a class of euclidean routing problems with general rout cost functions in most vehicle routing problems a given set of customers is to be partitioned into a collection of regions each of which is assigned to a single vehicle starting at a depot and returning there after visiting all of the region s customers exactly once in a route in this paper we consider problem settings where the cost of a route may depend on its length as well as m the number of points on the route according to some general function f m assumed to be nondecreasing and concave in we describe a class of o n logn or o n2 heuristics and show under mild probabilistic assumptions that the solutions generated are asymptotically optimal we also show that lower and upper bounds on the system wide costs may be computed with even simpler procedures and that these bounds are asymptotically tight under the same assumptions

submodularity and the traveling salesman problem in this paper we investigate the relationship between traveling salesman tour lengths and submodular functions this work is motivated by the one warehouse multi retailer inventory distribution problem with traveling salesman tour vehicle routing costs our goal is to find a submodular function whose values are close to those of optimal tour lengths through a central warehouse and a group of retailers our work shows that a submodular approximation to traveling salesman tour lengths whose error is bounded by a constant does not exist however we present heuristics that have errors which grow slowly with the number of retailers for the traveling salesman problem in the euclidean plane furthermore we perform computational tests that show that our submodular approximations of traveling salesman tour lengths have smaller errors than our theoretical worst case analysis would lead us to believe

propagator for finite range potentials the schrodinger equation in integral form is applied to the one dimensional scattering problem in the case of a general finite range nonsingular potential a simple expression for the laplace transform of the transmission propagator is obtained in terms of the associated fredholm determinant by means of matrix methods the particular form of the kernel and the peculiar aspects of the transmission problem play an important role the application to an array of delta potentials is shown

the primality of r1031 on decrit une technique permettant de demontrer que le nombre r1031 10 1031 1 9 compose uniquement de 1 est premier

on the distance constrained vehicle routing problem we analyze the vehicle routing problem with constraints on the total distance traveled by each vehicle two objective functions are considered minimize the total distance traveled by vehicles and minimize the number of vehicles used we demonstrate a close relationship between the optimal solutions for the two objective functions and perform a worst case analysis for a class of heuristics we present a heuristic that provides a good worst case result when the number of vehicles used is relatively small

reoptimization approaches for the vehicle routing problem with stochastic demands we consider the vehicle routing problem with stochastic demands vrpsd under reoptimization we develop and analyze a finite horizon markov decision process mdp formulation for the single vehicle case and establish a partial characterization of the optimal policy we also propose a heuristic solution methodology for our mdp named partial reoptimization based on the idea of restricting attention to a subset of all the possible states and computing an optimal policy on this restricted set of states we discuss two families of computationally efficient partial reoptimization heuristics and illustrate their performance on a set of instances with up to and including 100 customers comparisons with an existing heuristic from the literature and a lower bound computed with complete knowledge of customer demands show that our best partial reoptimization heuristics outperform this heuristic and are on average no more than 10 13 away from this lower bound depending on the type of instances

probabilistic analysis of a generalized bin packing problem and applications we give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs in this general class of packing problems objects are described by a given number of attribute values some attributes may be discrete others may be continuous bins are sets of objects and the collection of feasible bins is merely required to satisfy some general consistency properties we characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed or as the limit of a sequence of such linear program values we also provide bounds for the rate of convergence of the average cost to its asymptotic value the analysis suggests an a s asymptotically optimal heuristic that runs in linear time the heuristics can be designed to be asymptotically optimal while still running in polynomial time we also show that in several important cases the algorithm has both po

on fermat s quotient base two this paper extends the search for solutions of the congruence 2p1 0 modp2 to the limit p 6 109 no solution except the well known p 1093 and p 3511 was found in 1969 brillhart tonascia and weinberger 1 reported on the search for solutions of the congruence a 1 modp2 a 2 1 199 for a 2 a special effort was made to consider all p 3 109 only the two known solutions p 1093 and p 3511 were found having the occasional opportunity to use the illiac iv one of the projects decided upon was the recalculation and extension of the above result for a 2 by a somewhat different method the calculation was pushed to twice the above limit that is top 6 109 without finding any further solutions the parallel construction of the illiac iv makes it possible to look for 64 different values of p at the same time the speed of illiac iv is such that a range of 100000 numbers can be searched in one second thus the range for p 6 109 was broken up into 60 runs of 1000 seconds each the program was run as one of a few backlog problems over the past two years let n belong to one of the 64 residue classes that are prime to 240 and let 2m am nbm mod n2 where 0 am n 0 bm n for numbers n as large as 109 the number n2 is a doubly precise integer nevertheless the calculation of am and bm can be accomplished by single precision arithmetic in only o log m operations in fact one uses one or the other of the following two recurrences if m 2h 1 then am 2a2h and bm 2b2h mod n if m 2h and if a 2 rm nqm 0 rm n and if 2ahbh dm mod n then am rm and bm qm dm received march 30 1980 1980 mathematics subject classification primary 10 04 1981 american mathematical society 0025 5718 81 0000 0027 0 1 50 289 this content downloaded from 157 55 39 58 on tue 11 oct 2016 04 26 58 utc all use subject to http about jstor org terms

heuristics based on spacefilling curves for combinatorial problems in euclidean space we describe a family of heuristics to solve combinatorial problems such as routing and partitioning these heuristics exploit geometry but ignore specific distance measures consequently they are simple and fast but nonetheless fairly accurate and so seem well suited to operational problems where time or computing resources are limited we survey promising new application areas and show how procedures may be customized to reflect the structure of particular applications

approximation algorithms for optimal decision trees and adaptive tsp problems we consider the problem of constructing optimal decision trees given a collection of tests that can disambiguate between a set of m possible diseases each test having a cost and the a priori likelihood of any particular disease what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease this problem has been studied in several works with o log m approximations known in the special cases when either costs or probabilities are uniform in this paper we settle the approximability of the general problem by giving a tight o log m approximation algorithm we also consider a substantial generalization the adaptive traveling salesman problem given an underlying metric space a random subset s of vertices is drawn from a known distribution but s is initially unknown we get information about whether any vertex is in s only when it is visited what is a good adaptive strategy to visit all vertices in the random subset s while minimizing the expected dista

risky traveling salesman problem abstract in this paper we introduce a methodology for optimizing the expected cost of routing a single vehicle which has a probability of breaking down or failing to complete some of its tasks more specifically a calculus is devised for finding the optimal order in which each site should be visited

exact algorithms for the vehicle routing problem based on spanning tree and shortest path relaxations we consider the problem of routing vehicles stationed at a central facility depot to supply customers with known demands in such a way as to minimize the total distance travelled the problem is referred to as the vehicle routing problem vrp and is a generalization of the multiple travelling salesman problem that has many practical applications we present tree search algorithms for the exact solution of the vrp incorporating lower bounds computed from i shortest spanning k degree centre tree k dct and ii q routes the final algorithms also include problem reduction and dominance tests computational results are presented for a number of problems derived from the literature the results show that the bounds derived from the q routes are superior to those from k dct and that vrps of up to about 25 customers can be solved exactly

primality testing and jacobi sums we present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by adleman and rumely the new algorithm performs well in practice it is the first pnmality test in existence that can routinely handle numbers of hundreds of decimal digits

approximation algorithms for the maximum hamiltonian path problem with specified endpoint s abstract this paper deals with the problem of constructing hamiltonian paths of optimal weight called hpps t if the two endpoints are specified hpps if only one endpoint is specified we show that hpps t is 1 2 differential approximable and hpps is 2 3 differential approximable moreover we observe that these problems cannot be differential approximable better than 741 742 based upon these results we obtain new bounds for standard ratio a 1 2 standard approximation for m ax hpps t and a 2 3 for m ax hpps which can be improved to 2 3 for m ax hpps t a 2a all the edge weights are within an interval a 2a to 5 6 for m ax hpps a 2a and to 2 3 for m in hpps t a 2a to 3 4 for m in hpps a 2a

exact algorithms for the chance constrained vehicle routing problem we study the chance constrained vehicle routing problem ccvrp a version of the vehicle routing problem vrp with stochastic demands where a limit is imposed on the probability that each vehicle s capacity is exceeded a distinguishing feature of our proposed methodologies is that they allow correlation between random demands whereas nearly all existing exact methods for the vrp with stochastic demands require independent demands we first study an edge based formulation for the ccvrp in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers we then investigate the use of a branch and cut and price bcp algorithm while bcp algorithms have been considered the state of the art in solving the deterministic vrp few attempts have been made to extend this framework to the vrp with stochastic demands in contrast to the deterministic vrp we find that the pricing problem for the ccvrp problem is strongly mathcal np hard even when the routes being priced are allowed to have cycles we therefore propose a further relaxation of the routes that enables pricing via dynamic programming we also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust ccvrp problem numerical results indicate that the proposed methods can solve instances of ccvrp having up to 55 vertices

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

heuristics for the stochastic eulerian tour problem the stochastic eulerian tour problem setp seeks the eulerian tour of minimum expected length on an undirected eulerian graph when demand on the arcs that have to be serviced is probabilistic the setp is np hard and in this paper we develop three constructive heuristics for this problem the first two are greedy tour construction heuristics while the third is a sub tour concatenation heuristic our experimental results show that for grid networks the sub tour concatenation heuristic performs well when the probability of service of each edge is greater than 0 1 for euclidean networks as the number of edges increases the second heuristic performs the best among the three also the expected length of our overall best solution is lower than the expected length of a random tour by up to 10 on average for grid networks and up to 2 for euclidean networks

on the one commodity pickup and delivery traveling salesman problem with stochastic demands this paper studies how to set the vehicle capacity for traveling salesman problems where some of the customer demands are stochastic the analyses are done for the one commodity pickup and delivery tsp as this problem also includes the setting of the initial load the paper first considers feasibility issues this includes finding the smallest vehicle capacity and some initial load such that a given tour is feasible for all scenarios different variants are considered as a function of the time when information becomes available the paper then analyzes the case where some penalties are paid for routing a tour unable to handle customer demands various types of penalties are considered the paper studies properties of the minimal expected penalty of a given tour which are then used to provide approaches to find near optimal tours computational results are presented

probabilistic analysis of the capacitated vehicle routing problem with unsplit demands in the capacitated vehicle routing problem with unsplit demands the demand of a customer may not be divided over more than one vehicle the objective is to find tours for the vehicles such that the amount delivered by a vehicle does not exceed its capacity each customer receives its demand and the total distance traveled is as small as possible we find the asymptotic optimal solution value of the capacitated vehicle routing problem with unsplit demands for any distribution of the demands when customers are independently and identically distributed in a given region we also develop polynomial time heuristics for the problem and show that under certain assumptions on the distribution of customer locations and demands the heuristics produce solutions that are asymptotically optimal in addition we give a tight bound on the rate of convergence for the case when customers are uniformly distributed in a rectangular region

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

capacitated vehicle routing with nonuniform speeds the capacitated vehicle routing problem cvrp involves distributing identical items from a depot to a set of demand locations using a single capacitated vehicle we introduce the heterogeneous capacitated vehicle routing problem a generalization of cvrp to the setting of multiple vehicles having nonuniform speeds and present for it a constant factor approximation algorithm our main contribution is an approximation algorithm for the heterogeneous traveling salesman problem which is the special case of heterogeneous cvrp with uncapacitated vehicles given a metric denoting distances between vertices a depot r containing k vehicles having respective speeds i i 1k the objective in heterogeneous tsp is to find a tour for each vehicle starting and ending at r so that every vertex is covered in some tour and the maximum completion time is minimized the completion time of a vehicle is the distance traveled divided by its speed our algorithm relies on a new approximate minimum spanning tree constructi

propagator for narrow potential barriers the schrodinger equation in integral form is applied to the one dimensional scattering problem in the case of a narrow potential barrier since the kernel can be considered in a first approximation separable an explicit expression for the propagator is found by means of the complementary error function the problem of a particle confined in a half space and interacting with a narrow potential barrier is also considered and solved in an approximate way

generalized online routing new competitive ratios resource augmentation and asymptotic analyses we consider online routing optimization problems where the objective is to minimize the time needed to visit a set of locations under various constraints the problems are online because the set of locations are revealed incrementally over time we consider two main problems 1 the online traveling salesman problem tsp with precedence and capacity constraints and 2 the online tsp with m salesmen for both problems we propose online algorithms each with a competitive ratio of 2 for the m salesmen problem we show that our result is best possible we also consider polynomial time online algorithms we then consider resource augmentation where we give the online servers additional resources to offset the powerful offline adversary advantage in this way we address a main criticism of competitive analysis we consider the cases where the online algorithm has access to faster servers servers with larger capacities additional servers and or advanced information we derive improved competitive ratios we also give lower bounds on the competitive ratios under resource augmentation which in many cases are tight and lead to best possible results finally we study online algorithms from an asymptotic point of view we show that under general stochastic structures for the problem data unknown and unused by the online player the online algorithms are almost surely asymptotically optimal furthermore we provide computational results that show that the convergence can be very fast

approximating k forest with resource augmentation a primal dual approach in this paper we study the k forest problem in the model of resource augmentation in the k forest problem given an edge weighted graph g v e a parameter k and a set of m demand pairs subseteq v times v the objective is to construct a minimum cost subgraph that connects at least k demands the problem is hard to approximate the best known approximation ratio is o min sqrt n sqrt k furthermore k forest is as hard to approximate as the notoriously hard densest k subgraph problem

a probabilistic analysis of tour partitioning heuristics for the capacitated vehicle routing problem with unsplit demands in the capacitated vehicle routing problem with unsplit demands a customer s demand may not be divided over more than one vehicle we analyze the asymptotic performance of a class of heuristics called route first cluster second and in particular the empirically well studied sweep algorithm and optimal partitioning heuristic for any distribution of the demands and when customers are independent and identically distributed in a given region we show a strong relationship between this class of heuristics and the familiar next fit bin packing heuristic

a dynamic traveling salesman problem with stochastic arc costs we propose a dynamic traveling salesman problem tsp with stochastic arc costs motivated by applications such as dynamic vehicle routing in which the cost of a decision is known only probabilistically beforehand but is revealed dynamically before the decision is executed we formulate this as a dynamic program dp and compare it to static counterparts to demonstrate the advantage of the dynamic paradigm over an a priori approach we then apply approximate linear programming alp to overcome the dp s curse of dimensionality obtain a semi infinite linear programming lower bound and discuss its tractability we also analyze a rollout version of the price directed policy implied by our alp and derive worst case guarantees for its performance our computational study demonstrates the quality of a heuristically modified rollout policy using a computationally effective a posteriori bound

propagator for finite range potentials the case of reflection following a previous study on the transmission propagator for a finite range potential the problem of reflection is considered it is found that the laplace transform of the reflection propagator can be expressed in terms of the usual fredholm determinant and of a new similar determinant containing the peculiar characteristics of reflection as an example an array of delta potentials is considered moreover a possible application to the calculation of quantum traversal time is shown

a priori optimization consider a complete graph g v e in which each node is present with probability p we are interested in solving combinatorial optimization problems on subsets of nodes which are present with a certain probability we introduce the idea of a priori optimization as a strategy competitive to the strategy of reoptimization under which the combinatorial optimization problem is solved optimally for every instance we consider four problems the traveling salesman problem tsp the minimum spanning tree vehicle routing and traveling salesman facility location we discuss the applicability of a priori optimization strategies in several areas and show that if the nodes are randomly distributed in the plane the a priori and reoptimization strategies are very close in terms of performance we characterize the complexity of a priori optimization and address the question of approximating the optimal a priori solutions with polynomial time heuristics with provable worst case guarantees finally we use the tsp as an example to find practical solutions based on ideas of local optimality

structured partitioning problems in many important combinatorial optimization problems such as bin packing allocating customer classes to queueing facilities vehicle routing multi item inventory replenishment and combined routing inventory control an optimal partition into groups needs to be determined for a finite collection of objects each is characterized by a single attribute the cost is often separable in the groups and the group cost often depends on the cardinality and some aggregate measure of the attributes such as the sum or the maximum element an upper bound capacity may be specified for the cardinality of each group and the number of groups in the partition may either be fixed or variable the objects are indexed in nondecreasing order of their attribute values and within a given partition the groups are indexed in nondecreasing order of their cardinalities we identify easily verifiable analytical properties of the group cost function under which it is shown that an optimal partition exists of one of three increas

last mile shared delivery a discrete sequential packing approach we propose a model for optimizing the last mile delivery of n packages from a distribution center to their final recipients using a strategy that combines the use of ride sharing platforms e g

routing optimization under uncertainty we consider a class of routing optimization problems under uncertainty in which all decisions are made before the uncertainty is realized the objective is to obtain optimal routing solutions that would as much as possible adhere to a set of specified requirements after the uncertainty is realized these problems include finding an optimal routing solution to meet the soft time window requirements at a subset of nodes when the travel time is uncertain and sending multiple capacitated vehicles to different nodes to meet the customers uncertain demands we introduce a precise mathematical framework for defining and solving such routing problems in particular we propose a new decision criterion called the requirements violation rv index which quantifies the risk associated with the violation of requirements taking into account both the frequency of violations and their magnitudes whenever they occur the criterion can handle instances when probability distributions are known and ambiguity when dis

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

extension of the 2 p opt and 1 shift algorithms to the heterogeneous probabilistic traveling salesman problem abstract the probabilistic traveling salesman problem is a well known problem that is quite challenging to solve it involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability there are several proposed algorithms for the homogeneous version of the problem where all customers have identical probability of being realized from the literature the most successful approaches involve local search procedures with the most famous being the 2 p opt and 1 shift procedures proposed by bertsimas d j bertsimas l howell further results on the probabilistic traveling salesman problem european journal of operational research 65 1 1993 68 95 recently however evidence has emerged that indicates the equations offered for these procedures are not correct and even when corrected the translation to the heterogeneous version of the problem is not simple in this paper we extend the analysis and correction to the heterogeneous case we derive new expressions for computing the cost of 2 p opt and 1 shift local search moves and we show that the neighborhood of a solution may be explored in o n2 time the same as for the homogeneous case instead of o n3 as first reported in the literature

probabilistic combinatorial optimization problems on graphs a new domain in operational research abstract we present a new research domain in operational research probabilistic combinatorial optimization problems we survey and comment a number of results and we discuss several problems dealing with this domain as well as future research issues

different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is a topic of theoretical and practical importance in the study of stochastic network problems it provides researchers with a modeling framework for exploring the stochastic effects in routing problems this paper proposed three initial solution generators nn1 nn2 ran under a genetic algorithm ga framework for solving the ptsp a set of numerical experiments based on heterogeneous and homogeneous ptsp instances were conducted to test the effectiveness and efficiency of the proposed algorithms the results from the heterogeneous ptsp show that the average e t values obtained by the three generators under a ga framework are similar to those obtained by the previous best but with an average computation time saving of 50 2 as for the homogeneous ptsp instances nn1 is a relatively better generator among the three examined while ran consistently performs worse than the other two generators in terms of average e t values additionally as compared to previously reported studies no one single algorithm consistently outperformed the others across all homogeneous ptsp instances in terms of the best e t values the fact that no one initial solution generator consistently performs best in terms of the e t value obtained across all instances in heterogeneous cases and that the performance of each examined algorithm is dependent on the number of nodes n and probability p for homogeneous cases suggest the possibility of context dependent phenomenon finally to obtain valid results researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting ptsp experiments

incorporating inventory and routing costs in strategic location models we consider a supply chain design problem where the decision maker needs to decide the number and locations of the distribution centers dcs customers face random demand and each dc maintains a certain amount of safety stock in order to achieve a certain service level for the customers it serves the objective is to minimize the total cost that includes location costs and inventory costs at the dcs and distribution costs in the supply chain we show that this problem can be formulated as a nonlinear integer programming model for which we propose a lagrangian relaxation based solution algorithm by exploring the structure of the problem we find a low order polynomial algorithm for the nonlinear integer programming problem that must be solved in solving the lagrangian relaxation sub problems we present computational results for several instances of the problem with sizes ranging from 40 to 320 customers our results show the benefits of having an integrated supply chain design framework that includes location inventory and routing decisions in the same optimization model

a priori solution of a travelling salesman problem in which a random subset of the customers are visited consider a problem of routing through a set of n points on any given instance of the problem only a subset consisting of k out of n points 0 k n has to be visited with the number k random with known probability distribution we wish to find a priori a tour through all n points on any given instance the k points present will then be visited in the same order as they appear in the a priori tour the problem of finding such a tour of minimum length in the expected value sense is defined as a probabilistic traveling salesman problem ptsp what distinguishes one ptsp from another is the probability distribution or more generally the probability law that specifies the number k and the identity of the points that need to be visited on any given instance of the problem after motivating the problem by applications we first derive closed form expressions for computing efficiently the expected length of any given tour under very general probabilistic assumptions we then provide in a unified way

diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem this paper focuses on introducing a concept of diversified local search strategy under the scatter search framework for the probabilistic traveling salesman problem ptsp different combinations of three commonly used local search methods in the ptsp i e 1 shift 2 opt and 3 opt were used to investigate its effects a set of numerical experiments were conducted to test the validity of the proposed strategy based on randomly generated test instances the numerical results and the permutation test showed that the diversified local search strategy especially by combining 1 shift and 2 opt algorithms can most effectively solve the homogeneous and heterogeneous ptsp in most of the tested instances in comparison with the single local search strategy under scatter search framework

primality testing using elliptic curves we present a primality proving algorithm a probablistic primality test that produces short certificates of primality on prime inputs we prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes as a corollary we obtain an algorithm for generating large certified primes with distribution statistically close to uniform under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size the test is shown to run in expected polynomial time for all primes yielding a las vegas primality test our test is based on a new methodology for applying group theory to the problem of prime certification and the application of this methodology using groups generated by elliptic curves over finite fields we note that our methodology and methods have been subsequently used and improved upon most notably in the primality proving algorithm of adleman and huang using hyperelliptic curves and in practical primality provers using elliptic curves

tables of fibonacci and lucas factorizations we list the known prime factors of the fibonacci numbers fn for n 999 and lucas numbers ln for n 500 we discuss the various methods used to obtain these factorizations and primality tests and give some history of the subject

analysis of probabilistic combinatorial optimization problems in euclidean spaces probabilistic combinatorial optimization problems are generalized versions of deterministic combinatorial optimization problems with explicit inclusion of probabilistic elements in the problem definitions based on the probabilistic traveling salesman problem ptsp and on the probabilistic minimum spanning tree problem pmstp the objective of this paper is to give a rigorous treatment of the probabilistic analysis of these problems in the plane more specifically we present general finite size bounds and limit theorems for the objective functions of the ptsp and pmstp we also discuss the practical implications of these results and indicate some open problems

sharpening primes is in p for a large family of numbers we present algorithms that are deterministic primality tests for a large family of integers namely integers n 1 mod 4 for which an integer a is given such that the jacobi symbol a n 1 and integers n 1 mod 4 for which an integer a is given such that a n 1 a n 1 the algorithms we present run in 2 min k 2 log log n o log n 6 time where k v 2 n 1 is the exact power of 2 dividing n 1 when n 1 mod 4 and k v 2 n 1 if n 1 mod 4 the complexity of our algorithms improves up to o log n 4 when k 2 log log n we also give tests for a more general family of numbers and study their complexity

divisors in residue classes in this paper the following result is proved let r s and n be integers satisfying t i l 3 gcd r s 1 then there exist at most 11 positive divisors of n that arc congruent to r modulo s moreover there exists an efficient algorithm for determining all these divisors the bound 11 is obtained by means of a combinatorial model related to coding theory it is not known whether 11 is best possible in any case it cannot be replaced by 5 nor is it known whether similar results are true for significantly smaller values of log s log n the algorithm trcated in the paper has applications in computational number theory in this paper we prove the following theorem theorem let r s and n be integers satisfying o n n3 gcd r s 1 then there exist at most 11 positive divisors of n that are congruent to r niodulo s and there is a polynomial algorithm for determining all these divisors the algorithm referred to in the theorem is described in section 1 it is polynomial in the sense that the number of bit operations required by the algorithm is bounded by a polynomial function of the binary length of n more precisely we shall see that this number of bit operations is o log n 3 employing fast multiplication techniques we can improve this bound to o log n 2 e for every e 0 we mention two applications of the algorithm in several primality testing algorithms see 3 7 the number n to be tested is subjected to a collection of pseudo prime tests if n does not pass all these tests it is composite if n does pass all these tests one knows that each divisor of n lies in one of a small and explicitly known set of residue classes modulo an auxiliary number s in the latter case all divisors of n can easily be found if s satisfies the condition s n 2 our algorithm shows that the same can be done if s satisfies the weaker condition s nl 3 in special cases this observation was already made in 2 theorems 5 and 17 the second application is to the related problem of factoring n choosing s to be a suitable integer exceeding n 3 and applying our algorithm to all residue classes rmod s we obtain an algorithm that factors n in time o n l 3 e for every e 0 the same bound was achieved by lehman 6 and conjecturally by pinter 9 by methods that are similar in spirit there exist better factoring methods both in theory and in practice see 7 but this application indicates at least that it may be difficult to extend the algorithm to significantly smaller values of s received april 27 1983 1980 mathematics subject classificcation primary loh20 ioa25 94b25

the mixed capacitated general routing problem under uncertainty abstract we study the general routing problem defined on a mixed graph and with stochastic demands the problem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clients whose demand is not deterministically known since each vehicle has a limited capacity the demand uncertainty occurring at some clients affects the satisfaction of the capacity constraints that hence become stochastic the contribution of this paper is twofold firstly we present a chance constrained integer programming formulation of the problem for which a deterministic equivalent is derived the introduction of uncertainty into the problem poses severe computational challenges addressed by the design of a branch and cut algorithm for the exact solution of limited size instances and of a heuristic solution approach exploring promising parts of the search space the effectiveness of the solution approaches is shown on a probabilistically constrained version of the benchmark instances proposed in the literature for the mixed capacitated general routing problem

a priori optimization of the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is defined on a graph g v e where v is the vertex set and e is the edge set each vertex vi has a probability pi of being present with each edge vi vj is associated a distance or cost cij in a first stage an a priori hamiltonian tour on g is designed the list of present vertices is then revealed in a second stage the a priori tour is followed by skipping the absent vertices the ptsp consists of determining a first stage solution that minimizes the expected cost of the second stage tour the problem is formulated as an integer linear stochastic program and solved by means of a branch and cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function problems involving up to 50 vertices are solved to optimality

a one parameter quadratic base version of the baillie psw probable prime test the well known baillie psw probable prime test is a combination of a rabin miller test and a true i e with d n 1 lucas test arnault mentioned in a recent paper that no precise result is known about its probability of error grantham recently provided a probable prime test rqft with probability of error less than 1 7710 and pointed out that the lack of counter examples to the baillie psw test indicates that the true probability of error may be much lower in this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter tu t mod t2 ut 1 and define the base counting functions b n u 0 u n n is a psp tu and sb n u 0 u n n is an spsp tu then we give explicit formulas to compute b n and sb n and prove that for odd composites n b n n 2 and sb n n 8 and point out that these are best possible finally based on one parameter quadratic base pseudoprimes we provide a probable prime test called the one parameter quadratic base test opqbt which passed by all primes 5 and passed by an odd composite n p1r1 p2r2 psrs p1 p2 ps odd primes with probability of error n we give explicit formulas to compute n and prove that n 1 n4 3 for n nonsquare free with s 1 1 n2 3 for n square free with s 2 1 n2 7 for n square free with s 3 1 8s 4 166 p1 1 for n square free with s even 4 1 16s 5 119726 for n square free with s odd 5 1 4s i 1s 1 pi2 ri 1 otherwise i e for n nonsquare free with s 2 the running time of the opqbt is asymptotically 4 times that of a rabin miller test for worst cases but twice that of a rabin miller test for most composites we point out that the opqbt has clear finite group field structure and nice symmetry and is indeed a more general and strict version of the baillie psw test comparisons with gantham s rqft are given

primality test for numbers m with a large power of 5 dividing m 4 1 the quintic reciprocity law is used to produce an algorithm that runs in polynomial time and that determines the primality of numbers m such that m4 1 is divisible by a power of 5 which is larger that m provided that a small prime p p 1 mod 5 is given such that m is not a fifth power modulo p the same test equations are used for all such m a sufficiency test together with its probability of succeeding in determining primality is given when the condition on m modulo p is omitted

implementing the asymptotically fast version of the elliptic curve primality proving algorithm the elliptic curve primality proving ecpp algorithm is one of the current fastest practical algorithms for proving the primality of large numbers its running time cannot be proven rigorously but heuristic arguments show that it should run in time o log n 5 to prove the primality of n an asymptotically fast version of it attributed to j o shallit runs in time o log n 4 the aim of this article is to describe this version in more details leading to actual implementations able to handle numbers with several thousands of decimal digits

bounds and heuristics for capacitated routing problems in a capacitated routing problem the objective is to minimize the total distance travelled by vehicles of limited capacity to serve a set of customers that are located in the euclidean plane we develop asymptotically optimal bounds and heuristics for this problem under the assumption that the capacity of a vehicle is expressed in terms of an upper bound on the number of customers that it can serve the analysis culminates in an algorithm that for a given capacity and given will find a solution with relative error at most in time polynomial in the number of customers

a hybrid differential evolution algorithm to vehicle routing problem with fuzzy demands in this paper the vehicle routing problem with fuzzy demands vrpfd is considered and a fuzzy chance constrained program model is designed based on fuzzy credibility theory then stochastic simulation and differential evolution algorithm are integrated to design a hybrid intelligent algorithm to solve the fuzzy chance constrained program model moreover the influence of the dispatcher preference index on the final objective of the problem is discussed using stochastic simulation and the best value of the dispatcher preference index is obtained

uncertain multiobjective traveling salesman problem abstract traveling salesman problem is a fundamental combinatorial optimization model studied in the operations research community for nearly half a century yet there is surprisingly little literature that addresses uncertainty and multiple objectives in it a novel tsp variation called uncertain multiobjective tsp umtsp with uncertain variables on the arc is proposed in this paper on the basis of uncertainty theory and a new solution approach named uncertain approach is applied to obtain pareto efficient route in umtsp considering the uncertain and combinatorial nature of umtsp a new abc algorithm inserted with reverse operator crossover operator and mutation operator is designed to this problem which outperforms other algorithms through the performance comparison on three benchmark tsps finally a new benchmark umtsp case study is presented to illustrate the construction and solution of umtsp which shows that the optimal route in deterministic tsp can be a poor route in umtsp

technical note approximation algorithms for vrp with stochastic demands we consider the vehicle routing problem with stochastic demands vrpsd we give randomized approximation algorithms achieving approximation guarantees of 1 for split delivery vrpsd and 2 for unsplit delivery vrpsd here is the best approximation guarantee for the traveling salesman problem these bounds match the best known for even the respective deterministic problems altinkemer k b gavish 1987 heuristics for unequal weight delivery problems with a fixed error guarantee oper res lett 6 4 149 158 altinkemer k b gavish 1990 heuristics for delivery problems with constant error guarantees transportation res 24 4 294 297 we also show that the cyclic heuristic for split delivery vrpsd achieves a constant approximation ratio as conjectured in bertsimas bertsimas d j 1992 a vehicle routing problem with stochastic demand oper res 40 3 574 585

solving a class of stochastic mixed integer programs with branch and price we begin this paper by identifying a class of stochastic mixed integer programs that have column oriented formulations suitable for solution by a branch and price algorithm b p we then survey a number of examples and use a stochastic facility location problem sflp for a detailed demonstration of the relevant modeling and solution techniques computational results with a scenario representation of uncertain costs demands and capacities show that b p can be orders of magnitude faster than solving the standard formulation by branch and bound we also demonstrate how b p can solve sflp exactly as exactly as a deterministic mixed integer program when demands and other parameters can be represented as certain types of independent random variables e g independent normal random variables with integer means and variances

implementation of a new primality test an implementation of the cohen lenstra version see ibid vol 42 p 297 330 1984 of the adleman pomerance rumely primality test see l m adleman c pomerance and r s rumely ann of math vol 117 p 173 206 1983 is presented primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

a tabu search heuristic for the vehicle routing problem with stochastic demands and customers this paper considers a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands a tabu search heuristic is developed for this problem comparisons with known optimal solutions on problems whose sizes vary from 6 to 46 customers indicate that the heuristic produces an optimal solution in 89 45 of cases with an average deviation of 0 38 from optimality

heuristic and lower bound for a stochastic location routing problem in this article a stochastic location routing problem is defined and cast as a two stage model in a first stage the set of plants and a family of routes are determined in a second stage a recourse action is applied to adapt these routes to the actual set of customers to visit once they are known a two phase heuristic is developed an initial feasible solution is built by solving a sequence of subproblems and an improvement phase is then applied a lower bound based on bounding separately different parts of the cost of any feasible solution is also developed computational results are reported

characterization of the probabilistic traveling salesman problem we show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem the probabilistic traveling salesman must decide on an a priori order in which to visit n cities randomly distributed over a unit square before learning that some cities can be omitted we find the optimized average length of the pruned tour follows e l over bar pruned rootnp 0 872 0 105p f np where p is the probability of a city needing to be visited and f np 1 as np infinity the average length of the a priori tour before omitting any cities is found to follow e l a priori rootn pbeta p where beta p 1 1 25 0 82 ln p is measured for 0 05less than or equal topless than or equal to0 6 scaling arguments and indirect measurements suggest that beta p tends towards a constant for p 0 03 our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths exploiting the sampling error to provide the analog of thermal fluctuations in simulated thermal annealing the method has general application to the optimization of functions whose cost to evaluate rises with the precision required

a vehicle routing problem with stochastic demand we consider a natural probabilistic variation of the classical vehicle routing problem vrp in which demands are stochastic given only a probabilistic description of the demand we need to design routes for the vrp motivated by applications in strategic planning and distribution systems rather than resolving the problem when the demand becomes known we propose to construct an a priori sequence among all customers of minimal expected total length we analyze the problem using a variety of theoretical approaches we find closed form expressions and algorithms to compute the expected length of an a priori sequence under general probabilistic assumptions based on these expressions we find upper and lower bounds for the probabilistic vrp and the vrp re optimization strategy in which we find the optimal route at every instance we propose heuristics and analyze their worst case performance as well as their average behavior using techniques from probabilistic analysis our results suggest that our approac

finding isomorphisms between finite fields we show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time every finite field has cardinality p for some prime number p and some positive integer conversely if p is a prime number and a positive integer then there exists a field of cardinality p and any two fields of cardinality p are isomorphic these results are due to e h moore 1893 10 in the present paper we are interested in an algorithmic version of his theorem in particular of the uniqueness part we say that a finite field is explicitly given if for some basis of the field over its prime field we know the product of any two basis elements expressed in the same basis let more precisely p be a prime number and a positive integer then by explicit data for a finite field of cardinality p we mean a system of elements ajjk k x of the prime field fp z pz suchthat f becomes a field with the ordinary addition and multiplication by elements of fp and the multiplication determined by n

finding strong pseudoprimes to several bases define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic primality testing algorithm which is not only easier to implement but also faster than either the jacobi sum test or the elliptic curve test thanks to pornerance et al and jaeschke m are known for 1 m 8 upper bounds for 9 10 and 1 were given by jaeschke in this paper we tabulate all strong pseudoprimes spsp s n 10 24 to the first ten prime bases 2 3 29 which have the form n pq with p q odd primes and q 1 k p 1 k 2 3 4 there are in total 44 such numbers six of which are also spsp 31 and three numbers are spsp s to both bases 31 and 37 as a result the upper bounds for 10 and 11 are lowered from 28 and 29 decimal digit numbers to 22 decimal digit numbers and a 24 decimal digit upper bound for 12 is obtained the main tools used in our methods are the biquadratic residue characters and cubic residue characters we propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases comparisons of effectiveness with both jaeschke s and arnault s methods are given