a property of left linear rewrite systems preserving recognizability abstract let r be a left linear rewrite system over sign r and let g sign r where g 1 sign r and 0 sign r we show that r effectively preserves recognizability if and only if r effectively preserves recognizability

bottom up tree pushdown automata classification and connection with rewrite systems we define different types of bottom up tree pushdown automata and study their connections with rewrite systems along this line of research we complete and generalize the results of gallier book and salomaa we define the notion of a tail reduction free trf rewrite system using the decidability of ground reducibility we prove the decidability of the trf property monadic rewrite systems of book gallier and salomaa become a natural particular case of trf rewrite systems we associate a deterministic bottom up tree pushdown automaton with any left linear trf rewrite system finally we generalize monadic rewrite systems by introducing the notion of a semi monadic rewrite system and show that like a monadic rewrite system it preserves recognizability

solving symmetric vehicle routing problems asymmetrically abstract the vehicle routing problem can be regarded as a traveling salesman problem with additional constraints algorithms based on assignment relaxations provide better solutions for the symmetric traveling salesman problem if they are used on an asymmetric transformed distance matrix as shown by jonker et al in a paper in operations research the generalization of such a transformation to the distance matrix of symmetric vehicle routing problems is described the approach is illustrated within a heuristic algorithm although it can also be the basis for an exact algorithm for a number of standard problems computational results are given that are competitive to results of other algorithms

optimal routing under capacity and distance restrictions this paper describes an integer linear programming algorithm for vehicle routing problems involving capacity and distance constraints the method uses constraint relaxation and a new class of subtour elimination constraints two versions of the algorithm are presented depending upon the nature of the distance matrix exact solutions are obtained for problems involving up to sixty cities

on dual solutions of the linear assignment problem abstract for the linear assignment problem we describe how to obtain different dual solutions it turns out that a shortest path algorithm can be used to compute such solutions with several interesting properties that enable to do better post optimality analysis two examples illustrate how different dual solutions can be used in the context of the traveling salesman problem

an improved approximation algorithm for the maximum tsp in this paper we consider the maximum traveling salesman problem with c parameterized triangle inequality for c 12 1 which means that the edge weights in the given complete graph g v e w satisfy w uv c w ux w xv for all distinct nodes u x v v for the maximum traveling salesman problem with c parameterized triangle inequality r hassin and s rubinstein gave a constant factor approximation algorithm with polynomial running time they achieved a performance ratio c only for c 12 57 in 8 which is the best known result we design a k c 1 2 ck c approximation algorithm for the maximum traveling salesman problem with c parameterized triangle inequality by using a similar idea but very different method to that in 11 where k min c i i 1 2 m c 1 c 2 c m is an optimal solution of the minimum cycle cover in g which is better than the c approximation algorithm for almost all c 12 1

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

network distance characteristics that affect computational effort in p median location problems abstract in solving location models the effort expended and the quality of the solutions obtained often varies significantly from one problem instance to another our conjecture is that this is not a random occurrence but instead can be correlated with characteristics of the network upon which the model is constructed in this paper we examine a variety of approaches for generating networks for the p median model the types of networks studied include those based on euclidean inter point distances shortest paths developed from a predefined network and randomly generated distance matrices in addition to the process by which the network is constructed we consider the distribution of distances symmetry and the satisfaction of the triangle inequality all of these characteristics influence the effort required to solve the p median model we have found however that the triangle inequality has the most significant impact

an addendum on sensitivity analysis of the optimal assignment we point out that sensitivity results for the linear assignment problem can be produced by a shortest path based approach in a straightforward manner and as efficient as finding an optimal solution

linear generalized semi monadic rewrite systems effectively preserve recognizability abstract we introduce the notion of the generalized semi monadic rewrite system which is a generalization of well known rewrite systems the ground rewrite system the monadic rewrite system and the semi monadic rewrite system we show that linear generalized semi monadic rewrite systems effectively preserve recognizability we show that a tree language l is recognizable if and only if there exists a rewrite system r such that r r 1 is a linear generalized semi monadic rewrite system and that l is the union of finitely many r classes we show several decidability and undecidability results on rewrite systems effectively preserving recognizability and on generalized semi monadic rewrite systems for example we show that for a rewrite system r effectively preserving recognizability it is decidable if r is locally confluent moreover we show that preserving recognizability and effectively preserving recognizability are modular properties of linear collapse free rewrite systems finally as a consequence of our results on trees we get that restricted right left overlapping string rewrite systems effectively preserve recognizability

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

monadic thue systems abstract certain infinite thue systems over a finite alphabet are studied in particular systems s e such that for each a e the set u u a s is a context freelanguage the syntactic structure of sets of ancestors and sets of descendants is considered as well as that of unions of congruence classes taken over infinite context free languages or regular sets the common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable even for finite systems the word problem for confluent systems of this type is shown to be tractable the question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation

descendants of a recognizable tree language for prefix constrained linear monadic term rewriting with position cutting strategy abstract for a tree language l a finite set z of regular path languages and a set s of z prefix constrained linear monadic term rewriting rules over the position cutting descendant of l for s is the set s l of trees reachable from a tree in l by rewriting in s by position cutting strategy if l is recognizable then s l is recognizable as well moreover if s is finite then we can construct a tree automaton recognizing s l for a recognizable tree language l and a finite set z of regular path languages we study the set d z l of position cutting descendants of l for all sets of z prefix constrained linear monadic term rewriting rules over we show that d z l is finite and that if l is given by a tree automaton a and each element of z is given by an automaton then we can construct a set r 1 r k of z prefix constrained linear monadic term rewriting systems over such that d z l r 1 l r k l

a best possible heuristic for the k center problem in this paper we present a 2 approximation algorithm for the k center problem with triangle inequality this result is best possible since for any 2 the existence of approximation algorithm would imply that p np it should be noted that no approximation algorithm for any constant has been reported to date linear programming duality theory provides interesting insight to the problem and enables us to derive in o e log e time a solution with value no more than twice the k center optimal value a by product of the analysis is an o e algorithm that identifies a dominating set in g2 the square of a graph g the size of which is no larger than the size of the minimum dominating set in the graph g the key combinatorial object used is called a strong stable set and we prove the np completeness of the corresponding decision problem

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

classical model selection via simulated annealing summary the classical approach to statistical analysis is usually based upon finding values for model parameters that maximize the likelihood function model choice in this context is often also based on the likelihood function but with the addition of a penalty term for the number of parameters though models may be compared pairwise by using likelihood ratio tests for example various criteria such as the akaike information criterion have been proposed as alternatives when multiple models need to be compared in practical terms the classical approach to model selection usually involves maximizing the likelihood function associated with each competing model and then calculating the corresponding criteria value s however when large numbers of models are possible this quickly becomes infeasible unless a method that simultaneously maximizes over both parameter and model space is available we propose an extension to the traditional simulated annealing algorithm that allows for moves that not only change parameter values but also move between competing models this transdimensional simulated annealing algorithm can therefore be used to locate models and parameters that minimize criteria such as the akaike information criterion but within a single algorithm removing the need for large numbers of simulations to be run we discuss the implementation of the transdimensional simulated annealing algorithm and use simulation studies to examine its performance in realistically complex modelling situations we illustrate our ideas with a pedagogic example based on the analysis of an autoregressive time series and two more detailed examples one on variable selection for logistic regression and the other on model selection for the analysis of integrated recapture recovery data

some undecidability results concerning the property of preserving regularity a finite string rewriting system r preserves regularity if and only if it preserves regularity where is the alphabet containing exactly those letters that have occurrences in the rules of r this proves a conjecture of gyenizse and vagvolgyi 1997 in addition some undecidability results are presented that generalize results of gilleron and tison 1995 from term rewriting systems to string rewriting systems it follows that the property of being regularity preserving is undecidable for term rewriting systems thus answering another question of gyenizse and vagvolgyi 1997 finally it is shown that it is undecidable in general whether a finite lengthreducing and confluent string rewriting system yields a regular set of normal forms for each regular language

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

a restricted lagrangean approach to the traveling salesman problem we describe an algorithm for the asymmetric traveling salesman problem tsp using a new restricted lagrangean relaxation based on the assignment problem ap the lagrange multipliers are constrained so as to guarantee the continued optimality of the initial ap solution thus eliminating the need for repeatedly solving ap in the process of computing multipliers we give several polynomially bounded procedures for generating valid inequalities and taking them into the lagrangean function with a positive multiplier without violating the constraints so as to strengthen the current lower bound upper bounds are generated by a fast tour building heuristic when the bound strengthening techniques are exhausted without matching the upper with the lower bound we branch by using two different rules according to the situation the usual subtour breaking disjunction and a new disjunction based on conditional bounds we discuss computational experience on 120 randomly generated asymmetric tsp s with up to 325 cities the maximum time used for any single problem being 82 seconds this is a considerable improvement upon earlier methods though the algorithm discussed here is for the asymmetric tsp the approach can be adapted to the symmetric tsp by using the 2 matching problem instead of ap

triangulation and embedding using small sets of beacons concurrent with recent theoretical interest in the problem of metric embedding a growing body of research in the networking community has studied the distance matrix defined by node to node latencies in the internet resulting in a number of recent approaches that approximately embed this distance matrix into low dimensional euclidean space there is a fundamental distinction however between the theoretical approaches to the embedding problem and this recent internet related work in addition to computational limitations internet measurement algorithms operate under the constraint that it is only feasible to measure distances for a linear or near linear number of node pairs and typically in a highly structured way indeed the most common framework for internet measurements of this type is a beacon based approach one chooses uniformly at random a constant number of nodes beacons in the network each node measures its distance to all beacons and one then has access to only these measurements for the remainder of the algorithm moreover beacon based algorithms are often designed not for embedding but for the more basic problem of triangulation in which one uses the triangle inequality to infer the distances that have not been measured here we give algorithms with provable performance guarantees for beacon based triangulation and embedding we show that in addition to multiplicative error in the distances performance guarantees for beacon based algorithms typically must include a notion of slack a certain fraction of all distances may be arbitrarily distorted for metric spaces of bounded doubling dimension which have been proposed as a reasonable abstraction of internet latencies we show that triangulation based distance reconstruction with a constant number of beacons can achieve multiplicative error 1 on a 1 e fraction of distances for arbitrarily small constants and e for this same class of metric spaces we give a beacon based embedding algorithm that achieves constant distortion on a 1 e fraction of distances this provides some theoretical justification for the success of the recent global network positioning algorithm of ng and zhang 2002 and it forms an interesting contrast with lower bounds showing that it is not possible to embed all distances in a doubling metric space with constant distortion we also give results for other classes of metric spaces as well as distributed algorithms that require only a sparse set of distances but do not place too much measurement load on any one node

a closure property of regular languages note given a semigroup homomorphism x p y where p y is the powerset of y we show that the inverse image k 1 of a regular set k y is regular too this theorem has several consequences generalizing previously known results

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

some decision problems about controlled rewriting systems abstract a description of deterministic context free languages by some kind of finitely generated right congruences the sets of relations generating such right congruences are called controlled rewriting systems or c systems has been given previously in the first part of this paper we study several decision problems on c systems namely the confluence problem the equivalence problem the refinement problem the class equivalence problem and the class inclusion problem in the second part of the paper we deduce from the main theorem given previously a characterisation of dcfls by means of finitely generated congruences the sets of relations generating such congruences are now ordinary finite semi thue systems we study three decision problems on finite semi thue systems namely the class equivalence problem the word problem for the syntactic congruence of one class and the partial confluence problem all the problems are investigated in several classes of c systems or semi thue systems for every class we give an answer which may be yes the problem is decidable no the problem is undecidable or eq which means that the problem is recursively equivalent to the equivalence problem for dpda

the undecidability of the semi unification problem abstract the semi unification problem sup is a natural generalization of both first order unification and matching the problem arises in various branches of computer science and logic although several special cases of sup are known to be decidable the problem in general has been open for several years we show that sup in general is undecidable by reducing what we call the boundedness problem of turing machines to sup the undecidability of this boundedness problem is established by a technique developed in the mid 1960s to prove related results about turing machines

a reduction approach to the repeated assignment problem abstract we consider the repeated assignment problem rap which is a k fold repetition of the n n linear assignment problem lap with the additional requirement that no assignment can be repeated more than once in actual applications k is typically much smaller than n first we derive upper and lower bounds respectively by a heuristic together with local search and an efficient method solving the continuous relaxation the latter also solves a lagrangian relaxation such that the related pegging test to fix variables at zero or one decomposes into k independent pegging tests to laps these can be solved exactly by transforming them into all pairs shortest path problems together with these procedures we also employ a virtual pegging test and reduce rap in size numerical experiments show that the reduced instances with k n can be solved exactly using standard mip solvers

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

term rewriting restricted to ground terms we study rewriting of ground terms for an arbitrary term rewrite system r over a ranked alphabet we restrict the rewriting relation r to ground terms we introduce the relation r g r t t we show that r g r t t we show that for a given term rewrite system r and a given ground term rewrite system s over a ranked alphabet it is decidable if r g s we show that for a given left linear right ground term rewrite system r over a ranked alphabet it is decidable if there is a ground term rewrite system s over such that r g s

algorithms for generation of ramanujan graphs other expanders and related ldpc codes expander graphs are highly connected sparse finite graphs the property of being an expander seems significant in many of these mathematical computational and physical contexts for practical applications it is very important to construct expander and ramanujan graphs with given regularity and order in general constructions of the best expander graphs with a given regularity and order is no easy task in this paper we present algorithms for generation of ramanujan graphs and other expanders we describe properties of obtained graphs in comparison to previously known results we present a method to obtain a new examples of irregular ldpc codes based on described graphs and we briefly describe properties of this codes

a note on dual solutions of the assignment problem in connection with the traveling salesman problem abstract the algorithm of christofides is considered for the determination of a lower bound for the traveling salesman problem there exist many investigations for improving this bound but these are only heuristic proposals it is shown that the subgradients of the bound function can be easily calculated and a subgradient procedure is developed for computing a sharp bound

triangle inequality and symmetry in connection with the assignment and the traveling salesman problem abstract triangle inequality and symmetry play an important role in the treatment of the traveling salesman problem tsp examplarily it is shown how to transform the distance matrix in order to use a given heuristic algorithm furthermore related performance bounds will be given and studied the corresponding transformation originates in the dual assignment problem finally results concerning the exact case are mentioned as reducing the dimension of the original tsp and as well as the restricted use of non optimal edges the paper represents a survey of the above mentioned topic

towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem the investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory the aim of this paper is twofold first we introduce the notion of stability of approximation algorithms this notion is shown to be of practical as well as of theoretical importance especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit polynomial time approximation secondly we apply our concept to the study of the traveling salesman problem tsp we show how to modify the christofides algorithm for tsp to obtain efficient approximation algorithms with constant approximation ratio for every instance of tsp that violates the triangle inequality by a multiplicative constant factor this improves the result of andreae and bandelt siam j discrete math 8 1995 1

the effects of triangle inequality on the vehicle routing problem abstract routing problems often utilize experimental networks to represent real world scenarios however most ignore the inclusion of triangle inequality violations a phenomenon resulting from delays or rounding errors within a network this work evaluates the effect of both frequency the number of violations and severity the degree of intensity of a violation of triangle inequality and evaluates both solution quality and solution time based on simulated annealing ant colony optimization and savings algorithm methods findings indicate that while both frequency and severity degrade solution quality increased levels of frequency and severity together result in significant adverse affects to solution quality solution time however is not impacted by the presence of triangle inequality violations within the network this information should encourage practitioners to identify delays and maintain the presence of triangle inequality violations in a network to ensure accuracy of solution quality

on the relationship of approximation algorithms for the minimum and the maximum traveling salesman problem abstract we consider the relationship between the minimum and the maximum traveling salesman problem the paper is based on the idea of applying heuristics for the maximum traveling salesman problem to the minimum traveling salesman problem numerical results confirm the efficiency of the proposed method

on the refinement of bounds of heuristics algorithms of the traveling salesman poblem a number of heuristics for the traveling salesman problem tsp rely on the assumption that the triangle inequality ti is satisfied when ti does not hold the paper proposes a transformation such that for the transformed problem the ti holds consequently the bounds obtained for heuristics are valid with appropriate modification moreover for a tsp satisfying ti the same transformation strengthens such bounds the transformation essentially maps the problem into one that is minimal with respect to the property that ti holds for the symmetric tsp the transformation is particularly simple for an application of the transformation in the asymmetric case we need the dual solution of an assignment problem

a chernoff bound for random walks on expander graphs we consider a finite random walk on a weighted graph g we show that the fraction of time spent in a set of vertices a converges to the stationary probability pi a with error probability exponentially small in the length of the random walk and the square of the size of the deviation from pi a the exponential bound is in terms of the expansion of g and improves previous results of d aldous probab engrg inform sci 1 1987 pp 33 46 l lovasz and m simonovits it random structures algorithms 4 1993 pp 359 412 m ajtai j komlos and e szemeredi deterministic simulation of logspace in proc 19th acm symp on theory of computing 1987 we show that taking the sample average from one trajectory gives a more efficient estimate of pi a than the standard method of generating independent sample points from several trajectories using this more efficient sampling method we improve the algorithms of jerrum and sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic ising system and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

some undecidability results for non monadic church rosser thue systems abstract in this paper the computational power of finite church rosser thue systems that are non monadic is explored in particular it is shown that there exists a finite non monadic church rosser thue system t such that the set of linear sentences that are true under the interpretation given by t is undecidable this contrasts with the situation for finite church rosser thue systems that are monadic in addition some problems are presented that are decidable for finite monadic church rosser thue systems but that are undecidable in general for finite non monadic church rosser thue systems finally it is shown how degrees of complexity can be encoded in finite non monadic church rosser thue systems

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

the symmetric traveling salesman problem and edge exchanges in minimal 1 trees abstract in this article the effect of exchanging edges inside a minimal 1 tree with edges outside is analysed in combination with an upper bound this analysis enables the elimination of variables in the symmetric traveling salesman problem after discussion on a number of improvements for this analysis the implementation is described in a traveling salesman algorithm based on the 1 tree relaxation computational results show the advantages of the edges exchanges for euclidean problems up to 120 cities as well as for random table problems up to 200 cities

reductions in tree replacement systems abstract certain kinds of church rosser tree replacement systems reduction systems are studied most results presented here generalize results proved for string rewriting systems by book 1982 the concept of a tree rewriting system with a norm is introduced and studied a new tree reducing machine which reduces every input tree to an irreducible tree is defined as a consequence when the norm under consideration is the size of the tree the number of nodes the word problem for finite church rosser reduction systems is decidable in linear time in the size of the input tree church rosser monadic reduction systems generalizing the monadic thue systems of book 1982 and book et al 1982 are also studied it is shown that every congruence class and every finite union of congruence classes defined by such a system is accepted by a deterministic bottom up tree pushdown automation this new form of a tree automation is briefly studied

technical note a note on parametric analysis in linear assignment a classic application of the linear assignment problem is the assignment of people to jobs or jobs to people in this context it is interesting to measure competition for jobs and to generate a suitable list of jobs from which a person can choose the length of the list is a parameter a known list generation procedure is based on an interior point method followed by a parametric analysis we describe a more efficient procedure exploiting linear assignment theory and shortest path computations further we propose an alternative list generation procedure based on a special type of dual values for the linear assignment problem

solving some lexicographic multi objective combinatorial problems abstract the lexicographic multi objective optimization problem r lmop with r sum or bottleneck objectives optimizes a first objective then as far as a choice remains a second one then a third one and so on a general solution scheme from literature is based on scaling for the case that only sum criteria are involved the complexity is o rlogn t n m with t n m the complexity of optimization on combinatorial sum problem instances with m arcs and n nodes for cases with at least one bottleneck criterion the complexity is at least o rnlog2n t n m we describe a solution scheme that is suited to solve the shortest path r lmop the minimum spanning tree r lmop and the linear assignment r lmop the complexity is in all cases o r t n m

losing recognizability we consider the ranked alphabet s consisting of a binary symbol we give a rewrite system r over s such that r effectively preserves recognizability on any ranked alphabet obtained by adding finitely many nullary symbols to s however r does not preserve recognizability on the ranked alphabet obtained by adding one unary and one nullary symbol to s

an additive bounding procedure for the asymmetric travelling salesman problem in this paper new lower bounds for the asymmetric travelling salesman problem are presented based on spanning arborescences the new bounds are combined in an additive procedure whose theoretical performance is compared with that of the balas and christofides procedure 1981 both procedures have been imbedded in a simple branch and bound algorithm and experimentally evaluated on hard test problems

new formulations for the uncapacitated multiple allocation hub location problem in this paper we review the integer linear formulations of the uncapacitated multiple allocation hub location problem we study the scope of validity of these formulations and give new ones that generalize the older formulations our formulations allow one or two visits to hubs and include a more general cost structure that needs not satisfy the triangle inequality we prove that the constraints defined by cliques of a related intersection graph are tighter constraints than the classical ones we also discuss a pre processing of the problem which is very useful for reducing its size finally we check the strength of the new formulations and compare them with others in the literature by solving instances of two commonly used data sets the cab civil aeronautics board and ap australian post and also randomly generated instances our computational results clearly show that our formulations outperform all others previously used for small and medium problems

approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs a directed multigraph is said to be d regular if the indegree and outdegree of every vertex is exactly d by hall s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity we prove that if the d regular multigraph does not contain more than d 2 copies of any 2 cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2 cycle occurs in at most one component of each pair our proof is constructive and gives a polynomial algorithm to find such a decomposition since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs we also give an alternative simpler algorithm to extract a single such pair this combinatorial theorem then comes handy in rounding a fractional solution of an lp relaxation of the maximum traveling salesman problem tsp problem the first stage of the rounding procedure obtains two cycle covers that do not share a 2 cycle with weight at least twice the weight of the optimal solution then we show how to extract a tour from the 2 cycle covers whose weight is at least 2 3 of the weight of the longest tour this improves upon the previous 5 8 approximation with a simpler algorithm utilizing a reduction from maximum tsp to the shortest superstring problem we obtain a 2 5 approximation algorithm for the latter problem which is again much simpler than the previous one for minimum asymmetric tsp the same technique gives two cycle covers not sharing a 2 cycle with weight at most twice the weight of the optimum assuming triangle inequality we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0 842 log 2 n larger than optimal this improves upon a previous approximation algorithm with approximation guarantee of 0 999 log 2 n other applications of the rounding procedure are approximation algorithms for maximum 3 cycle cover factor 2 3 previously 3 5 and maximum asymmetric tsp with triangle inequality factor 10 13 previously 3 4

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

a new series of dense graphs of high girth let k 1 be an odd integer t b k 2 4 c and q be a prime power we construct a bipartite q regular edge transitive graph cd k q of order v 2qk t 1 and girth g k 5 if e is the the number of edges of cd k q then e v 1 k t 1 these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g g 5 g 6 11 12 for g 24 this represents a slight improvement on bounds established by margulis and lubotzky phillips sarnak for 5 g 23 g 6 11 12 it improves on or ties existing bounds