a polynomial time algorithm to solve the single item capacitated lot sizing problem with minimum order quantities and concave costs abstract this paper deals with the single item capacitated lot sizing problem with concave production and storage costs and minimum order quantity clsp moq in this problem a demand must be satisfied at each period t over a planning horizon of t periods this demand can be satisfied from the stock or by a production at the same period when a production is made at period t the produced quantity must be greater to than a minimum order quantity l and lesser than the production capacity u to solve this problem optimally a polynomial time algorithm in o t 5 is proposed and it is computationally tested on various instances

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

economic lot sizing an o n log n algorithm that runs in linear time in the wagner whitin case we consider the n period economic lot sizing problem where the cost coefficients are not restricted in sign in their seminal paper h m wagner and t m whitin proposed an o n2 algorithm for the special case of this problem where the marginal production costs are equal in all periods and the unit holding costs are nonnegative it is well known that their approach can also be used to solve the general problem without affecting the complexity of the algorithm in this paper we present an algorithm to solve the economic lot sizing problem in o n log n time and we show how the wagner whitin case can even be solved in linear time our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure furthermore we show how wagner and whitin s and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem

the γ filtration and the rost invariant let x be the variety of borel subgroups of a simple and strongly inner linear algebraic group g over a field k we prove that the torsion part of the second quotient of grothendieck s filtration on x is a cyclic group of order the dynkin index of g as a byproduct of the proof we obtain an explicit cycle that generates this cyclic group we provide an upper bound for the torsion of the chow group of codimension 3 cycles on x we relate the cycle with the rost invariant and the torsion of the respective generalized rost motives we use to obtain a uniform lower bound for the essential dimension of almost all simple linear algebraic groups grothendieck s celebrated filtration of the ring k0 x gives a way to estimate the chow group ch x of algebraic cycles on x modulo rational equivalence when x is a smooth projective variety over a field k namely by the riemannroch theorem without denominators 8 15 3 the i th chern class provides a well defined group homomorphism ci x ch x i 0 from the i th quotient of the filtration to the chow group of codimension i cycles onx observe that for i 0 it is the identity map and for i 1 it is an isomorphism identifying ch x with the picard group of x in the present paper we study these homomorphisms in the cases i 2 3 and x is a generically split projective homogeneous variety under a semisimple linear algebraic group g our core determines and bounds respectively the torsion subgroup of b and b for the variety of borel subgroups b of strongly inner g theorem 3 1 for instance we show that the torsion subgroup tors b is cyclic of order the dynkin index ofg and exhibit a generator for it definition 3 3 this fact together with the riemann roch theorem imply see 4 that the surjection tors b c2 torsch b can be viewed as a substitute of the key map q v h k q z 2 in the definition of the rost invariant 10 pp 126 127 indeed a theorem of peyremerkurjev 27 shows that torsch b can be identified with the kernel of the restriction h k q z 2 h k b q z 2 furthermore the order of c2 in torsch b equals to the order of the rost invariant of g see prop 3 2 our result gives bounds for the torsion in ch for generically split x see 5 and provides explicit generators of torsion subgroups of ch of certain generalized rost voevodsky motives note that typically one does not even know a priori if the torsion subgroup of ch x i 3 is finitely generated however determining the torsion subgroup determines ch x as an abelian group since the dimension of its free part ch x q can be easily computed

an ot2 algorithm for the ni g ni nd capacitated lot size problem in this paper we study a class of the capacitated dynamic lot size problem where over time the setup costs are nonincreasing the unit holding costs have arbitrary pattern the unit production costs are nonincreasing and the capacities are nondecreasing we investigate the properties of the optimal solution for the problem and develop the concept of candidate subplan it is proven that only the candidate subplans need to be examined in searching for an optimal solution a dynamic programming algorithm incorporating the concept of candidate subplan is then devised which has run time complexity of ot2

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

an o t3 algorithm for the capacitated lot sizing problem with minimum order quantities abstract this paper explores a single item capacitated lot sizing problem with minimum order quantity which plays the role of minor set up cost we work out the necessary and sufficient solvability conditions and apply the general dynamic programming technique to develop an o t 3 exact algorithm that is based on the concept of minimal sub problems an investigation of the properties of the optimal solution structure allows us to construct explicit solutions to the obtained sub problems and prove their optimality in this way we reduce the complexity of the algorithm considerably and confirm its efficiency in an extensive computational study

computational complexity of the capacitated lot size problem in this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings for the single item case new properties are introduced classes of problems solvable by polynomial time algorithms are identified and efficient solution procedures are given we show that special classes are np hard and that the problem with two items and independent setups is np hard under conditions similar to those where the single item problem is easy topics for further research are discussed in the last section

corrigendum to a polynomial time algorithm to solve the single item capacitated lot sizing problem with minimum order quantities and concave costs eur j oper res 222 2012 10 16

multicore curve based cryptoprocessor with reconfigurable modular arithmetic logic units over gf 2 n this paper presents a reconfigurable curve based cryptoprocessor that accelerates scalar multiplication of elliptic curve cryptography ecc and hyperelliptic curve cryptography hecc of genus 2 over gf 2n by allocating a copies of processing cores that embed reconfigurable modular arithmetic logic units malus over gf 2n the scalar multiplication of ecc hecc can be accelerated by exploiting instruction level parallelism ilp the supported field size can be arbitrary up to a n 1 1 the superscaling feature is facilitated by defining a single instruction that can be used for all field operations and point divisor operations in addition the cryptoprocessor is fully programmable and it can handle various curve parameters and arbitrary irreducible polynomials the cost performance and security trade offs are thoroughly discussed for different hardware configurations and software programs the synthesis results with a 0 13 mum cmos technology show that the proposed reconfigurable cryptoprocessor runs at 292 mhz whereas the field sizes can be supported up to 587 bits the compact and fastest configuration of our design is also synthesized with a fixed field size and irreducible polynomial the results show that the scalar multiplication of ecc over gf 2163 and hecc over gf 283 can be performed in 29 and 63 mus respectively

lower bounds for essential dimensions via orthogonal representations let us first recall what is the essential dimension of a functor cf br 97 and r 00 let k be a field and let f be a functor from the category of field extensions of k into the category of sets let f k be an extension and let be an element of f f if e is a field with k e f we say that comes from e if it belongs to the image of f e f f the essential dimension ed of is the minimum of the transcendence degrees e k for all e with k e f such that comes from e one has ed tr deg f if there is equality we say that is incompressible the essential dimension ed f of f is ed f max ed the maximum being taken over all pairs f with k f and f f

improved computation of square roots in specific finite fields in this paper we study exponentiation in the specific finite fields f with very special exponents such as those that occur in algorithms for computing square roots here q is a prime power q pk where k 1 and k is odd our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent to the best of our knowledge it is the first major improvement to the tonelli shanks algorithm for example the number of multiplications can be reduced to at least 60 percent on the average when p 1 mod 16 several numerical examples are given that show the speedup of the proposed methods

identity based encryption from the weil pairing we propose a fully functional identity based encryption ibe scheme the scheme has chosen ciphertext security in the random oracle model assuming a variant of the computational diffie hellman problem our system is based on bilinear maps between groups the weil pairing on elliptic curves is an example of such a map we give precise definitions for secure ibe schemes and give several applications for such systems

the economic lot sizing problem with an emission capacity constraint abstract we consider a generalisation of the lot sizing problem that includes an emission capacity constraint besides the usual financial costs there are emissions associated with production keeping inventory and setting up the production process because the capacity constraint on the emissions can be seen as a constraint on an alternative objective function there is also a clear link with bi objective optimisation we show that lot sizing with an emission capacity constraint is np hard and propose several solution methods our algorithms are not only able to handle a fixed plus linear cost structure but also more general concave cost and emission functions first we present a lagrangian heuristic to provide a feasible solution and lower bound for the problem for costs and emissions such that the zero inventory property is satisfied we give a pseudo polynomial algorithm which can also be used to identify the complete set of pareto optimal solutions of the bi objective lot sizing problem furthermore we present a fully polynomial time approximation scheme fptas for such costs and emissions and extend it to deal with general costs and emissions special attention is paid to an efficient implementation with an improved rounding technique to reduce the a posteriori gap and a combination of the fptases and a heuristic lower bound extensive computational tests show that the lagrangian heuristic gives solutions that are very close to the optimum moreover the fptases have a much better performance in terms of their actual gap than the a priori imposed performance and especially if the heuristic s lower bound is used they are very fast

modular polynomials via isogeny volcanoes we present a new algorithm to compute the classical modular polynomial l in the rings z x y and z mz x y for a prime l and any positive integer m our approach uses the graph of l isogenies to eciently compute l mod p for many primes p of a suitable form and then applies the chinese remainder theorem crt under the generalized riemann hypoth esis grh we achieve an expected running time of o l 3 logl 3 log logl and compute l mod m using o l 2 logl 2 l 2 logm space we have used the new algorithm to compute l with l over 5000 and l mod m with l over 20000 we also consider several modular functions g for which g is smaller than l allowing us to handle l over 60000

counting points on elliptic curves over f2m in 1985 schoof 136 presented a polynomial time algorithm for computing e f q the number of fq rational points on an elliptic curve e defined over the field f q the algorithm has a running time of 0 log8 q bit operations and is rather cumbersome in practice buchmann and muller 20 combined schoof s algorithm with shanks baby step giant step algorithm and were able to compute orders of curves over fp where p is a 27 decimal digit prime the algorithm took 4 5 hours on a sun 1 sparc station

improved exact algorithms to economic lot sizing with piecewise linear production costs abstract in this article we study a classical single item economic lot sizing problem where production cost functions are assumed to be piecewise linear the lot sizing problem is fundamental in lot sizing research and it is applicable to a wide range of production planning models the intractability of the problem is related to the value of m the number of different breakpoints of the production cost functions when m is arbitrary the problem is known to be np hard florian lenstra rinnooy kan 1980 however if m is fixed then it can be solved in polynomial time koca yaman akturk 2014 so far the most efficient algorithm to solve the problem has a complexity of o t 2 m 3 where t is the number of periods in the planning horizon koca et al 2014 in this article we present an improved exact algorithm for solving the problem where the complexity is reduced to o t m 2 log t as such it also improves upon the computational efficiency of solving some existing lot sizing problems which are important special cases of our model in order to achieve a more efficient implementation our algorithm makes use of a specific data structure named range minimum query rmq

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

meta heuristics for dynamic lot sizing a review and comparison of solution approaches proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve because these problems are so difficult various solution techniques have been proposed to solve them in the past decade meta heuristics such as tabu search genetic algorithms and simulated annealing have become popular and efficient tools for solving hard combinatorial optimization problems we review the various meta heuristics that have been specifically developed to solve lot sizing problems discussing their main components such as representation evaluation neighborhood definition and genetic operators further we briefly review other solution approaches such as dynamic programming cutting planes dantzig wolfe decomposition lagrange relaxation and dedicated heuristics this allows us to compare these techniques understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms finally we discuss general guidelines for computational experiments and illustrate these with several examples

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

lower bounds in lot sizing models a polyhedral study variable lower bounds in mixed integer programs are constraints with the general form x ly where xis a continuous variable and y is a binary or an integer variable this type of constraints is present in some lot sizing models where x represents the amount of some article produced by a machine and y 0 1 indicates whether the machine is set up for that articley 1 or not y 0 in these models production below some level is not allowed in order to make full use of resources in this paper we study from the polyhedral viewpoint the mixed integer models whose feasible set includes an additive constraint xj d and constraints lyj xj kyj where xj are continuous variables yj are integer or binary variables and k is a large constant we derive families of strong valid inequalities and show they are sufficient to describe completely the convex hulls of the sets of feasible solutions moreover we develop polynomial algorithms to solve the separation problem associated to each of the families obtained finally we incorporate the inequalities obtained in a cutting plane algorithm and test their ability to solve some multi item lot sizing problems

the single item green lot sizing problem with fixed carbon emissions we consider in this paper a single item lot sizing problem with a periodic carbon emission constraint in each period the carbon emission constraint defines an upper limit on the average emission per product different modes are available each one is characterized by its own cost and carbon emission parameters the problem consists in selecting the modes used in each period such that no carbon emission constraint is violated and the cost of satisfying all the demands on a given time horizon is minimized this problem has been introduced in absi et al 2013 and has been shown polynomially solvable when only unit carbon emissions are considered in this paper we extend the analysis for this constraint to the realistic case of a fixed carbon emission associated with each mode in addition to its unit carbon emission we establish that this generalization renders the problem np hard several dominant properties are presented and two dynamic programming algorithms are proposed we also establish that the problem can be solved in polynomial time for a fixed number of modes when carbon emission parameters are stationary

parametric combinatorial computing and a problem of program module distribution en calcul parametrique certaines parties de la description du probleme sont donnees comme des fonctions d un ou plusieurs parametres pouvant varier sur un grand nombre de valeurs des valeurs fixes des parametres specifient un probleme particulier l objectif du calcul parametrique etant de resoudre de maniere efficace de tels problemes en fonction des valeurs des parametres on propose une methode de calcul parametrique generale efficace pour une large classe de problemes combinatoires la methode est liee a une methode developpee par megiddo pour la resolution de problemes d optimisation de ratios on traite un exemple de distribution de modules de programme d ordinateur entre deux processeurs

computing modular correspondences for abelian varieties the aim of this paper is to give a higher dimensional equivalent of the classical modular polynomials phi ell x y if j is the j invariant associated to an elliptic curve e k over a field k then the roots of phi ell j x correspond to the j invariants of the curves which are ell isogeneous to e k denote by x 0 n the modular curve which parametrizes the set of elliptic curves together with a n torsion subgroup it is possible to interpret phi ell x y as an equation cutting out the image of a certain modular correspondence x 0 ell rightarrow x 0 1 times x 0 1 in the product x 0 1 times x 0 1 let g be a positive integer and overn in n g we are interested in the moduli space that we denote by mn of abelian varieties of dimension g over a field k together with an ample symmetric line bundle pol and a symmetric theta structure of type overn if ell is a prime and let overl ell ldots ell there exists a modular correspondence mln rightarrow mn times mn we give a system of algebraic equations defining the image of this modular correspondence we describe an algorithm to solve this system of algebraic equations which is much more efficient than a general purpose gr obner basis algorithm as an application we explain how this algorithm can be used to speed up the initialisation phase of a point counting algorithm

factorization with genus 2 curves the elliptic curve method ecm is one of the best factorization methods available it is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower we use special hyperelliptic curves and kummer surfaces to reduce the complexity of the algorithm our implementation gmp hecm is faster than gmp ecm for factoring large numbers

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

solving multi item capacitated lot sizing problems using variable redefinition mixed integer programming models are typically not used to solve realistic sized production scheduling problems because they require exorbitant solution times we impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy the linear programming relaxation of the new models is very effective in generating bounds we show that these bounds are equal to those that could be generated using lagrangian relaxation or column generation the linear programming bounds increase in effectiveness as the problems become larger perhaps of greatest significance is that practitioners can obtain our results using only standard off the shelf codes such as lindo or mpsx 370 we report computational experience in several computing environments hardware and software on problems with up to 200 products and 10 time periods 2000 0 1 variables

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method