approximation of pareto optima in multiple objective shortest path problems we study methods for approximating the set of pareto optimal paths in multiple objective shortest path problems known generalizations of standard shortest path methods will compute this set but can suffer from rapidly increasing computational and storage demands as problem size increases in an effort to avoid such difficulties we develop approximation methods that can estimate the pareto optima to any required degree of accuracy the approximation methods are fully polynomial that is they operate in time and space bounded by a polynomial in problem size and accuracy of approximation the greater the accuracy the more time required to reach a solution we show how approximation methods may be applied to yield fully polynomial approximation schemes for a variety of np complete single objective problems

fuzzy shortest paths abstract shortest paths are one of the simplest and most widely used concepts in nonfuzzy networks fuzzy shortest paths also have a variety of applications but are hampered by the possibility that the fuzzy shortest length may not correspond to an actual path in the network in this paper we present new models based on fuzzy shortest paths that circumvent this problem and we also give a general algorithm based on dynamic programming to solve the new models in addition we analyze the fuzzy shortest path algorithms in terms of submodular functions the analysis concepts developed for fuzzy shortest paths are also discussed in terms of general fuzzy mathematical programming

the maximum fuzzy weighted matching models and hybrid genetic algorithm a matching of a graph is an independent subset of the edges set and a maximum matching is a matching with as many edges in it as possible the maximum weighted matching problem is to find a maximum matching in a given graph such that the sum of the weights of the edges in it is maximum in this paper the concepts of expected maximum fuzzy weighted matching the maximum fuzzy weighted matching and the most maximum fuzzy weighted matching are initialized according to various decision criteria the maximum fuzzy weighted matching problem is formulated as expected value model chance constrained programming and dependent chance programming by using the credibility theory and the crisp equivalents are also given furthermore a hybrid genetic algorithm is designed for solving the proposed fuzzy programming models finally a numerical example is given

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

fuzzy shortest path problems incorporating interactivity among paths abstract this paper deals with a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length such a problem is ill posed because each arc cannot be identified as being either on the shortest path or not therefore based on the possibility theory we introduce the concept of degree of possibility that an arc is on the shortest path every pair of distinct paths from the source node to any other node is implicitly assumed to be noninteractive in the conventional approaches this assumption is unrealistic and involve inconsistencies to overcome this drawback we define a new comparison index between the sum of fuzzy numbers by considering interactivity among fuzzy numbers an algorithm is presented to determine the degree of possibility for each arc on a network finally this algorithm is evaluated by means of large scale numerical examples consequently we can find this approach is efficient even for real world practical networks

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

a shortest path problem on a network with fuzzy arc lengths abstract we concentrate on a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length introducing an order relation between fuzzy numbers based on fuzzy min a nondominated path or pareto optimal path from the specified node to every other node is defined an algorithm for solving the problem is developed on the basis of the multiple labeling method for a multicriteria shortest path as a result a number of nondominated paths can be obtained and is offered to a decision maker however a number of nondominated paths derived from large scale network may be too numerous for him to choose a preferable path due to this situation we propose a method to reduce the number of paths according to a possibility level the proposed algorithm is numerically evaluated on large scale random networks

the computational complexity of the criticality problems in a network with interval activity times abstract the paper analyzes the criticality in a network with interval activities duration times a natural generalization of the criticality notion for a path an activity and an event for the case of network with interval activity duration times is given the computation complexity of five problems linked to the introduced criticality notion is presented

on conservation and dual consistency for summation by parts based approximations of parabolic problems we consider the coupling of parabolic problems discretized using difference operators on summation by parts sbp form with interface conditions imposed weakly in 1 2 it was shown that conserv

on combinatorial optimization problems on matroids with uncertain weights in this paper the combinatorial optimization problem on weighted matroid is considered it is assumed that the weights in the problem are ill known and they are modeled as fuzzy intervals the optimality of solutions and the optimality of elements are characterized this characterization is performed in the setting of possibility theory a method of choosing a solution under uncertainty is also proposed

fuzzy location problems on networks location problems concern a wide set of fields where it is usually assumed that exact data are known however in real applications the location of the facility considered can be full of linguistic vagueness that can be appropriately modelled using networks with fuzzy values thus fuzzy location problems on networks arise this paper deals with their general formulation and the description of the ways to solve them namely we show the variety of problems that can be considered in this context and for some of them we propose the most operative approaches for their solution

a dual approach to solve the fuzzy linear programming problem abstract a concept of fuzzy objective based on the fuzzification principle is presented in accordance with this concept the fuzzy linear mathematical programming problem is easily solved a relationship of duality among fuzzy constraints and fuzzy objectives is given the dual problem of a fuzzy linear programming problem is also defined

critical path analysis in the network with fuzzy activity times a natural generalization of the criticality notion in a network with fuzzy activity times is given it consists in direct application of the extension principle of zadeh to the notion of criticality of a path an activity an event treated as a function of the activities duration times in the network there are shown some relations between the notion of fuzzy criticality introduced in the paper and the notion of interval criticality criticality in the network with interval activity times proposed by the authors in another paper two methods of calculation of the path degree of criticality according to the proposed concept of fuzzy criticality are presented

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

unified approach fuzzy graph problems we present a taxonomy of fuzzy graphs that treats fuzziness in vertex existence edge existence edge connectivity and edge weight within that framework we formulate some standard graph theoretic problems shortest paths and minimum cut for fuzzy graphs using a unified approach distinguished by its uniform application of guiding principles such as the construction of membership grades via the ranking of fuzzy numbers the preservation of membership grade normalization and the collapsing of fuzzy sets of graphs into fuzzy graphs finally we provide algorithmic solutions to these problems with examples

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

the shortest path problem on networks with fuzzy parameters shortest path problem where the costs have vague values is one of the most studied problems in fuzzy sets and systems area however due to its high computational complexity previously published algorithms present peculiarities and problems that need to be addressed e g they find costs without an existing path they determine a fuzzy solution set but do not give any guidelines to help the decision maker choose the best path they can only be applied in graphs with fuzzy non negative parameters etc in this paper one proposes an iterative algorithm that assumes a generic ranking index for comparing the fuzzy numbers involved in the problem in such a way that each time in which the decision maker wants to solve a concrete problem s he can choose or propose the ranking index that best suits that problem this algorithm that solves the above remarked drawbacks is based on the ford moore bellman algorithm for classical graphs and in concrete it can be applied in graphs with negative parameters and it can detect whether there are negative circuits for the sake of illustrating the performance of the algorithm in the paper it has been here developed using only certain order relations but it is not restricted at all to use these comparison relations exclusively the proposed algorithm is easy of understanding as the theoretical base of a decision support system oriented to solving this kind of problems

results related to threshold phenomena research in satisfiability lower bounds we present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form cnf formulas in a companion paper d achlioptas theoret comput sci this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal intuitive manner

shortest paths algorithms we conduct an extensive computational study of shortest paths algorithms including some very recent algorithms we also suggest new algorithms motivated by the experimental results and prove inter

on a multicriteria shortest path problem abstract multicriteria shortest path problems have not been treated intensively in the specialized literature despite their potential applications in fact a single objective function may not be sufficient to characterize a practical problem completely for instance in a road network several parameters as time cost distance etc can be assigned to each arc clearly the shortest path may be too expensive to be used nevertheless the decision maker must be able to choose some solution possibly not the best for all the criteria in this paper we present two algorithms for this problem one of them is an immediate generalization of the multiple labelling scheme algorithm of hansen for the bicriteria case based on this algorithm it is proved that any pair of nondominated paths can be connected by nondominated paths this result is the support of an algorithm that can be viewed as a variant of the simplex method used in continuous linear multiobjective programming a small example is presented for both algorithms

paths of least resistance in possibilistic production systems abstract we introduce the idea of possibilistic production systems we provide a general algorithm for finding a path from an initial node to a goal node we show that the use of a heuristic function can help in the efficient implementation of the search process we see that a significant role in determining potential optimal paths is played by concentrating on the most difficult activity to accomplish

shortest path problem with uncertain arc lengths uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths with help from the operational law of uncertainty theory this paper gives the uncertainty distribution of the shortest path length also it investigates solutions to the a shortest path and the most shortest path in an uncertain network it points out that there exists an equivalence relation between the a shortest path in an uncertain network and the shortest path in a corresponding deterministic network which leads to an effective algorithm to find the a shortest path and the most shortest path roughly speaking this algorithm can be broken down into two parts constructing a deterministic network and then invoking the dijkstra algorithm

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

minimizing energy below the glass thresholds focusing on the optimization version of the random k satisfiability problem the max k sat problem we study the performance of the finite energy version of the survey propagation algorithm we show that a simple linear time backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states a comparative numerical study on one of the most efficient local search procedures is also given

phase transitions and the search problem we describe how techniques that were originally developed in statistical mechanics can be applied to search problems that arise commonly in artificial intelligence this approach is useful for understanding the typical behavior of classes of problems in particular these techniques predict that abrupt changes in computational cost analogous to physical phase transitions should occur universally as heuristic effectiveness or search space topology is varied we also present a number of open questions raised by these studies

survey propagation at finite temperature application to a sourlas code as a toy model in this paper we investigate a finite temperature generalization of survey propagation by applying it to the problem of finite temperature decoding of a biased finite connectivity sourlas code for temperatures lower than the nishimori temperature we observe that the result is a shift of the location of the dynamical critical channel noise to larger values than the corresponding dynamical transition for belief propagation as suggested recently by migliorini and saad for ldpc codes we show how the finite temperature 1rsb sp gives accurate results in the regime where competing approaches fail to converge or fail to recover the retrieval state

project network analysis with fuzzy activity times abstract triangular fuzzy numbers are used to represent project activities whose duration times are uncertain and cannot be represented stochastically a comprehensive path analysis method is devised using fuzzy arithmetic and a fuzzy number comparison method to determine fuzzy project completion time and the degrees of criticality of each network path possibility theory is then used to determine the possibilities of the project completion given the fuzzy project completion time the existing composite method and the new comprehensive comparison method are compared and contrasted

computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using α cuts we are concerned with the design of a model and an algorithm for computing a shortest path in a network having various types of fuzzy arc lengths first we develop a new technique for the addition of various fuzzy numbers in a path using a cuts by proposing a linear least squares model to obtain membership functions for the considered additions then using a recently proposed distance function for comparison of fuzzy numbers we present a dynamic programming method for finding a shortest path in the network examples are worked out to illustrate the applicability of the proposed model

the use of fuzzy variables in pert abstract in this paper we present a method called fpert for estimating a project completion time in the situation when activity duration times in the project network model are given in the form of fuzzy variables fuzzy sets on time space theoretical foundations of the method as well as results of calculations derived from a simple example are included

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb