constants and label equivalence a decision procedure for reflexive regular splicing languages abstract a structural characterization of reflexive splicing languages has been recently given in p bonizzoni c de felice r zizza the structure of reflexive regular splicing languages via schutzenberger constants theoretical computer science 334 2005 71 98 and p bonizzoni g mauri regular splicing languages and subclasses theoretical computer science 340 2005 349 363 showing surprising connections between long standing notions in formal language theory the syntactic monoid and schutzenberger constant and the splicing operation in this paper we provide a procedure to decide whether a regular language is a reflexive splicing language based on the above mentioned characterization that is given in terms of a finite set of constants for the language the procedure relies on the notion of label equivalence that induces a finite refinement of the syntactic monoid of a regular language l a finite set of representatives for label equivalent classes of constant words in l is defined and it is proved that such a finite set provides the splice sites of splicing rules generating language l

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

an alternative definition of splicing in this paper we propose a new definition of the language generated by a splicing system motivated by both biochemical and mathematical considerations the main feature of the new definition is that by applying a splicing rule we not only create new strings but also allow for the removal of the strings entering the rule this behaviour seems to correspond better to biochemical reality and is in fact used as a tool in several experimental dna computations we show that using this new definition finite extended h systems can generate all recursively enumerable languages even a weaker version of these h systems defined using the new notion of delay is shown to be strictly more powerful than h systems defined in the traditional way

a modified reproducing kernel method for solving burgers equation arising from diffusive waves in fluid dynamics abstract as we known reproducing kernel method rkm has been presented for solving differential equations for initial and boundary value problems however the direct application of the rkm presented in the previous works cannot produce good numerical results for burgers equation to solve this problem this paper give a modified reproducing kernel method by piecewise technique the exact solution is given by reproducing kernel functions in a series expansion form the approximation solution is expressed by n term summation of reproducing kernel functions the three numerical experiments results show that the piecewise method is more easily implemented and effective some numerical results are also compared with the results obtained by other methods

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

on the separation of maximally violated mod k cuts separation is of fundamental importance in cutting plane based techniques for integer linear programming ilp in recent decades a considerable research effort has been devoted to the definition of effective separation procedures for families of well structured cuts in this paper we address the separation of chvatal rank 1 inequalities in the context of general ilp s of the form min c t x ax b x integer where a is an m n integer matrix and b an m dimensional integer vector in particular for any given integer k we study mod k cuts of the form t ax t b for any 0 1 k k 1 k m such that t a is integer following the line of research recently proposed for mod 2 cuts by applegate bixby chvatal and cook 1 and fleischer and tardos 19 we restrict to maximally violated cuts i e to inequalities which are violated by k 1 k by the given fractional point we show that for any given k such a separation requires o mn min m n time applications to both the symmetric and asymmetric tsp are discussed in particular for any given k we propose an o v 2 e time exact separation algorithm for mod k cuts which are maximally violated by a given fractional symmetric or asymmetric tsp solution with support graph g v e this implies that we can identify a maximally violated cut for the symmetric tsp whenever a maximally violated extended comb inequality exists finally facet defining mod k cuts for the symmetric and asymmetric tsp are studied

polynomial time separation of a superclass of simple comb inequalities the comb inequalities are a well known class of facet inducing inequalities for the traveling salesman problem defined in terms of certain vertex sets called the handle and the teeth we say that a comb inequality is simple if the following holds for each tooth either the intersection of the tooth with the handle has cardinality one or the part of the tooth outside the handle has cardinality one or both the simple comb inequalities generalize the classical 2 matching inequalities of edmonds edmonds j 1965 maximum matching and a polyhedron with 0 1 vertices j res nat bur standards69b 125 130 and also the so called chvatal comb inequalities in 1982 padberg and rao padberg m w m r rao 1982 odd minimum cut sets and b matchings math oper res 7 67 80 gave a polynomial time separation algorithm for the 2 matching inequalities i e an algorithm for testing if a given fractional solution to an lp relaxation violates a 2 matching inequality we extend this significantly by giving a polynomial time separation algorithm for a class of valid inequalities which includes all simple comb inequalities

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

numerical solution of one dimensional burgers equation using reproducing kernel function this paper presents a numerical method for one dimensional burgers equation by the hopf cole transformation and a reproducing kernel function abbreviated as rkf the numerical solution is given as explicit integral expressions with the rkf at each time step so that the computation is fully parallel the stability and error estimates are derived numerical results for some test problems are presented and compared with the exact solutions some numerical results are also compared with the results obtained by other methods the present method is easily implemented and effective

the domino inequalities facets for the symmetric traveling salesman polytope adam letchford defines in 4 the domino parity inequalities for the symmetric traveling salesman polytope stsp and gives a polynomial algorithm for the separation of such constraints when the support graph is planar generalizing a result of fleischer and tardos 2 for maximally violated comb inequalities naddef in 5 gives a set of necessary conditions for such inequalities to be facet defining for the stsp these conditions lead to the domino inequalities and it is shown in 5 that one does not lose any facet inducing inequality restricting the domino parity inequalities to domino inequalities except maybe for some very particular case we prove here that all the domino inequalities are facet inducing for the stsp settling the conjecture given in 5 as a by product we will also have a complete proof that the comb inequalities are facet inducing

using reproducing kernel for solving a class of singularly perturbed problems this paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of partial differential equations with a singularly perturbed term in a reproducing kernel space two numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

on the domino parity inequalities for the stsp one method which has been used very successfully for finding optimal and provably good solutions for large instances of the symmetric travelling salesman problem stsp is the branch and cut method this method requires knowledge of classes of useful valid inequalities for the polytope associated with the stsp as well as efficient separation routines for these classes of inequalities recently a new class of valid inequalities called the domino parity inequalites were introduced for the stsp an efficient separation routine is known for these constraints if certain conditions are satisfied by the point to be separated this separation routine has never been implemented or tested we present several performance enhancements for this separation routine and discuss our implementation of this improved algorithm we test our implementation and provide results which we believe demonstrate the practical usefulness of these constraints and the separation routine for the stsp within a branch and cut framework

binary clutter inequalities for integer programs we introduce a new class of valid inequalities for general integer linear programs called binary clutter bc inequalities they include the 0 frac 1 2 cuts of caprara and fischetti as a special case and have some interesting connections to binary matroids binary clutters and gomory corner polyhedra we show that the separation problem for bc cuts is strongly hard in general but polynomially solvable in certain special cases as a by product we also obtain new conditions under which 0 frac 1 2 cuts can be separated in polynomial time these ideas are then illustrated using the traveling salesman problem tsp as an example this leads to an interesting link between the tsp and two apparently unrelated problems the t join and max cut problems

a novel method for numerical simulation of sand motion model in beach formation based on fractional taylor jumarie series expansion and piecewise interpolation technique abstract in this paper for the first time fractional taylor jumarie series expansion is used to solve a glass of time fractional delay partial differential equation by piecewise interpolation reproducing kernel method rkm this class of equations describe sand motion model in beach formation the aim of this work is to obtain more accurate numerical solution by fractional taylor jumarie series expansion and piecewise interpolation technique three numerical experiments are provided to show the advantage of this method the results show the characteristics of sand motion the research in this paper is the theoretical basis for further sand motion study

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

on the separation of split cuts and related inequalities the split cuts of cook kannan and schrijver are general purpose valid inequalities for integer programming which include a variety of other well known cuts as special cases to detect violated split cuts one has to solve the associated separation problem the complexity of split cut separation was recently cited as an open problem by cornuejols li cl01 in this paper we settle this question by proving strong completeness of separation for split cuts as a by product we also show completeness of separation for several other classes of inequalities including the mir inequalities of nemhauser and wolsey and some new inequalities which we call balanced split cuts and binary split cuts we also strengthen completeness results of caprara fischetti cf96 for cuts and eisenbrand e99 for chvatal gomory cuts to compensate for this bleak picture we also give a positive result for the symmetric travelling salesman problem we show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

toward solving the steiner travelling salesman problem on urban road maps using the branch decomposition of graphs the steiner travelling salesman problem stsp is an important issue in intelligent transportation systems and has various practical applications such as travelling and parcel delivery in this study we consider the stsp in real world road maps i e given a road map with a set of service points or intersections selection of the representative vertices of a graph g to determine a closed route so a vehicle can visit each service point at least once with the minimum cost it has been theoretically proved that the stsp and its ancestor travelling salesman problem are np hard however it has been empirically demonstrated that they can be solved efficiently on planar graphs with small branchwidths therefore we propose a branch decomposition based extraction algorithm for the stsp in planar graphs our algorithm can solve the stsp in a planar graph g with a time complexity of 2o bw g no 1 where bw g is the branchwidth of g in the case of planar graphs with a small branchwidth our algorithm performs efficiently we evaluated our algorithm by applying it to real world urban road maps which demonstrated the suitability of our method for real world applications

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

a faster strongly polynomial minimum cost flow algorithm in this paper we present a new strongly polynomial time algorithm for the minimum cost flow problem based on a refinement of the edmonds karp scaling technique our algorithm solves the uncapacitated minimum cost flow problem as a sequence of on log n shortest path problems on networks with n nodes and m arcs and runs in on log nm n log n time using a standard transformation this approach yields an om log nm n log n algorithm for the capacitated minimum cost flow problem this algorithm improves the best previous strongly polynomial time algorithm due to z galil and e tardos by a factor of n2 m our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds say m is much less than m in this case the running time of the algorithm is om n log nm n log n

a novel numerical scheme for solving burgers equation abstract in this paper we propose a novel numerical scheme for solving burgers equation the scheme is based on a cubic spline quasi interpolant and multi node higher order expansion which make the algorithm simple and easy to implement the numerical experiments show that the proposed method produces high accurate results

regular splicing languages and subclasses recent developments in the theory of finite splicing systems have revealed surprising connections between long standing notions in the formal language theory and splicing operation more precisely the syntactic monoid and schutzenberger constant have strong interaction with the investigation of regular splicing languages this paper surveys results of structural characterization of classes of regular splicing languages based on the above two notions and discusses basic questions that motivate further investigations in this field in particular we improve the result in 6 that provides a structural characterization of reflexive symmetric splicing languages by showing that it can be extended to the class of all reflexive splicing languages this is the larger class for which a characterization is known

numerical solutions of nonlinear burgers equation with modified cubic b splines collocation method abstract in this paper a numerical method is proposed to approximate the solution of the nonlinear burgers equation the method is based on collocation of modified cubic b splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range we apply modified cubic b splines for spatial variable and derivatives which produce a system of first order ordinary differential equations we solve this system by using ssp rk43 or ssp rk54 scheme this method needs less storage space that causes less accumulation of numerical errors the numerical approximate solutions to the burgers equation have been computed without transforming the equation and without using the linearization illustrative eleven examples are included to demonstrate the validity and applicability of the technique easy and economical implementation is the strength of this method

separation algorithms for classes of stsp inequalities arising from a new stsp relaxation the problem ofseparating a class of inequalities that are valid for the symmetric traveling salesman problem stsp consists of devising an efficient method that given a fractional pointx for a relaxation of the stsp either finds an inequality in the given class of stsp inequalities that is violated byx or determines that there are no such violated inequalities we can define important classes of stsp inequalities by performing naddef s and rinaldi s 1993 node lifting operation on stsp inequalities defined on small graphs we present efficient methods forexactly separating large classes of stsp inequalities that arise from an stsp relaxation related to node lifting in particular we show how to find efficiently the most violated inequality in the class of stsp inequalities having abackbone set with a constant numberk of vertices and thus separate in polynomial time the class of all stsp inequalities whose backbone set has at mostk vertices

existence of constants in regular splicing languages in spite of wide investigations of finite splicing systems in formal language theory basic questions such as their characterization remain unsolved it has been conjectured that a necessary condition for a regular language l to be a splicing language is that l must have a constant in the schutzenberger sense we prove this longstanding conjecture to be true the result is based on properties of strongly connected components of the minimal deterministic finite state automaton for a regular splicing language using constants of the corresponding languages we also provide properties of transitive automata and path automata

computing the order of a locally testable automaton a locally testable language is a language with the property that for some positive integer j whether or not a string x is in the language depends on 1 the prefix and suffix of x of length j 1 and 2 the set of substrings of x of length j without regard to the order in which these substrings occur or the number of times each substring occurs for any j for which this is true it is said that the language is j testable for a given locally testable language the smallest such number j is called the order of the language locally testable languages are regular and therefore these concepts apply to the finite automata that recognize the languages the authors show that computing the order of a given locally testable deterministic automaton is np hard and present a polynomial time epsilon approximation algorithm for computing it in addition an upper bound of 2n 2 1 on the order of a locally testable automaton of n states is obtained and the co np completeness of the problem of whether for a given j a given deterministic automaton is j testable is proven

a chebyshev spectral collocation method for solving burgers type equations in this paper we elaborated a spectral collocation method based on differentiated chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations the problem is reduced to a system of ordinary differential equations that are solved by runge kutta method of order four numerical results for the nonlinear evolution equations such as 1d burgers kdv burgers coupled burgers 2d burgers and system of 2d burgers equations are obtained the numerical results are found to be in good agreement with the exact solutions numerical computations for a wide range of values of reynolds number show that the present method offers better accuracy in comparison with other previous methods moreover the method can be applied to a wide class of nonlinear partial differential equations

the symmetric traveling salesman polytope revisited in this paper we present a tour of the symmetric traveling salesman polytope focusing on inequalities that can be defined on sets of nodes most widely known inequalities are of this type many papers have appeared that give increasingly complex valid inequalities for this polytope but little intuition on why these inequalities are valid has been given to help in understanding these inequalities we develop an intuition into their validity by giving a unifying way of defining them through a sequential lifting procedure this procedure is based on lifting the slack variables associated with subtour elimination inequalities defined on sets of nodes called teeth we apply this procedure to some known classes of valid inequalities for the traveling salesman polytope tsp respectively comb brush star path and bipartition inequalities where the lifting coefficients are sequence independent for comb star and bipartition inequalities we provide new and noninductive proofs of validity directly inspired by this lifting procedure we also give an example where a facet defining inequality is derived from the lifting procedure but where the lifting coefficients are sequence dependent we finally study the ladder inequalities and show that they can be generated by an extension of the general procedure where the lifted variables are different from the slack variables of subtour elimination inequalities

efficient separation routines for the symmetric traveling salesman problem i general tools and comb separation this is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the traveling salesman problem via branch and cut in this paper we give the basic ideas behind these heuristics and design heuristics for comb separation the second paper will deal with more complex inequalities such as clique tree star or path and ladder inequalities and give computational results

separating maximally violated comb inequalities in planar graphs the traveling salesman problem tsp is a benchmark problem in combinatorial optimization it was one of the very first problems used for developing and testing approaches to solving large integer programs including cutting plane algorithms and branch and cut algorithms much of the research in this area has been focused on finding new classes of facets for the tsp polytope and much less attention has been paid to algorithms for separating from these classes of facets in this paper we consider the problem of finding violated comb inequalities if there are no violated subtour constraints in a fractional solution of the tsp a comb inequality may not be violated by more than 1 given a fractional solution in the subtour elimination polytope whose graph is planar we either find a violated comb inequality or determine that there are no comb inequalities violated by 1 our algorithm runs in on mcn time where mcn is the time to compute a cactus representation of all minimum cuts of a weighted planar graph on n vertices

numerical solution of the burgers equation by automatic differentiation abstract we compute the solution of the one dimensional burgers equation by marching the solution in time using a taylor series expansion our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non linear algebraic equations instead we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the taylor series we illustrate the effectiveness of our method by solving four test problems with known exact solutions the numerical solutions we obtain are in excellent agreement with the exact solutions while being superior to other previously reported numerical solutions

complexity theory for splicing systems this paper proposes a notion of time complexity in splicing systems the time complexity of a splicing system at length n is defined to be the smallest integer t such that all the words of the system having length n are produced within t rounds for a function t from the set of natural numbers to itself the class of languages with splicing system time complexity t n is denoted by spltime f n this paper presents fundamental properties of spltime and explores its relation to classes based on standard computational models both in terms of upper bounds and in terms of lower bounds as to upper bounds it is shown that for any function t n spltime t n is included in 1 nspace t n i e the class of languages accepted by a t n space bounded non deterministic turing machine with one way input head expanding on this result it is shown that 1 nspace t n is characterized in terms of splicing systems it is the class of languages accepted by a t n space uniform family of extended splicing systems having production time o t n with the additional property that each finite automaton appearing in the family of splicing systems has at most a constant number of states as to lower bounds it is shown that for all functions t n logn all languages accepted by a pushdown automaton with maximal stack height t x for a word x are in spltime t n from this result it follows that the regular languages are in spltime o logn and that the context free languages are in spltime o n

separating a superclass of comb inequalities in planar graphs many classes of valid and facet inducing inequalities are known for the family of polytopes associated with thesymmetric travelling salesman problem stsp includingsubtour elimination 2 matching andcomb inequalities for a given class of inequalities anexact separation algorithm is a procedure which given an lp relaxation vectorx finds one or more inequalities in the class which are violated byx or proves that none exist such algorithms are at the core of the highly successfulbranch and cut algorithms for the stsp however whereas polynomial time exact separation algorithms are known for subtour elimination and 2 matching inequalities the complexity of comb separation is unknown apartial answer to the comb problem is provided in this paper we define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx e 0 is planar the separation algorithm runs in o n3 time wheren is the number of vertices in the graph

the structure of reflexive regular splicing languages via schutzenberger constants the splicing operation was introduced in 1987 by head as a mathematical model of the recombination of dna molecules under the influence of restriction and ligases enzymes this operation allows us to define a computing language generating device called a splicing system other variants of this original definition were also proposed by paun and pixton respectively the computational power of splicing systems has been thoroughly investigated nevertheless an interesting problem is still open namely the characterization of the class of regular languages generated by finite splicing systems in this paper we will solve the problem for a special class of finite splicing systems termed reflexive splicing systems according to each of the definitions of splicing given by paun and pixton this special class of systems contains in perticular finite head splicing systems the notion of a constant given by schutzenberger once again intervenes