generalized domino parity inequalities for the symmetric traveling salesman problem we extend the work of letchford letchford a n 2000 separating a superclass of comb inequalities in planar graphs math oper res 25 443 454 by introducing a new class of valid inequalities for the traveling salesman problem called the generalized domino parity gdp constraints just as letchford s domino parity constraints generalize comb inequalities gdp constraints generalize the most well known multiple handle constraints including clique tree bipartition path and star inequalities furthermore we show that a subset of gdp constraints containing all of the clique tree inequalities can be separated in polynomial time provided that the support graph g is planar and provided that we bound the number of handles by a fixed constant h

ornstein zernike theory for finite range ising models above t c we derive a precise ornstein zernike asymptotic formula for the decay of the two point function 0 x in the general context of finite range ising type models on d the proof relies in an essential way on the a priori knowledge of the strict exponential decay of the two point function and by the sharp characterization of phase transition due to aizenman barsky and fernandez goes through in the whole of the high temperature region c as a byproduct we obtain that for every c the inverse correlation length is an analytic and strictly convex function of direction

exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

domination by product measures we consider families of 0 1 valued random variables indexed by the vertices of countable graphs with bounded degree first we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site the probability of seeing a 1 at this site is at least a value p which is large enough then this random field dominates a product measure with positive density moreover the density of this dominated product measure can be made arbitrarily close to 1 provided that p is close enough to 1 next we address the issue of obtaining the critical value of p defined as the threshold above which the domination by positive density product measures is assured for the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units this critical value is shown to be 1 k k k 1 k 1 and a discontinuous transition is shown to occur similar critical values of p are found for other classes of probability measures on 0 1 z for the class of k dependent measures the critical value is again 1 k k k 1 k 1 with a discontinuous transition for the class of two block factors the critical value is shown to be 1 2 and a continuous transition is shown to take place in this case thus both the critical value and the nature of the transition are different in the two block factor and 1 dependent cases

the droplet in the tube a case of phase transition in the canonical ensemble we consider the 2 dimensional ising model with ferromagnetic nearest neighbour interaction at inverse temperature lets n t be the total magnetization inside ann n square box per be the gibbs state in with periodic b c andm be the spontaneous magnetization we show the existence of the limit psi varrho mathop lim limits n to infty left frac 1 beta n right ln mu lambda per s n n varrho for m provided is large enough it turns out that the quantity is closely related to the wulf construction and the dependence of the function on is singular

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

variational problems with percolation dilute spin systems at zero temperature we study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of convergence and techniques mixing geometric measure theory and percolation while scaling to zero the lattice spacing the limit is not trivial above a percolation threshold since the lattice energies are not equi coercive a suitable notion of limit magnetization must be defined which can be characterized by two phases separated by an interface the macroscopic surface tension at this interface is characterized through a first passage percolation formula which highlights interesting connections between variational problems and percolation issues a companion result on the asymptotic description on energies defined on paths in a dilute environment is also given

harmonic crystal on the wall a microscopic approach a three dimensional winterbottom type construction in the regime of partial wetting is derived in a scaling limit of a gas of microscopic gaussian sos droplets under the fixed volume constraint the proof is based on a coarse graining of the random microscopic region wetted by the crystal random walk representations of various quantities related to free massless fields and a stability analysis of the torsional rigidity problem

gibbs ensembles of nonintersecting paths we consider a family of determinantal random point processes on the two dimensional lattice and prove that members of our family can be interpreted as a kind of gibbs ensembles of nonintersecting paths examples include probability measures on lozenge and domino tilings of the plane some of which are non translation invariant the correlation kernels of our processes can be viewed as extensions of the discrete sine kernel and we show that the gibbs property is a consequence of simple linear relations satisfied by these kernels the processes depend on infinitely many parameters which are closely related to parametrization of totally positive toeplitz matrices

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

learning conditional preference networks conditional preference networks cp nets have recently emerged as a popular language capable of representing ordinal preference relations in a compact and structured manner in this paper we investigate the problem of learning cp nets in the well known model of exact identification with equivalence and membership queries the goal is to identify a target preference ordering with a binary valued cp net by interacting with the user through a small number of queries each example supplied by the user or the learner is a preference statement on a pair of outcomes in this model we show that acyclic cp nets are not learnable with equivalence queries alone even if the examples are restricted to swaps for which dominance testing takes linear time by contrast acyclic cp nets are what is called attribute efficiently learnable when both equivalence queries and membership queries are available we indeed provide a learning algorithm whose query complexity is linear in the description size of the target concept but only logarithmic in the total number of attributes interestingly similar properties are derived for tree structured cp nets in the presence of arbitrary examples our learning algorithms are shown to be quasi optimal by deriving lower bounds on the vc dimension of cp nets in a nutshell our results reveal that active queries are required for efficiently learning cp nets in large multi attribute domains

on local behaviour of the phase separation line in the 2d ising model the aim of this note is to discuss some statistical properties of the phase separation line in the 2d low temperature ising model we prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation the limiting gaussian measure corresponds to a scaled brownian bridge with direction dependent parameters up to the temperature factor the variances of local increments of this limiting process are inversely proportional to the stiffness

polyhedron of triangle free simple 2 matchings in subcubic graphs a simple 2 matching in an edge weighted graph is a subgraph all of whose nodes have degree 0 1 or 2 we consider the problem of finding a maximum weight simple 2 matching that contains no triangles which is closely related to a class of relaxations of the traveling salesman problem tsp our main results are for graphs with maximum degree 3 a complete description of the convex hull of incidence vectors of triangle free simple 2 matchings and a strongly polynomial time algorithm for the above problem our system requires the use of a type of comb inequality introduced by grotschel and padberg for the tsp polytope that has 0 1 2 coefficients and hence is more general than the well known blossom inequality used in edmonds characterization of the simple 2 matching polytope

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

small travelling salesman polytopes we introduce a new class of valid inequalities for the polytope of the symmetric travelling salesman problem we also gave complete characterizations of the polytope for 6 and 7 cities for the latter case the new inequalities are needed these results are related to work of r z norman in the 1950 s

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

decision making with multiple objectives using gai networks this paper deals with preference representation on combinatorial domains and preference based recommendation in the context of multicriteria or multiagent decision making the alternatives of the decision problem are seen as elements of a product set of attributes and preferences over solutions are represented by generalized additive decomposable gai utility functions modeling individual preferences or criteria thanks to decomposability utility vectors attached to solutions can be compiled into a graphical structure closely related to junction trees the so called gai network using this structure we present preference based search algorithms for multicriteria or multiagent decision making although such models are often non decomposable over attributes we actually show that gai networks are still useful to determine the most preferred alternatives provided preferences are compatible with pareto dominance we first present two algorithms for the determination of pareto optimal elements then the second of these algorithms is adapted so as to directly focus on the preferred solutions we also provide results of numerical tests showing the practical efficiency of our procedures in various contexts such as compromise search and fair optimization in multicriteria or multiagent problems

equivalences of the large deviation principle for gibbs measures and critical balance in the ising model in this paper we obtain the equivalence of the large deviation principle for gibbs measures with and without an external field for the ising model the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube b h ash 0 for variousb we find a critical balance at a valueb0 ofb

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

constrained variational problem with applications to the ising model we continue our study of the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh initiated in our earlier work we strengthen further a result previously proven by martirosyan at low enough temperature which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the system if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the system in our earlier work this result was extended to every subcritical value of the temperature here for every subcritical value of the temperature we show the existence of a critical valueb0 t which separates the two regimes specified above we also find the asymptotic shape of the region occupied by the phase in the second regime which turns out to be a squeezed wulff shape the main step in our study is the solution of the variational problem of finding the curve minimizing the wulff functional which curve is constrained to the unit square other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field extended to all temperatures below the critical one

approximation of subadditive functions and convergence rates in limiting shape results for a nonnegative subadditive function h on mathbb z d with limiting approximation g x lim n h nx n it is of interest to obtain bounds on the discrepancy between g x and h x typically of order x nu with nu 1 for certain subadditive h x particularly those which are expectations associated with optimal random paths from 0 to x in a somewhat standardized way a more natural and seemingly weaker property can be established every x is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound we show that this convex hull property implies the desired bound for all x applications include rates of convergence in limiting shape results for first passage percolation standard and oriented and longest common subsequences and bounds on the error in the exponential decay approximation to the off axis connectivity function for subcritical bernoulli bond percolation on the integer lattice

parameterized ceteris paribus preferences over atomic conjunctions under conservative semantics we consider a propositional language for describing parameterized ceteris paribus preferences over atomic conjunctions such preferences are only required to hold when the alternatives being compared agree on a specified subset of propositional variables regarding the expressivity of the language in question we show that a parameterized preference statement is equivalent to a conjunction of an exponential number of classical non parameterized ceteris paribus statements next we present an inference system for parameterized statements and prove that the problem of checking the semantic consequence relation for such statements is conp complete we propose an approach based on formal concept analysis to learning preferences from data by showing that ceteris paribus preferences valid in a particular model correspond to implications of a special formal context derived from this model the computation of a complete preference set is then reducible to the computation of minimal hypergraph transversals finally we adapt a polynomial time algorithm for abduction using horn clauses represented by their characteristic models to the problem of determining preferences over new alternatives from preferences over given alternatives with ceteris paribus preferences as the underlying model

ornstein zernike theory for the bernoulli bond percolation on mathbb z d we derive a precise ornstein zernike asymptotic formula for the decay of the two point function p p 0 x of the bernoulli bond percolation on the integer lattice z d in any dimension d 2 in any direction x and for any subcritical value of p p c d

on the 2d ising wulff crystal near criticality we study the behavior of the two dimensional ising model in a finite box at temperatures that are below but very close to the critical temperature in a regime where the temperature approaches the critical point and simultaneously the size of the box grows fast enough we establish a large deviation principle that proves the appearance of a round wulff crystal

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

dobrushin kotecký shlosman theorem up to the critical temperature we develop a non perturbative version of the dobrushin koteck shlosman theory of phase separation in the canonical 2d ising ensemble the results are valid for all temperatures below critical

finite size scaling for the 2d ising model with minus boundary conditions we study the magnetization m l h for the ising model on a large but finite lattice square under the minus boundary conditions using known large deviation results evaluating the balance between the competing effects of the minus boundary conditions and the external magnetic field h we describe the details of its dependence on h as exemplified by the finite size rounding of the infinite volume magnetization discontinuity and its shift with respect to the infinite volume transition point

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

solving the discrete multiple criteria problem using convex cones an interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi concave increasing utility or value function the method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions the maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker responses are used to eliminate alternatives using convex cones based on expressed preferences and then a new set of weights is found that satisfies the indicated preferences the requisite theory and proofs as well as a detailed numerical example are included in addition the results of some computational experiments to test the effectiveness of the method are described

cp and ocf networks a comparison network approaches are used to structure partition and display formalisms in the area of knowledge representation as well as decision making known approaches are for instance ocf networks bayesian style networks where every variable is annotated with a conditional ranking table and cp networks directed acyclic networks with local preferences annotated at each vertex the structures of these networks are similar but their semantics seem to be quite different in this paper we discuss if ocf networks can be used to model the information of cp networks and vice versa to answer this question we investigate which restrictions and conditions have to be presupposed to either of the approaches such that one structure can be used to generate the other

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

critical region for droplet formation in the two dimensional ising model we study the formation dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence specifically we consider the 2d ising model in volumes of size l2 inverse temperature c and overall magnetization conditioned to take the value m l2 2m vl where c 1 is the critical temperature m m is the spontaneous magnetization and vl is a sequence of positive numbers we find that the critical scaling for droplet formation dissolution is when vl3 2l 2 tends to a definite limit specifically we identify a dimensionless parameter proportional to this limit a non trivial critical value c and a function such that the following holds for c there is a single wulff shaped droplet containing a fraction c 2 3 of the magnetization deficit and there are no other droplets beyond the scale of log l moreover and are related via a universal equation that apparently is independent of the details of the system

a general approximation technique for constrained forest problems we present a general approximation technique for a large class of graph problems our technique mostly applies to problems of covering at minimum cost the vertices of a graph with trees cycles or paths satisfying certain requirements in particular many basic combinatorial optimization problems fit in this framework including the shortest path minimum cost spanning tree minimum weight perfect matching traveling salesman and steiner tree problems our technique produces approximation algorithms that run in o n 2 log n time and come within a factor of 2 of optimal for most of these problems for instance we obtain a 2 approximation algorithm for the minimum weight perfect matching problem under the triangle inequality our running time of o n 2 log n time compares favorably with the best strongly polynomial exact algorithms running in o n 3 time for dense graphs a similar result is obtained for the 2 matching problem and its variants we also derive the first approximation algorithms for many np complete problems including the non fixed point to point connection problem the exact path partitioning problem and complex location design problems moreover for the prize collecting traveling salesman or steiner tree problems we obtain 2 approximation algorithms therefore improving the previously best known performance guarantees of 2 5 and 3 respectively math programming 59 1993 pp 413 420

maximum cardinality simple 2 matchings in subcubic graphs a simple 2 matching in a graph is a subset of edges m such that every node is incident with at most two edges in m a simple 2 matching is called ck restricted k 2 if it contains no cycles of length k or less we begin by considering the problem of finding maximum cardinality simple 2 matchings in subcubic graphs i e graphs whose nodes have degree at most 3 for this problem we present a min max theorem a simple polynomial time algorithm and some other results then by generalizing an approach due to vornberger we show the following for subcubic graphs the maximum cardinality c3 and c4 restricted simple 2 matching problems are polynomial time solvable there exist min max theorems for these problems and for k 5 the maximum cardinality ck restricted simple 2 matching problems are np hard

the 2d ising model near criticality a fk percolation analysis we study the 2d ising model defined on finite boxes at temperatures that are below but very close from the critical point when the temperature approaches the critical point and the size of the box grows fast enough we establish large deviations estimates on fk percolation events that concern the phenomenon of phase coexistence

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

preference structures and their numerical representations abstract this paper presents a selective survey of numerical representations of preference structures from the perspective of the representational theory of measurement it reviews historical contributions to ordinal additive and expected utility theories then describes recent contributions in these areas

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

shannon mcmillan theorems for discrete random fields along curves and lower bounds for surface order large deviations the notion of a surface order specific entropy hc p of a two dimensional discrete random field p along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c existence is shown by proving a corresponding shannon mcmillan theorem we obtain a representation of hc p as a mixture of specific entropies along the tangent lines of c as an application the specific entropy along curves is used to refine follmer and ort s lower bound for the large deviations of the empirical field of an attractive gibbs measure from its ergodic behaviour in the phase transition regime

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

a weighted kt t free t factor algorithm for bipartite graphs for a simple bipartite graph and an integer t 2 we consider the problem of finding a minimum weight kt t free t factor which is a t factor containing no complete bipartite graph kt t as a subgraph when t 2 this problem amounts to the square free 2 factor problem in a bipartite graph for the unweighted square free 2 factor problem a combinatorial algorithm is given by hartvigsen and the weighted version of the problem is np hard for general t pap designed a combinatorial algorithm for the unweighted version and makai gave a dual integral description of kt t free t matchings for a certain case where the weight vector is vertex induced on any subgraph isomorphic to kt t for this class of weight vectors we propose a strongly polynomial algorithm to find a minimum weight kt t free t factor the algorithm adapts the unweighted algorithms of hartvigsen and pap and a primal dual approach to the minimum cost flow problem the algorithm is fully combinatorial and thus provides a dual integrality theorem which is tantamount to makai s one

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

restricted 2 factor polytopes the optimal k restricted 2 factor problem consists of finding in a complete undirected graph k n a minimum cost 2 factor subgraph having degree 2 at every node with all components having more than k nodes the problem is a relaxation of the well known symmetric travelling salesman problem and is equivalent to it when k n 1 we study the k restricted 2 factor polytope we present a large class of valid inequalities called bipartition inequalities and describe some of their properties some of these results are new even for the travelling salesman polytope for the case k 3 the triangle free 2 factor polytope we derive a necessary and sufficient condition for such inequalities to be facet inducing

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

computational techniques for a simple theory of conditional preferences a simple logic of conditional preferences is defined with a language that allows the compact representation of certain kinds of conditional preference statements a semantics and a proof theory cp nets and tcp nets can be mapped into this logic and the semantics and proof theory generalise those of cp nets and tcp nets the system can also express preferences of a lexicographic kind the paper derives various sufficient conditions for a set of conditional preferences to be consistent along with algorithmic techniques for checking such conditions and hence confirming consistency these techniques can also be used for totally ordering outcomes in a way that is consistent with the set of preferences and they are further developed to give an approach to the problem of constrained optimisation for conditional preferences

dynamics of an unbounded interface between ordered phases we investigate the evolution of a single unbounded interface between ordered phases in two dimensional ising ferromagnets that are endowed with single spin flip zero temperature glauber dynamics we examine specifically the cases where the interface initially has either one or two corners in both examples the interface evolves to a limiting self similar form we apply the continuum time dependent ginzburg landau equation and a microscopic approach to calculate the interface shape for the single corner system we also discuss a correspondence between the interface and the young diagram that represents the partition of the integers

variable length analog of stavskaya process a new example of misleading simulation this article presents a new example intended to showcase limitations of computer simulations in the study of random processes with local interaction for this purpose we examine a new version of the well known stavskaya process which is a discrete time analog of the well known contact processes like the bulk of random processes studied till now the stavskaya process is constant length that is its components do not appear or disappear in the course of its functioning the process which we study here and call variable stavskaya vs is similar to stavskaya it is discrete time its states are bi infinite sequences whose terms take only two values denoted here as minus and plus and the measure concentrated in the configuration all pluses is invariant however it is a variable length which means that its components also called particles may appear and disappear under its action the operator vs is a composition of the following two operators the first operator called birth depends on

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

expressive efficiency of two kinds of specific cp nets cp nets conditional preference networks are a graphical model for compactly expressing conditional ceteris paribus all other things being equal preference statements on multi attribute domains in this paper we investigate the expressive efficiency of two kinds of binary valued cp nets the first kind is set structured cp nets and the second kind is equal difference cp nets for the first kind we prove that it can express 3 n 2 n preference relations with n preference rules and has an expressive efficiency of 3 n 2 n n for the second kind we show that it can express a total order of 2 n 1 2 n 1 preference relations with 2 n 1 preference rules and has an expressive efficiency of 2 n 1 for the future research we propose an open problem given an acyclic cp net n with the in degree sequence of d 1 d 2 d n how many preference relations can be expressed by n

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

dobrushin kotecký shlosman theorem for polygonal markov fields in the plane we consider the so called length interacting arak surgailis polygonal markov fields with v shaped nodes a continuum and isometry invariant process in the plane sharing a number of properties with the two dimensional ising model for these polygonal fields we establish a low temperature phase separation theorem in the spirit of the dobrushin koteck shlosman theory with the corresponding wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model as an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process following the ideas of fernandez ferrari and garcia

lower bounds for boundary roughness for droplets in bernoulli percolation we consider boundary roughness for the droplet created when supercritical two dimensional bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l 2 for large l the maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull we show that for large l this maximum is at least of order l 1 3 logl 2 3 this complements the upper bound of order l 1 3 logl 2 3 proved in al3 for the average local roughness the exponent 1 3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions

interactive approaches for discrete alternative multiple criteria decision making with monotone utility functions in this paper we develop interactive approaches for the discrete alternative multiple criteria decision making problem we develop an algorithm that finds the most preferred alternative of a decision maker dm assuming only that the dm has a monotonic utility function the algorithm divides the criteria space into a number of smaller subspaces and then uses the ideal points of these subspaces to eliminate alternatives we also develop a more efficient version of the algorithm for the more restrictive case of a monotonic quasiconcave utility function we present favorable computational results in terms of the required number of pairwise comparisons for both versions of the algorithm we then develop a general algorithm that first identifies the type of the dm s utility function and then employs the approach that is compatible with the identified utility function type we also present computational results for the general algorithm

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

isoperimetry in two dimensional percolation we study isoperimetric sets i e sets with minimal boundary for a prescribed volume on the unique infinite connected component of supercritical bond percolation on the square lattice in the limit of the volume tending to infinity properly scaled isoperimetric sets are shown to converge in the hausdorff metric to the solution of an isoperimetric problem in 2 with respect to a particular norm as part of the proof we also show that the anchored isoperimetric profile as well as the cheeger constant of the giant component in finite boxes scale to deterministic quantities this settles a conjecture of itai benjamini for the square lattice 2015 wiley periodicals inc

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

horn approximations of empirical data abstract formal ai systems traditionally represent knowledge using logical formulas sometimes however a model based representation is more compact and enables faster reasoning than the corresponding formula based representation the central idea behind our work is to represent a large set of models by a subset of characteristic models more specifically we examine model based representations of horn theories and show that there are large horn theories that can be exactly represented by an exponentially smaller set of characteristic models we show that deduction based on a set of characteristic models requires only polynomial time as it does using horn theories more surprisingly abduction can be performed in polynomial time using a set of characteristic models whereas abduction using horn theories is np complete finally we discuss algorithms for generating efficient representations of the horn theory that best approximates a general set of models

the spectral gap for the kawasaki dynamics at low temperature in this paper we analyze the convergence to equilibrium of kawasaki dynamics for the ising model in the phase coexistence region first we show in strict analogy with the nonconservative case that in any lattice dimension for any boundary condition and any positive temperature and particle density the spectral gap in a box of side l does not shrink faster than a negative exponential of the surface ld 1 then we prove that in two dimensions and for free boundary condition the spectral gap in a box of side l is smaller than a negative exponential of l provided that the temperature is below the critical one and the particle density satisfies where represents the particle density of the plus and minus phase respectively

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

a proof of the gibbs thomson formula in the droplet formation regime we study equilibrium droplets in two phase systems at parameter values corresponding to phase coexistence specifically we give a self contained microscopic derivation of the gibbs thomson formula for the deviation of the pressure and the density away from their equilibrium values which according to the interpretation of the classical thermodynamics appears due to the presence of a curved interface the general albeit heuristic reasoning is corroborated by a rigorous proof in the case of the two dimensional ising lattice gas

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

use of convex cones in interactive multiple objective decision making one approach for solving decision problems involving multiple objectives is interactive optimization methods based on this approach assess the decision maker s preference structure interactively typically based on pairwise comparisons and tradeoffs and guide the search process toward identifying improved solutions a desirable feature of such approaches that is based on minimizing the preference information requirements is fast convergence toward this end the use of convex cones as a preference structure representation has been proposed in the literature in this work new theory is developed that aids in further reducing preference information requirements and improving convergence new cones termed p cones are developed the efficiencies of solution alternatives are evaluated with respect to the p cones and these are termed p cone efficiencies acceleration and early termination procedures that are based on these efficiencies are proposed the procedures are presented within a solution framework for solving multiple objective linear programming molp problems along with computational results

fluctuations of shapes of large areas under paths of random walks we discuss statistical properties of random walks conditioned by fixing a large area under their paths we prove the functional central limit theorem invariance principle for these conditional distributions the limiting gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous gaussian random process obtained by an integral transformation of the white noise from the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one dimensional sos model

graphically structured value function compilation classical work on eliciting and representing preferences over multi attribute alternatives has attempted to recognize conditions under which value functions take on particularly simple and compact form making their elicitation much easier in this paper we consider preferences over discrete domains and show that for a certain class of simple and intuitive qualitative preference statements one can always generate compact value functions consistent with these statements these value functions maintain the independence structure implicit in the original statements for discrete domains these representation theorems are much more general than previous results however we also show that it is not always possible to maintain this compact structure if we add explicit ordering constraints among the available outcomes

on criticality for competing influences of boundary and external field in the ising model we continue a study of schonmann 1994 schonmann and shlosman 1996 and greenwood and sun 1997 regarding the competing influences of boundary conditions and external field for the ising model we find a critical point b0 in the competing influences for low temperature in dimension d 2a7e 2

possibilistic preference networks this paper studies the use of product based possibilistic networks for representing preferences in multidimensional decision problems this approach uses symbolic possibility weights and defines a partial preference order among solutions to a set of conditional preference statements on the domains of discrete decision variables in the case of boolean decision variables this partial ordering is shown to be consistent with the preference ordering induced by the ceteris paribus assumption adopted in cp nets namely by completing the possibilistic net ordering with suitable constraints between products of symbolic weights all cp net preferences can be recovered computing procedures for comparing solutions are provided the flexibility and representational power of the approach is stressed

the wulff construction and asymptotics of the finite cluster distribution for two dimensional bernoulli percolation we consider two dimensional bernoulli percolation at densityp p c and establish the following results

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

geometric variational problems of statistical mechanics and of combinatorics we present the geometric solutions of the various extremal problems of statistical mechanics and combinatorics together with the wulff construction which predicts the shape of the crystals we discuss the construction which exhibits the shape of a typical young diagram and of a typical skyscraper

fluctuations of the phase boundary in the 2d ising ferromagnet we discuss some statistical properties of the phase boundary in the 2d low temperature ising ferromagnet in a box with the two component boundary conditions we prove the weak convergence in c 0 1 of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them the limiting gaussian measure coincides with the conditional distribution of certain gaussian process obtained by the integral transformation of the white noise