems in the interpretation and generalization of this solution

mystery of point charges we discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in rn this question goes back to j c maxwell 10 and m morse

clustering analysis of the ground state structure of the vertex cover problem vertex cover is one of the classical np complete problems in theoretical computer science a vertex cover of a graph is a subset of vertices such that for each edge at least one of the two endpoints is contained in the subset when studied on erd os r enyi random graphs with connectivity c one observes a threshold behavior in the thermodynamic limit the size of the minimal vertex cover is independent of the specific graph recent analytical studies show that on the phase boundary for small connectivities cle the system is replica symmetric while for larger connectivities replica symmetry breaking occurs this change coincides with a change of the typical running time of algorithms from polynomial to exponential to understand the reasons for this behavior and to compare with the analytical results we numerically analyze the structure of the solution landscape for this purpose we have also developed an algorithm which allows the calculation of the backbone without the need to enumerate all solutions we study exact solutions found with a branch and bound algorithm as well as configurations obtained via a monte carlo simulation we analyze the cluster structure of the solution landscape by direct clustering of the states by analyzing the eigenvalue spectrum of correlation matrices and by using a hierarchical clustering method all results are compatible with a change at c e for small connectivities the solutions are collected in a finite small number of clusters while the number of clusters diverges slowly with system size for larger connectivities and replica symmetry breaking but not one step replica symmetry breaking 1 rsb occurs

number of guards needed by a museum a phase transition in vertex covering of random graphs in this letter we study the np complete vertex cover problem on finite connectivity random graphs when the allowed size of the cover set is decreased a discontinuous transition in solvability and typical case complexity occurs this transition is characterized by means of exact numerical simulations as well as by analytical replica calculations the replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e where replica symmetry becomes locally unstable

heterogeneous k core versus bootstrap percolation on complex networks we introduce the heterogeneous k core which generalizes the k core and contrast it with bootstrap percolation vertices have a threshold r i that may be different at each vertex if a vertex has fewer than r i neighbors it is pruned from the network the heterogeneous k core is the subgraph remaining after no further vertices can be pruned if the thresholds r i are 1 with probability f or k ensuremath geqslant 3 with probability 1 ensuremath f the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation which is an activation process we show that there are two types of transitions in this heterogeneous k core process the giant heterogeneous k core may appear with a continuous transition and there may be a second discontinuous hybrid transition we compare critical phenomena critical clusters and avalanches at the heterogeneous k core and bootstrap percolation transitions we also show that the network structure has a crucial effect on these processes with the giant heterogeneous k core appearing immediately at a finite value for any fg0 when the degree distribution tends to a power law p q q ensuremath ensuremath gamma with ensuremath gamma l3

covering problems and core percolations on hypergraphs we introduce two generalizations of core percolation in graphs to hypergraphs related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs respectively we offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments we find that for several real world hypergraphs their two cores tend to be much smaller than those of their null models suggesting that covering problems in those real world hypergraphs can actually be solved in polynomial time

sinr diagrams convexity and its applications in wireless networks the rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal to interference noise ratio sinr model for a collection of simultaneously transmitting stations in the plane it is possible to identify a reception zone for each station consisting of the points where its transmission is received correctly the resulting sinr diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard sinr diagrams appear to be fundamental to understanding the behavior of wireless networks and may play a key role in the development of suitable algorithms for such networks analogous perhaps to the role played by voronoi diagrams in the study of proximity queries and related issues in computational geometry so far however the properties of sinr diagrams have not been studied systematically and most algorithmic studies in wireless networking rely on simplified graph based models such as the unit disk graph udg model which conveniently abstract away interference related complications and make it easier to handle algorithmic issues but consequently fail to capture accurately some important aspects of wireless networks this article focuses on obtaining some basic understanding of sinr diagrams their properties and their usability in algorithmic applications specifically we have shown that assuming uniform power transmissions the reception zones are convex and relatively well rounded these results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks

polynomial iterative algorithms for coloring and analyzing random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find with a one step replica symmetry breaking approximation the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c in c d c q in which ground states spontaneously divide into an exponential number of clusters furthermore we extended our considerations to the case of single instances showing consistent results this leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region i e when c in c d c q

generalization of core percolation on complex networks we introduce a k leaf removal algorithm as a generalization of the so called leaf removal algorithm in this pruning algorithm vertices of degree smaller than k together with their first nearest neighbors and all incident edges are progressively removed from a random network as the result of this pruning the network is reduced to a subgraph which we call the generalized k core gk core performing this pruning for the sequence of natural numbers k we decompose the network into a hierarchy of progressively nested gk cores we present an analytical framework for description of gk core percolation for undirected uncorrelated networks with arbitrary degree distributions configuration model to confirm our results we also derive rate equations for the k leaf removal algorithm which enable us to obtain the structural characteristics of the gk cores in another way also we apply our algorithm to a number of real world networks and perform the gk core decomposition for them

a singly exponential stratification scheme for real semi algebraic varieties and its applications abstract this paper describes an effective procedure for stratifying a real semi algebraic set into cells of constant description size the attractive feature of our method is that the number of cells produced is singly exponential in the number of input variables this compares favorably with the doubly exponential size of collins decomposition unlike collins construction however our scheme does not produce a cell complex but only a smooth stratification nevertheless we are able to apply our results in interesting ways to problems of point location and geometric optimization

detecting the solution space of vertex cover by mutual determinations and backbones to solve the combinatorial optimization problems especially the minimal vertex cover problem with high efficiency is a significant task in theoretical computer science and many other subjects aiming at detecting the solution space of vertex cover a new structure named mutual determination is defined and discovered for vertex cover on general graphs which results in the emergence of strong correlations among the unfrozen nodes based on the backbones and mutual determinations with node ranks by leaf removal we propose a mutual determination and backbone evolution algorithm to achieve the reduced solution graph which provides a graphical expression of the solution space of vertex cover by this algorithm the whole solution space and detailed structures such as backbones can be obtained strictly when there is no leaf removal core on the given graph compared with the current algorithms the mutual determination and backbone evolution algorithm performs as well as the replica symmetry one in a certain interval but has a small gap higher than the replica symmetric breaking one and has a relatively small error for the exact results the algorithm with the mutual determination provides a new viewpoint to solve vertex cover and understand the organizations of the solution spaces and the reduced solution graph gives an alternative way to catch detailed information of the ground steady states

the topology of wireless communication this article studies the topological properties of wireless communication maps and their usability in algorithmic design we consider the sinr model which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise to describe the behavior of a multistation network we use the convenient representation of a reception map which partitions the plane into reception zones one per station and the complementary region of the plane where no station can be heard sinr diagrams have been studied in avin et al 2009 for the specific case where all stations use the same power it was shown that the reception zones are convex hence connected and fat and this was used to devise an efficient algorithm for the fundamental problem of point location here we consider the more general and common case where transmission energies are arbitrary or nonuniform under that setting the reception zones are not necessarily convex or even connected this poses the algorithmic challenge of designing efficient point location techniques for the nonuniform setting as well as the theoretical challenge of understanding the geometry of sinr diagrams e g the maximal number of connected components they might have our key result exhibits a striking contrast between d and d 1 dimensional maps for a network embedded in d dimensional space specifically it is shown that whereas the d dimensional map might be highly fractured drawing the map in one dimension higher heals the zones which become connected in fact hyperbolically connected we also provide bounds for the fatness of reception zones subsequently we consider algorithmic applications and propose a new variant of approximate point location

dynamical replica analysis of processes on finitely connected random graphs i vertex covering we study the stochastic dynamics of ising spin models with random bonds interacting on finitely connected poissonian random graphs we use the dynamical replica method to derive closed dynamical equations for the joint spin field probability distribution and solve these within the replica symmetry ansatz although the theory is developed in a general setting with a view to future applications in various other fields in this paper we apply it mainly to the dynamics of the glauber algorithm extended with cooling schedules when running on the so called vertex cover optimization problem our theoretical predictions are tested against both monte carlo simulations and known results from equilibrium studies in contrast to previous dynamical analyses based on deriving closed equations for only a small number of scalar order parameters the agreement between theory and experiment in the present study is nearly perfect

nonlinear self adjointness of a generalized fifth order kdv equation the new concepts of self adjoint equations formulated by ibragimov and gandarias are applied to a class of fifth order evolution equations then from ibragimov s theorem on conservation laws conservation laws for the generalized kawahara equation simplified kahawara equation and modified simplified kawahara equation are established

computational complexity arising from degree correlations in networks we apply a bethe peierls approach to statistical mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices using the nondeterministic polynomial time hard optimization problem of finding minimal vertex covers on these graphs we show that such correlations may lead to a qualitatively different solution structure as compared to uncorrelated networks this results in a higher complexity of the network in a computational sense simple heuristic algorithms fail to find a minimal vertex cover in the highly correlated case whereas uncorrelated networks seem to be simple from the point of view of combinatorial optimization

integrating factors adjoint equations and lagrangians integrating factors and adjoint equations are determined for linear and non linear differential equations of an arbitrary order the new concept of an adjoint equation is used for construction of a lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables the method is illustrated by considering several equations traditionally regarded as equations without lagrangians noether s theorem is applied to the maxwell equations

convex partitions of polyhedra a lower bound and worst case optimal algorithm the problem of partitioning a polyhedron into a minimum number of convex pieces is known to be np hard we establish here a quadratic lower bound on the complexity of this problem and we describe an algorithm that produces a number of convex parts within a constant factor of optimal in the worst case the algorithm is linear in the size of the polyhedron and cubic in the number of reflex angles since in most applications areas the former quantity greatly exceeds the latter the algorithm is viable in practice

correlations in connected random graphs we study the properties of the giant connected component in random graphs with arbitrary degree distribution we concentrate on the degree degree correlations we show that the adjoining nodes in the giant connected component are correlated and derive analytic formulas for the joint nearest neighbor degree probability distribution using those results we describe correlations in maximal entropy connected random graphs we show that connected graphs are disassortative and that correlations are strongly related to the presence of one degree nodes leaves we propose an efficient algorithm for generating connected random graphs we illustrate our results with several examples

observability transition in multiplex networks abstract we extend the observability model to multiplex networks composed of two network layers we present mathematical frameworks valid under the treelike ansatz able to describe the emergence of the macroscopic cluster of mutually observable nodes in both synthetic and real world multiplex networks we show that the observability transition in synthetic multiplex networks is discontinuous in real world multiplex networks instead edge overlap among layers is responsible for the disappearance of any sign of abruptness in the emergence of the macroscopic cluster of mutually observable nodes

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

core organization of directed complex networks the recursive removal of leaves dead end vertices and their neighbors from an undirected network results when this pruning algorithm stops in a so called core of the network this specific subgraph should be distinguished from k cores which are principally different subgraphs in networks if the vertex mean degree of a network is sufficiently large the core is a giant cluster containing a finite fraction of vertices we find that generalization of this pruning algorithm to directed networks provides a significantly more complex picture of cores by implementing a rate equation approach to this pruning procedure for directed uncorrelated networks we identify a set of cores progressively embedded into each other in a network and describe their birth points and structure

coloring random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c e c d c q in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms

statistical mechanics perspective on the phase transition in vertex covering offinite connectivity random graphs the vertex cover problem is studied for random graphs gn cn having n vertices and cn edges exact numerical results are obtained by a branch and bound algorithm it is found that a transition in the coverability at a c dependent threshold x xc c appears where xn is the cardinality of the vertex cover this transition coincides with a sharp peak of the typical numerical e1ort which is needed to decide whether there exists a cover with xn vertices or not for small edge concentrations c two less than arrows one inside the other 0 5 a cluster expansion is performed giving very accurate results in this regime these results are extended using methods developed in statistical physics the so called annealed approximation reproduces a rigorous bound on xc c which was known previously the main part of the paper contains an application of the replica method within the replicasymmetric ansatz the threshold xc c and various statistical properties of minimal vertex covers can be calculated for c e 2 the results show an excellent agreement with the numerical findings at average vertex degree 2c e an instability of the simple replica symmetric solution occurs

optimal vertex cover for the small world hanoi networks the vertex cover problem on the hanoi networks hn3 and hn5 is analyzed with an exact renormalization group and parallel tempering monte carlo simulations the grand canonical partition function of the equivalent hard core repulsive lattice gas problem is recast first as an ising like canonical partition function which allows for a closed set of renormalization group equations the flow of these equations is analyzed for the limit of infinite chemical potential at which the vertex cover problem is attained the relevant fixed point and its neighborhood are analyzed and nontrivial results are obtained both for the coverage as well as for the ground state entropy density which indicates the complex structure of the solution space using special hierarchy dependent operators in the renormalization group and monte carlo simulations structural details of optimal configurations are revealed these studies indicate that the optimal coverages or packings are not related by a simple symmetry using a clustering analysis of the solutions obtained in the monte carlo simulations a complex solution space structure is revealed for each system size nevertheless in the thermodynamic limit the solution landscape is dominated by one huge set of very similar solutions

random graph coloring statistical physics approach the problem of vertex coloring in random graphs is studied using methods of statistical physics and probability our analytical results are compared to those obtained by exact enumeration and monte carlo simulations we critically discuss the merits and shortcomings of the various methods and interpret the results obtained we present an exact analytical expression for the two coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and k body edges

phase transition for cutting plane approach to vertex cover problem we study the vertex cover problem which is a nondeterministic polynomial time hard optimization problem and a prototypical model exhibiting phase transitions on random graphs such as erd s renyi er random graphs these phase transitions coincide with changes of the solution space structure e g for the er ensemble at connectivity c e 2 7183 from replica symmetric to replica symmetry broken for the vertex cover problem the typical complexity of exact branch and bound algorithms which proceed by exploring the landscape of feasible configurations also changes close to this phase transition from easy to hard in this work we consider an algorithm which has a completely different strategy the problem is mapped onto a linear programming problem augmented by a cutting plane approach hence the algorithm operates in a space outside the space of feasible configurations until the final step where a solution is found here we show that this type of algorithm also exhibits an easy hard transition around c e which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem

a polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing in this paper we prove several extremal results for geometrically defined hypergraphs in particular we establish an improved lower bound single exponentially decreasing in k on the best constant delta 0 such that the vertex classes p 1 ldots p k of every k partite k uniform semialgebraic hypergraph h p 1 cup cdots cup p k e with e ge varepsilon pi j 1 k p i have for 1 leq i leq k delta p i element subsets p i subseteq p i satisfying p 1 times cdots times p k subseteq e the best previously known lower bound on delta due to bukh and hubard decreased triple exponentially fast in k we give three geometric applications of our results in particular we establish the following strengthening of the so called same type lemma of barany and valtr any disjoint finite sets p 1 ldots p k subset mathbb r d k d have for 1 leq i leq k subsets p i of size at least 2 o d 3k log k p i with the property that every k tuple formed by taking on

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

semialgebraic range reporting and emptiness searching with applications in a typical range emptiness searching resp reporting problem we are given a set p of n points in mathbb r d and we wish to preprocess it into a data structure that supports efficient range emptiness resp reporting queries in which we specify a range sigma which in general is a semialgebraic set in mathbb r d of constant description complexity and we wish to determine whether p cap sigma emptyset or to report all the points in p cap sigma range emptiness searching and reporting arise in many applications and have been treated by matousek comput geom theory appl 2 1992 pp 169 186 in the special case where the ranges are half spaces bounded by hyperplanes as shown in matousek s work the two problems are closely related and they have solutions for the case of half spaces with similar performance bounds in this paper we extend the analysis to general semialgebraic ranges and show how to adapt matousek s technique without the need to linearize the ranges into a higher dimensional space this yields more efficient solutions to several useful problems and we demonstrate the new technique in four applications with the following results i an algorithm for ray shooting amid balls in mathbb r 3 which uses o n storage and o n preprocessing we use the notation o n gamma to mean an upper bound of the form c varepsilon n gamma varepsilon which holds for any varepsilon 0 where c varepsilon is a constant that depends on varepsilon and answers a query in o n 2 3 time improving the previous bound of o n 3 4 ii an algorithm that preprocesses in o n time a set p of n points in mathbb r 3 into a data structure with o n storage so that for any query line ell or for that matter any simply shaped convex set the point of p farthest from ell can be computed in o n 1 2 time this in turn yields an algorithm that computes the largest area triangle spanned by p in time o n 26 11 as well as nontrivial algorithms for computing the largest perimeter or largest height triangle spanned by p iii an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that for any query alpha fat triangle delta we can determine in o 1 time whether delta cap p is empty alternatively we can report in o 1 o k time the points of delta cap p where k delta cap p iv an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that given any query semidisk c or a circular cap larger than a semidisk we can determine in o 1 time whether c cap p is empty or report the k points in c cap p in o 1 o k time adapting the recent techniques of b aronov and s har peled siam j comput 38 2008 pp 899 921 b aronov s har peled and m sharir on approximate halfspace range counting and relative epsilon approximations in proceedings of the 23rd acm symposium comput geom 2007 pp 327 336 b aronov and m sharir siam j comput 39 2010 pp 2704 2725 we can turn our solutions into efficient algorithms for approximate range counting with small relative error for the cases mentioned above our technique is closely related to the notions of nearest or farthest neighbor generalized voronoi diagrams and of the union or intersection of geometric objects where sharper bounds on the combinatorial complexity of decompositions of complements of these structures yield faster range emptiness searching or reporting algorithms

clustering determines the dynamics of complex contagions in multiplex networks we present the mathematical analysis of generalized complex contagions in a class of clustered multiplex networks the model is intended to understand spread of influence or any other spreading process implying a threshold dynamics in setups of interconnected networks with significant clustering the contagion is assumed to be general enough to account for a content dependent linear threshold model where each link type has a different weight for spreading influence that may depend on the content e g product rumor political view that is being spread using the generating functions formalism we determine the conditions probability and expected size of the emergent global cascades this analysis provides a generalization of previous approaches and is especially useful in problems related to spreading and percolation the results present nontrivial dependencies between the clustering coefficient of the networks and its average degree in particular several phase transitions are shown to occur depending on these descriptors generally speaking our findings reveal that increasing clustering decreases the probability of having global cascades and their size however this tendency changes with the average degree there exists a certain average degree from which on clustering favors the probability and size of the contagion by comparing the dynamics of complex contagions over multiplex networks and their monoplex projections we demonstrate that ignoring link types and aggregating network layers may lead to inaccurate conclusions about contagion dynamics particularly when the correlation of degrees between layers is high

k core percolation on multiplex networks we generalize the theory of k text ensuremath mathrm core percolation on complex networks to mathbf k text ensuremath mathrm core percolation on multiplex networks where mathbf k ensuremath equiv k 1 k 2 k m multiplex networks can be defined as networks with vertices of one kind but m different types of edges representing different types of interactions for such networks the mathbf k text ensuremath mathrm core is defined as the largest subgraph in which each vertex has at least k i edges of each type i 1 2 m we derive self consistency equations to obtain the birth points of the mathbf k text ensuremath mathrm cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution to clarify our general results we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of erd ifmmode mbox h o else h o fi s r enyi and scale free multiplex networks we find hybrid phase transitions at the emergence points of mathbf k text ensuremath mathrm cores except the 1 1 core for which the transition is continuous we apply the mathbf k text ensuremath mathrm core decomposition algorithm to air transportation multiplex networks composed of two layers and obtain the size of k 1 k 2 cores

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments