a class of label correcting methods for the k shortest paths problem in this paper we deal with the problem of finding the firstk shortest paths from a single origin node to all other nodes of a directed graph in particular we define the necessary and sufficient conditions for a set of distance label vectors on the basis of which we propose a class of methods which can be viewed as an extension of the generic label correcting method for solving the classical single origin all destinations shortest path problem the data structure used is characterized by a set ofk lists of candidate nodes and the proposed methods differ in the strategy used to select the node to be extracted at each iteration the computational results show that 1 some label correcting methods are generally much faster than the double sweep method of shier 1979 2 the most efficient node selection strategies used for solving the classical single origin all destinations shortest path problem have proved to be effective also in the case of thek shortest paths

efficient parallel algorithms for string editing and related problems the string editing problem for input strings x and y consists of transforming x into y by performing a series of weighted edit operations on x of overall minimum cost an edit operation on x can be the deletion of a symbol from x the insertion of a symbol in x or the substitution of a symbol of x with another symbol this problem has a well known o x y time sequential solution efficient pram parallel algorithms for the string editing problem are given if m min x y and n max x y then the crew bound is o log m log n time with o mn log m processors the crow bound is o log n log log m 2 time with o mn log log m processors in all algorithms space is o mn

enhancing spl epsiv approximation algorithms with the optimal linear scaling factor finding a least cost path subject to a delay constraint in a network is an np complete problem and has been extensively studied many works reported in the literature tackle this problem by using epsiv approximation schemes and scaling techniques i e by mapping link costs into integers or at least discrete numbers a solution which satisfies the delay constraint and has a cost within a factor of the optimal one that can be computed with pseudopolynomial computational complexity in this paper having observed that the computational complexities of the epsiv approximation algorithms using the linear scaling technique are linearly proportional to the linear scaling factor we investigate the issue of finding the optimal the smallest linear scaling factor to reduce the computational complexities and propose two algorithms the optimal linear scaling algorithm olsa and the transformed olsa we analytically show that the computational complexities of our proposed algorithms are very low as compared with those of epsiv approximation algorithms therefore incorporating the two algorithms can enhance the epsiv approximation algorithms by granting them a practically important capability self adaptively picking the optimal linear scaling factors in different networks as such epsiv approximation algorithms become more flexible and efficient

optimal lp 1 p rates of decay to linear diffusion waves for nonlinear evolution equations with ellipticity and dissipation abstract in this paper we are concerned with the large time decay estimates of solutions to the cauchy problem of nonlinear evolution equations with ellipticity and damping different end states of initial data are considered the optimal rate of decay to the linear diffusion waves is obtained the optimal rate of decay of solutions to the linearized system plays a crucial role in the analysis

asymptotic behavior of solution to nonlinear evolution equations with damping in this paper we consider the global existence and the asymptotic behavior of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x xx t x 0 r t 1 x 2 x xx with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 4 1 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the parabolic system defined by 2 4 which are obtained by the convection diffusion equations 2 1 and solutions to cauchy problem e and i tend asymptotically to the convection diffusion system with exponential rates precisely speaking we derive the asymptotic profile of e by gauss kernel g t x as follows 2 2e 1 4 t xg y t 1 1 sin 2 y 0 cos 2 y 0 dy e 1 t 2e 1 4 t rg t y cos 2 y 1 sin 2 y sin 2 y cos 2 y u0 x y v0 x y dy lp rx e 1 4 to 1 t 1 1p the same problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 nishihara k nishihara asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity z angew math phys 57 4 2006 604 614 for the case of 0 0

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

enumerating k best paths in length order in dags abstract we address the problem of finding the k best paths connecting a given pair of nodes in a directed acyclic graph dag with arbitrary lengths one of the main results in this paper is the proof that a tree representing the k th shortest path is obtained by an arc exchange in one of the previous k 1 trees each of which contains a previous best path an o m k n log k time and o k m space algorithm is designed to explicitly determine the k shortest paths in a dag with n nodes and m arcs the algorithm runs in o m kn time using o k m space in dags with integer length arcs empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported

scaling algorithms for the shortest paths problem we describe a new method for designing scaling algorithms for the single source shortest paths problem and use this method to obtain an o sqrt n m log n algorithm for the problem here n and m is the number of nodes and arcs in the input network and n is essentially the absolute value of the most negative arc length and arc lengths are assumed to be integral this improves previous bounds for the problem the method extends to related problems

kinetic relations and the propagation of phase boundaries in solids this paper treats the dynamics of phase transformations in elastic bars the specific issue studied is the compatibility of the field equations and jump conditions of the one dimensional theory of such bars with two additional constitutive requirements a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition a special elastic material with a piecewise linear non monotonic stress strain relation is considered and the riemann problem for this material is analyzed for a large class of initial data it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities

the modified integer round up property of the one dimensional cutting stock problem a linear integer minimization problem ip has the modified integer round up property mirup if the optimal value of any instance of ip is not greater than the optimal value of the corresponding lp relaxation problem rounded up plus one the aim of this paper is to investigate numerically whether the mirup holds for the one dimensional cutting stock problem the computational experiments carried out with a lot of randomly generated instances of the one dimensional cutting stock problem show that for all instances an integer solution fulfills the mirup moreover in most cases the optimal value equals the round up optimal value of the corresponding lp relaxation similarly the approach proposed to verify the mirup is usable as a new heuristic procedure for solving one dimensional cutting stock problems this heuristic always leads to very good solutions being optimal in the most cases

global decaying solution to dissipative nonlinear evolution equations with ellipticity abstract we establish the global existence and decaying results for the cauchy problem of nonlinear evolution equations e t 1 x x xx t 1 x 2 x xx forinitial data with different end states i x 0 x 0 0 x 0 x as x which displays the complexity in between ellipticity and dissipation although the nonlinear term x appears in equation e 1 which makes calculations more complicated due to smoothing effect of the parabolic operator we detail its regularity property and decay estimates when t 0 for the higher order spatial derivatives despite its relatively lower regularity of the initial data and we also discuss the decay estimates furthermore we do not restrict l1 bound on the initial data 0 x 0 x as in 2

nonlinear stability for dissipative nonlinear evolution equations with ellipticity we study the cauchy problem for a set of nonlinear evolution equations which has been proposed for the investigation of nonlinear interaction between ellipticity and dissipation the zero equilibrium is linearly stable for coefficients within certain range we further establish the global existence nonlinear stability and optimal decay rate of the solution for coefficients within the same range though with suitable restriction to initial data

constructing the nearly shortest path in crossed cubes a graph g is panconnected if each pair of distinct vertices u v v g are joined by a path of length l for all d g u v

combinatorial optimization problems with uncertain costs and the owa criterion in this paper a class of combinatorial optimization problems with uncertain costs is discussed the uncertainty is modeled by specifying a discrete scenario set containing k distinct cost scenarios the ordered weighted averaging owa for short aggregation operator is applied to choose a solution some well known criteria used in decision making under uncertainty such as the maximum minimum average hurwicz and median are special cases of owa furthermore by using owa the traditional robust min max approach to combinatorial optimization problems with uncertain costs can be generalized the computational complexity and approximability of the problem of minimizing owa for the considered class of problems are investigated and some new positive and negative results in this area are provided these results remain valid for many basic problems such as network or resource allocation problems

an algorithm for finding k minimum spanning trees this paper presents an algorithm for finding k minimum spanning trees in an undirected graph the required time is o km min n 2 m log log n and the space is o k m where n is the number of vertices and m is the number of edges the algorithm is based on three subroutines the first two subroutines are used to obtain the second minimum spanning tree in o min n 2 m alpha m n steps where alpha m n is tarjan s inverse of ackermann s function 12 which is very slowly growing the third one obtains the kth minimum spanning tree in o m steps when the jth minimum spanning trees for j 1 2 cdots k 1 are given

an optimal algorithm for selection in a min heap abstract an o k time algorithm is presented for selecting the kth smallest element in a binary min heap of size n k the approach uses recursively defined data structures that impose a hierarchical grouping on certain elements in the heap the result establishes a further example of a partial order for which the kth smallest element can be determined in time proportional to the information theory lower bound two applications to a resource allocation problem and to the enumeration of the k smallest spanning trees are identified

ordered weighted average optimization in multiobjective spanning tree problem abstract multiobjective spanning tree problems are studied in this paper the ordered median objective function is used as an averaging operator to aggregate the vector of objective values of feasible solutions this leads to the ordered weighted average spanning tree problem a nonlinear combinatorial optimization problem different mixed integer linear programs are proposed based on the most relevant minimum cost spanning tree models in the literature these formulations are analyzed and several enhancements presented their empirical performance is tested over a set of randomly generated benchmark instances the results of the computational experiments show that the choice of an appropriate formulation allows to solve larger instances with more objectives than those previously solved in the literature

a two objective mathematical model without cutting patterns for one dimensional assortment problems this paper considers a one dimensional cutting stock and assortment problem one of the main difficulties in formulating and solving these kinds of problems is the use of the set of cutting patterns as a parameter set in the mathematical model since the total number of cutting patterns to be generated may be very huge both the generation and the use of such a set lead to computational difficulties in solution process the purpose of this paper is therefore to develop a mathematical model without the use of cutting patterns as model parameters we propose a new two objective linear integer programming model in the form of simultaneous minimization of two contradicting objectives related to the total trim loss amount and the total number of different lengths of stock rolls to be maintained as inventory in order to fulfill a given set of cutting orders the model does not require pre specification of cutting patterns we suggest a special heuristic algorithm for solving the presented model the superiority of both the mathematical model and the solution approach is demonstrated on test problems

an efficient time and space k point to point shortest simple paths algorithm abstract we address the problem identifying the k best point to point simple paths connecting a given pair of nodes in a directed network with arbitrary lengths the main result in this paper is the proof that a path tree containing the kth point to point shortest simple path can be obtained by using one of the previous k 1 path trees containing each one of the previous k 1 best point to point shortest simple paths the proof implies that at most n single source shortest path computations re optimizations in a network with non negative length arcs are made in each iteration of the method in the optimistic case this strategy only needs o m time to compute the best neighbor associated with a path tree that is the second shortest simple path given a shortest simple path the algorithm runs in o k nf n m cmax time and uses o k m space to determine the k point to point shortest simple paths in a directed network with n nodes m arcs and maximum absolute length cmax here o f n m cmax is the best time needed to determine the shortest simple paths connecting a source node with any other non source node in a network with non negative length arcs we improve required space in yen s algorithm by a multiplicative factor of o n2 for each best solution moreover our algorithm runs in the optimistic case in o kf n m cmax time this affirmation is confirmed by an experimental study where o k shortest paths are used to determine the k point to point shortest simple paths in two versions of our algorithm

the minmax regret robust shortest path problem in a finite multi scenario model the robust shortest path problem is a network optimization problem that can be defined to deal with uncertainty of costs associated with the arcs of a network two models have been considered for the robust shortest path problem interval data and discrete data sets this work addresses the robust shortest path problem with a minmax regret objective function on a finite multi scenario model this problem consists in finding an optimal path in the sense that it has the minimum maximum deviation from the shortest one over all scenarios with this goal some properties of the problem and of its optimal solutions are derived these results allow to introduce three approaches a labeling algorithm an algorithm based on ranking loopless paths and a hybrid algorithm which ranks loopless paths in a suitable way to apply the early elimination of useless solutions the algorithms are tested on random networks and compared with a previous method for the same problem the obtained computational results are reported and discussed they show that the labeling and the hybrid approaches outperform the others

a branch and cut and price algorithm for one dimensional stock cutting and two dimensional two stage cutting abstract the one dimensional cutting stock problem 1d csp and the two dimensional two stage guillotine constrained cutting problem 2d 2cp are considered in this paper the gilmore gomory models of these problems have very strong continuous relaxations providing a good bound in an lp based solution approach in recent years there have been several efforts to attack the one dimensional problem by lp based branch and bound with column generation called branch and price and by general purpose c hv a tal g omory cutting planes in this paper we investigate a combination of both approaches i e the lp relaxation at each branch and price node is strengthened by c hv a tal g omory and g omory mixed integer cuts the branching rule is that of branching on variables of the gilmore gomory formulation tests show that for 1d csp general purpose cuts are useful only in exceptional cases however for 2d 2cp their combination with branching is more effective than either approach alone and mostly better than other methods from the literature

efficient shortest path simplex algorithms we consider the specialization of the primal simplex algorithm to the problem of finding a tree of directed shortest paths from a given node to all other nodes in a network of n nodes or finding a directed cycle of negative length two efficient variants of this shortest path simplex algorithm are analyzed and shown to require at most n 1 n 2 2 pivots and o n3 time

improving space efficiency with path length prediction for finding k shortest simple paths finding mbi k shortest simple paths in a directed graph is a fundamental problem in many engineering applications most existing algorithms such as yen s algorithm and its variants have polynomial worst case time complexity but their average case running time is very high the heuristic algorithm mps can run significantly faster in practice however it requires an excessive amount of memory space in this paper we provide a new heuristic algorithm that achieves high space efficiency while maintaining similar average case running time we first propose a sidetrack representation of path with which a path can be stored in mbi o bf 1 space we then show how to categorize a candidate path as either partial or complete and restrict the number of paths added to the queue in addition we provide an empirical equation that can very accurately predict the mbi k th shortest path length provided that a much smaller number of shortest paths have been found extensive experiments prove that our algorithm can achieve an mbi o n speedup in practice over yen s algorithm in comparison with mps it runs up to three times faster and uses less space by an order of magnitude

incremental elicitation of choquet capacities for multicriteria choice ranking and sorting problems this paper proposes incremental preference elicitation methods for multicriteria decision making with a choquet integral the choquet integral is an evaluation function that performs a weighted aggregation of criterion values using a capacity function assigning a weight to any coalition of criteria thus enabling positive and or negative interactions among them and covering an important range of possible decision behaviors however the specification of the capacity involves many parameters which raises challenging questions both in terms of elicitation burden and guarantee on the quality of the final recommendation in this paper we investigate the incremental elicitation of the capacity through a sequence of preference queries questions selected one by one using a minimax regret strategy so as to progressively reduce the set of possible capacities until the regret the worst case loss due to reasoning with only partially specified capacities is low enough we propose a new approach designed to efficiently compute minimax regret for the choquet model and we show how this approach can be used in different settings 1 the problem of recommending a single alternative 2 the problem of ranking alternatives from best to worst and 3 sorting several alternatives into ordered categories numerical experiments are provided to demonstrate the practical efficiency of our approach for each of these situations

computational study of a column generation algorithm for bin packing and cutting stock problems this paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem the main focus of the research is to study the extend to which standard branch and bound enhancement features such as variable fixing the tightening of the formulation with cutting planes early branching and rounding heuristics can be usefully incorporated in a branch and price algorithm we review and compare lower bounds for the cutting stock problem we propose a pseudo polynomial heuristic we discuss the implementation of the important features of the integer programming column generation algorithm and in particular the implementation of the branching scheme our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems

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

space efficient frameworks for top k string retrieval the inverted index is the backbone of modern web search engines for each word in a collection of web documents the index records the list of documents where this word occurs given a set of query words the job of a search engine is to output a ranked list of the most relevant documents containing the query however if the query consists of an arbitrary string which can be a partial word multiword phrase or more generally any sequence of characters then word boundaries are no longer relevant and we need a different approach in string retrieval settings we are given a set de d1 d2 d3 m dd of d strings with n characters in total taken from an alphabet set l r and the task of the search engine for a given query pattern p of length p is to report the most relevant strings in d containing p the query may also consist of two or more patterns the notion of relevance can be captured by a function score p dr which indicates how relevant document dr is to the pattern p some example score functions are the frequency of pattern occurrences proximity between pattern occurrences or pattern independent pagerank of the document the first formal framework to study such kinds of retrieval problems was given by muthukrishnan lsoda 2002r he considered two metrics for relevance frequency and proximity he took a threshold based approach on these metrics and gave data structures that use o n log n words of space we study this problem in a somewhat more natural top k framework here k is a part of the query and the top k most relevant highest scoring documents are to be reported in sorted order of score we present the first linear space framework i e using o n words of space that is capable of handling arbitrary score functions with near optimal o p p klog k query time the query time can be made optimal o ppk if sorted order is not necessary further we derive compact space and succinct space indexes for some specific score functions this space compression comes at the cost of higher query time at last we extend our framework to handle the case of multiple patterns apart from providing a robust framework our results also improve many earlier results in index space or query time or both

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant abstract in this paper we consider an initial boundary value problem for some nonlinear evolution system with dissipation and ellipticity we establish the global existence and furthermore obtain the l p p 2 decay rates of solutions corresponding to diffusion waves the analysis is based on the energy method and pointwise estimates

tighter relaxations for the cutting stock problem in the cutting stock problem csp a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions in many cases an optimal solution can be obtained quickly in this way but for instances which do not possess the integer round up property the optimality of the solution obtained cannot be verified by means of the lp bound in order to overcome this non satisfactory situation two tighter relaxations of the csp are proposed and results of theoretical and numerical investigations are presented

on computing all suboptimal alignments abstract naor and brutlag d naor d brutlag proceedings of the fourth symposium on combinational pattern matching lecture notes in computer science 684 1993 179 196 proposed a new compact representation for suboptimal alignments the kernel of that representation is a minimal directed acyclic graph dag containing all suboptimal alignments in this paper a flexible space saving scheme for computing such a dag is proposed in spite of the need for storing the dag these methods require very little additional working space for two sequences of lengths m and m m n the general scheme runs in o mn log 1 gu time and o m m n space for arbitrarily small 0 mn log log m using only o n space a variant of the method restricts the log log m factor to affect only grid points lying between suboptimal alignments it is also shown that a running time of o mn can be achieved by using only o m 1 e n space for arbitrarily small constant e 0 to exploit the computed dag a variant of aho corasick pattern matching machine a v aho m j corasick comm acm 18 1975 333 340 is employed to locate all occurrences of specified patterns and then a path is found in the dag that maximizes the sum of the scores of the non overlapping patterns occurring in it an example illustrates the utility

convergence to diffusion waves for nonlinear evolution equations with different end states in this paper we consider the global existence and the asymptotic decay of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x f x xx t 1 x x xx with initial data i x 0 0 x 0 x asx where and are positive constants such that 1 s 4 1 s is defined in 1 14 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the diffusion waves defined by 2 5 which are obtained by the diffusion equations 2 1 solutions to cauchy problem e and i tend asymptotically to those diffusion waves with exponential rates the analysis is based on the energy method the similar problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 for the case of 0 0

accelerating column generation for variable sized bin packing problems abstract in this paper we study different strategies to stabilize and accelerate the column generation method when it is applied specifically to the variable sized bin packing problem or to its cutting stock counterpart the multiple length cutting stock problem many of the algorithms for these problems discussed in the literature rely on column generation processes that are known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables in the sequel we introduce new dual optimal inequalities and explore the principle of model aggregation as an alternative way of controlling the progress of the dual variables two algorithms based on aggregation are proposed the first one relies on a row aggregated lp while the second one solves iteratively sequences of doubly aggregated models working with these approximations in the various stages of an iterative solution process has proven to be an effective way of achieving faster convergence the computational experiments were conducted on a broad range of instances many of them published in the literature they show a significant reduction of the number of column generation iterations and computing time

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

an enhanced k sp algorithm with pruning strategies to solve the constrained shortest path problem a new exact algorithm for the constrained shortest path problem is presented extensive experiments comparing the behavior of the new scheme and the recent best method from the literature are provided the proposed approach solves the constrained shortest path problem in usa road networks containing millions of nodes and arcs we propose an exact method to solve the constrained shortest path csp problem the new approach takes advantage of the binary partition strategy of a recent k shortest paths k sp algorithm that allows posing numerous additional path constraints in the model without extra difficulty we adapt and incorporate several pruning strategies from the literature on the csp problem in this scheme and obtain a robust and scalable algorithm the method is compared with the current best algorithm to solve the csp problem and produces significant speedups in a wide range of network configurations an example is given where the csp problem is solved in road networks containing a million nodes and arcs

decay rates of solutions to the cauchy problem for dissipative nonlinear evolution equations abstract in this paper we study the global existence and the asymptotic behavior of the solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x x x t 1 x x x x with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 1 through constructing a correct function x t defined by 2 13 and using the energy method we show sup x r x t x x x t 0 as t and the solutions decay with exponential rates the same problem was studied by tang and zhao s q tang h j zhao nonlinear stability for dissipative nonlinear evolution equations with ellipticity j math anal appl 233 1999 336 358 for the case of 0 0

a note on boundary layer of a nonlinear evolution system with damping and diffusions abstract in this paper we consider an initial boundary value problem for a nonlinear evolution system with damping and diffusions on the strip our main purpose is to investigate zero diffusion limits which formally yield parabolic hyperbolic coupled equations resulting from parabolic parabolic coupled equations it is shown that the boundary layer thickness is of the order o with 0 3 4 as the diffusion parameter goes to zero

the labeled multi bernoulli filter this paper proposes a generalization of the multi bernoulli filter called the labeled multi bernoulli filter that outputs target tracks moreover the labeled multi bernoulli filter does not exhibit a cardinality bias due to a more accurate update approximation compared to the multi bernoulli filter by exploiting the conjugate prior form for labeled random finite sets the proposed filter can be interpreted as an efficient approximation of the delta generalized labeled multi bernoulli filter it inherits the advantages of the multi bernoulli filter in regards to particle implementation and state estimation it also inherits advantages of the delta generalized labeled multi bernoulli filter in that it outputs labeled target tracks and achieves better performance

on the bicriterion minimal cost minimal label spanning tree problem we address a bicriterion spanning tree problem relevant in some application fields such as telecommunication networks or transportation networks each edge is assigned with a cost value and a label such as a color the first criterion intends to minimize the total cost of the spanning tree the summation of its edge costs while the second intends to get the solution with a minimal number of different labels since these criteria in general are conflicting criteria we developed an algorithm to generate the set of non dominated spanning trees computational experiments are presented and results discussed

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

nonlinear stability and optimal decay rate for a multidimensional generalized kuramoto sivashinsky system we study the cauchy problem for a generalized kuramoto sivashinsky system in multidimension the zero equilibrium is linearly stable for coefficients within a certain range we further establish the global existence nonlinear stability and optimal decay rate of the solution for coefficients within the same range though with suitable restriction to the initial data c 2000 academic press

three stage approaches for optimizing some variations of the resource constrained shortest path sub problem in a column generation context this paper presents a unified three stage approach tsa comprising preprocessing setup and iterative solution stages for solving several variations of the resource constrained shortest path problem rcsp tsa is designed specially for column generation applications in which sub problems must be solved repetitively the first two stages are implemented one time and only the third stage need be applied repetitively in a companion paper the authors proposed a tsa for solving rcsp on an acyclic graph with upper bound resource limitation constraints this paper shows that a tsa can be designed to solve each of several related problems shortest path with equality resource limitation constraints shortest path with resource windows resource constrained k shortest path and multiple resource multiple choice knapsack a numerical example demonstrates application of a tsa to design an international assembly system and its supply chain using branch and price with multiple choice knapsack sub problems computational results show that our tsa can solve this sub problem effectively in such a column generation environment

finding multi constrained multiple shortest paths numerous algorithms have been proposed for the well known multi constrained shortest path mcsp problem but very few have good practical performance in case of two or more constraints in this paper we propose a new lagrangian relaxation algorithm to solve a generalized version of the mcsp problem where we search for multiple shortest paths subject to multiple constraints as in some related work our algorithm first identifies the lower and upper bounds and then tries to close the gap with a path enumeration procedure however our algorithm is based on the recognition that the lagrange multipliers found by existing methods usually do not give the best search direction for minimizing path enumerations even though they can provide near optimized lower bounds we provide a solution to meet both of these goals and incorporate feasibility checks into a state of the art k shortest paths algorithm to further reduce path enumerations through experiments on various networks including the most challenging benchmark instances we show that our algorithm can solve a significantly larger number of instances to optimality with less computational cost often by one or two orders of magnitude when compared with the best known algorithm in the literature

on the k shortest simple paths problem in weighted directed graphs we present the first approximation algorithm for finding the k shortest simple paths connecting a pair of vertices in a weighted directed graph that breaks the barrier of mn it is deterministic and has a running time of o k m sqrt n n 3 2 log n where m is the number of edges in the graph and n is the number of vertices let s t in v the length of the i th simple path from s to t computed by our algorithm is at most frac 3 2 times the length of the i th shortest simple path from s to t the best algorithms for computing the exact k shortest simple paths connecting a pair of vertices in a weighted directed graph are due to yen management sci 17 1970 1971 pp 712 716 and lawler management sci 18 1971 1972 pp 401 405 the running time of their algorithms using modern data structures is o k mn n 2 log n both algorithms are from the early 70s although this problem and other variants of the k shortest path problem has drawn a lot of attention during the last three and a half decades the o k mn n 2 log n bound is still unbeaten

transducing markov sequences a markov sequence is a basic statistical model representing uncertain sequential data and it is used within a plethora of applications including speech recognition image processing computational biology radio frequency identification rfid and information extraction the problem of querying a markov sequence is studied under the conventional semantics of querying a probabilistic database where queries are formulated as finite state transducers specifically the complexity of two main problems is analyzed the first problem is that of computing the confidence probability of an answer the second is the enumeration of the answers in the order of decreasing confidence with the generation of the top k answers as a special case or in an approximate order thereof in particular it is shown that enumeration in any subexponential approximate order is generally intractable even for some fixed transducers and a matching upper bound is obtained through a proposed heuristic due to this hardness a special consideration is given to restricted yet common classes of transducers that extract matches of a regular expression subject to prefix and suffix constraints and it is shown that these classes are indeed significantly more tractable

choquet based optimisation in multiobjective shortest path and spanning tree problems this paper is devoted to the search of choquet optimal solutions in finite graph problems with multiple objectives the choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives we first present a condition on preferences name hereafter preference for interior points that characterizes preferences favouring compromise solutions a natural attitude in various contexts such as multicriteria optimisation robust optimisation and optimisation with multiple agents within choquet expected utility theory this condition amounts to using a submodular capacity and a convex utility function under these assumptions we focus on the fast determination of choquet optimal paths and spanning trees after investigating the complexity of these problems we introduce a lower bound for the choquet integral computable in polynomial time then we propose different algorithms using this bound either based on a controlled enumeration of solutions ranking approach or an implicit enumeration scheme branch and bound finally we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes

large time profile of solutions for a dissipative nonlinear evolution system with conservational form we examine the cauchy problem for a nonlinear dissipative evolution system with conservation form with initial condition where and are positive constants such that and 1 we establish the global existence and decay rate of the solution subject to the parameter restriction the optimal decay rate is obtained if 0 0 l1 r r2 furthermore the global existence of the solution to the same problem has been studied in jian and chen 1998 acta math sin 14 17 34 without giving the decay rate and optimal decay order of the solution in which 0 0 h1 r r2 l1 r r2

k a heuristic search algorithm for finding the k shortest paths we present a directed search algorithm called k for finding the k shortest paths between a designated pair of vertices in a given directed weighted graph k has two advantages compared to current k shortest paths algorithms first k operates on the fly which means that it does not require the graph to be explicitly available and stored in main memory portions of the graph will be generated as needed second k can be guided using heuristic functions we prove the correctness of k and determine its asymptotic worst case complexity when using a consistent heuristic to be the same as the state of the art o m nlogn k with respect to both runtime and space where n is the number of vertices and m is the number of edges of the graph we present an experimental evaluation of k by applying it to route planning problems as well as counterexample generation for stochastic model checking the experimental results illustrate that due to the use of heuristic on the fly search k can use less time and memory compared to the most efficient k shortest paths algorithms known so far

on the k shortest path trees problem abstract we address the problem of finding the k best path trees connecting a source node with any other non source node in a directed network with arbitrary lengths the main result in this paper is the proof that the k th shortest path tree is adjacent to at least one of the previous k 1 shortest path trees consequently we design an o f n m c max km time and o k m space algorithm to determine the k shortest path trees in a directed network with n nodes m arcs and maximum absolute length c max where o f n m c max is the best time needed to solve the shortest simple paths connecting a source node with any other non source node

construction of the similarity matrix for the spectral clustering method numerical experiments abstract spectral clustering is a powerful method for finding structure in a dataset through the eigenvectors of a similarity matrix it often outperforms traditional clustering algorithms such as k means when the structure of the individual clusters is highly non convex its accuracy depends on how the similarity between pairs of data points is defined two important items contribute to the construction of the similarity matrix the sparsity of the underlying weighted graph which depends mainly on the distances among data points and the similarity function when a gaussian similarity function is used the choice of the scale parameter can be critical in this paper we examine both items the sparsity and the selection of suitable s based either directly on the graph associated to the dataset or on the minimal spanning tree mst of the graph an extensive numerical experimentation on artificial and real world datasets has been carried out to compare the performances of the methods

efficient calculation of the most reliable pair of link disjoint paths in telecommunication networks abstract in transmission networks an important routing problem is to find a pair of link disjoint paths which optimises some performance measure in this paper the problem of obtaining the most reliable pair of link disjoint paths assuming the reliability of the links are known is considered this is a non linear optimisation problem it is further introduced the constraint that the length of the paths should not exceed a certain number of links which makes the efficient resolution of the problem more complex in this paper two variants of a novel exact algorithm for finding the most reliable pair of link disjoint paths with lengths constraints are presented also a computational study on the performance of the variants is described

a bicriterion shortest path algorithm abstract among the network models one of the more popular is the so called shortest path problem this model is used whenever it is intended to minimize a linear function which represents a distance between a predetermined pair of nodes in a given network often a single objective function is not sufficient to completely characterize some real world problems for instance in a road network two parameters as cost and time can be assigned to each arc clearly the fastest path may be too costly nevertheless the decision maker must choose one solution possibly not the best for both criteria in this paper we present an algorithm for this problem with this algorithm a special set of paths the set of pareto optimal paths is determined one objective for any pareto optimal path can not be improved without worsening the other one

fast computation of bounds for two terminal network reliability in this paper an algorithm for the fast computation of network reliability bounds is proposed the evaluation of the network reliability is an intractable problem for very large networks and hence approximate solutions based on reliability bounds have assumed importance the proposed bounds computation algorithm is based on an efficient bdd representation of the reliability graph model and a novel search technique to find important minpaths mincuts to quickly reduce the gap between the reliability upper and lower bounds furthermore our algorithm allows the control of the gap between the two bounds by controlling the overall execution time therefore a trade off between prediction accuracy and computational resources can be easily made in our approach the numerical results are presented for large real example reliability graphs to show the efficacy of our approach

subcubic equivalences between path matrix and triangle problems we say an algorithm on n n matrices with integer entries in l m mr or n node graphs with edge weights from l m m is truly subcubic if it runs in o n3 poly log m time for some 0 we define a notion of subcubic reducibility and show that many important problems on graphs and matrices solvable in o n3 time are equivalent under subcubic reductions namely the following weighted problems either all have truly subcubic algorithms or none of them do the all pairs shortest paths problem on weighted digraphs apsp detecting if a weighted graph has a triangle of negative total edge weight listing up to n2 99 negative triangles in an edge weighted graph finding a minimum weight cycle in a graph of non negative edge weights the replacement paths problem on weighted digraphs finding the second shortest simple path between two nodes in a weighted digraph checking whether a given matrix defines a metric verifying the correctness of a matrix product over the min semiring finding a maximum subarray in a given matrix therefore if apsp cannot be solved in n3 e time for any e 0 then many other problems also need essentially cubic time in fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization verifying a matrix product over the same structure and corresponding triangle detection problems over the structure these equivalences simplify prior work on subcubic algorithms for all pairs path problems since it now suffices to give appropriate subcubic triangle detection algorithms other consequences of our work are new combinatorial approaches to boolean matrix multiplication over the or and semiring abbreviated as bmm we show that practical advances in triangle detection would imply practical bmm algorithms among other results building on our techniques we give two improved bmm algorithms a derandomization of the combinatorial bmm algorithm of bansal and williams focs 09 and an improved quantum algorithm for bmm

weighting factor extensions for finite multiple objective vector minimization problems abstract this paper develops two procedures for finding the efficient set of finite action multiple objective problems the first gives an exact characterization of the efficient set in terms of the weighted q th powers of the objective functions for a fixed q and gives a lower bound for q the second gives an exact characterization of the efficient set in the t th order minima of the weighted objective functions some consideration is given to problems whose feasible action sets correspond to the vertices of certain polytopes and connectedness conjectures are given for which no counterexamples have yet been found

asymptotics of dissipative nonlinear evolution equations with ellipticity different end states in this paper we consider the global existence and asymptotic behaviors of solutions to the cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects e t 1 x xx t 1 x 2 x xx with initial data i x 0 0 x 0 x as x where and are positive constants such that 1 4 1 under the assumption that is sufficiently small we show that if the initial data is a small perturbation of the diffusion waves defined by 2 5 which are obtained by the diffusion equations 2 1 solutions to cauchy problem e and i tend asymptotically to those diffusion waves with exponential rates the analysis is based on the energy method the same problem was studied by tang and zhao j math anal appl 233 1999 336 358 for the case of 0 0

optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity abstract we derive the optimal convergence rates to diffusion wave for the cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects t 1 x x x t 1 x x x x subject to the initial data with end states x 0 0 x 0 x as x where and are positive constants such that 1 4 1 introducing the auxiliary function to avoid the difference of the end states we show that the solutions to the reformulated problem decay as t with the optimal decay order the decay properties of the solution in the l 2 sense which are not optimal were already established in paper c j zhu z y zhang h yin convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping and with different end states acta math sinica english ed in press the main element of this paper is to obtain the optimal decay order in the sense of l p space for 1 p which is based on the application of fourier analysis and interpolation inequality under some suitable restrictions on coefficients and moreover we discuss the asymptotic behavior of the solution to general system 1 1 at the end however the optimal decay rates of the solution to general system 1 1 remains unknown

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

inequality measures and equitable approaches to location problems abstract location problems can be considered as multiple criteria models where for each client spatial unit there is defined an individual objective function which measures the effect of a location pattern with respect to the client satisfaction e g it expresses the distance or travel time between the client and the assigned facility this results in a multiple criteria model taking into account the entire distribution of individual effects distances moreover the model enables us to introduce the concept of equitable efficiency which links location problems with theories of inequality measurement in this paper special attention is paid to solution concepts based on the bicriteria optimization of the mean distance and the absolute inequality measures the restrictions for the trade offs are identified which guarantee that the bicriteria approaches comply with the concept of equitable efficiency these results are further generalized to bicriteria approaches not using directly the trade off technique

efficient implementation of graph algorithms using contraction the component merging problem is a new graph problem versions of this problem appear as bottlenecks in various graph algorithms a new data structure solves this problem efficiently and two special cases of the problem have even more efficient solutions based on other data structures the performance of the data structures is sped up by introducing a new algorithmic tool called packets the algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures specifically trees can be used simultaneously as a priority queue and a concatenable queue similarly f heaps support some kinds of split operations with no loss of efficiency an immediate application of the solution to the simplest version of the merging problem is an o t m n algorithm for finding minimum spanning trees in undirected graphs without using f heaps where t m n m log 2 log 2 log d n the graph has n vertices and m edges and d max m n 2 packets also improve the f heap minimum spanning tree algorithm giving the fastest algorithm currently known for this problem the efficient solutions to the merging problem and the new observation about f heaps lead to an o n t m n n log n algorithm for finding a maximum weighted matching in general graphs this settles an open problem posed by tarjan 15 p 123 where the weaker bound of o nm log n 2 m was conjectured

shortest path games abstract we study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network such networks model among other things efficient development of a commuter rail system for a growing metropolitan area we motivate and define these games and provide reasonable conditions for the corresponding rail application we show that the core of a shortest path game is nonempty and satisfies the given conditions but that the shapley value for these games may lie outside the core however we show that the shortest path game is convex for the special case of tree networks and we provide a simple polynomial time formula for the shapley value in this case in addition we extend our tree results to the case where users of the network travel to nodes other than the source finally we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games where nodes are added to the network sequentially over time

heuristic methods for delay constrained least cost routing using spl kappa shortest paths the delay constrained least cost dclc problem is to find the least cost path in a graph subject to a delay constraint first formulated in the context of routing in computer networks the dclc model also applies to problems of path planning and other decision problems dclc is np complete many heuristic methods have been proposed for it this note presents two new methods based on the spl kappa shortest path spl kappa sp approach these heuristic methods one centralized the other distributed are both polynomial in numerical experiments the proposed algorithms almost always find the optimal paths

data structures for on line updating of minimum spanning trees with applications data structures are presented for the problem of maintaining a minimum spanning tree on line under the operation of updating the cost of some edge in the graph for the case of a general graph mai

combination of piecewise geodesic paths for interactive segmentation minimum cost paths have been extensively studied theoretical tools for interactive image segmentation the existing geodesically linked active contour glac model which basically consists of a set of vertices connected by paths of minimal cost blends the benefits of minimal paths and region based active contours this results in a closed piecewise smooth curve over which an edge or region energy functional can be formulated as an important shortcoming the glac in its initial formulation does not guarantee the curve to be simple consistent with respect to the purpose of segmentation in this paper we draw our inspiration from the glac and other boundary based interactive segmentation algorithms in the sense that we aim to extract a contour given a set of user provided points by connecting these points using paths the key idea is to select a combination among a set of possible paths such that the resulting structure represents a relevant closed curve instead of considering minimal paths only we switch to a more general formulation which we refer to as admissible paths these basically correspond to the roads travelling along the bottom of distinct valleys between given endpoints we introduce a novel term to favor the simplicity of the generated contour as well as a local search method to choose the best combination among possible paths

convergence rates of vanishing diffusion limit on nonlinear hyperbolic system with damping and diffusion the aim of this paper is to study the global unique solvability on c solution to the cauchy problem of a special 2 2 nonlinear hyperbolic system with damping and diffusion furthermore we also investigate the convergence rates as the diffusion parameter goes to zero it is shown that the convergence rates in c norm is of the order o the novelty of analysis taken in the present article is to obtain the uniform in a priori estimates on hs norm for any s n which are implemented by introducing induction we focus on two cases in this paper both and since the uniform in a priori estimates obtained are distinctly different for different cases

solving biobjective combinatorial max ordering problems by ranking methods and a two phases approach abstract in this paper we propose a new method to solve biobjective combinatorial optimization problems of the max ordering type the method is based on the two phases method and ranking algorithms to efficiently construct k best solutions for the underlying single objective combinatorial problem we show that the method overcomes some of the difficulties of procedures proposed earlier we illustrate this by an example and discuss the difficulties in extending it to more than two objectives

on partial differential equations related to lorenz system partial differential equations are constructed such that a truncation scheme as adopted by lorenz j atmos sci 20 130 1963 will lead to lorenz equations the partial differential equations are much simpler than those of the rayleigh bernard problem and are essentially of the mixed type various aspects of the partial differential equations are explored it is suggested that the switching back and forth between the elliptic and hyperbolic regimes represents the chaotic behavior of the system in the context of partial differential equations

k optimal solution sets for some polynomially solvable scheduling problems abstract a k optimal solution set s 1 s k of a given scheduling problem is a set of schedules such that no schedule s which is not contained in is better than any si i 1 k k optimal sets are useful in offering alternatives to a single optimal schedule in this paper algorithms are presented which compute k optimal sets for various scheduling problems for which the optimal schedule can be computed in polynomial time

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

local existence of solutions to dissipative nonlinear evolution equations with mixed types abstract in this paper we consider the local existence of solutions to the cauchy problems for the following nonlinear evolution equations with mixed types t 1 x x x t 1 x 2 x x x with initial data x 0 0 x 0 x as x where and are positive constants satisfying 1 1 through constructing an approximation solution sequence we obtain the local existence by using the contraction mapping principle

finding the k shortest paths we give algorithms for finding the k shortest paths not required to be simple connecting a pair of vertices in a digraph our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges in time o m n log n k we can also find the k shortest paths from a given source s to each vertex in the graph in total time o m n log n kn we describe applications to dynamic programming problems including the knapsack problem sequence alignment maximum inscribed polygons and genealogical relationship discovery

the stochastic trim loss problem the cutting stock problem csp is one of the most fascinating problems in operations research the problem aims at determining the optimal plan to cut a number of parts of various length from an inventory of standard size material so to satisfy the customers demands the deterministic csp ignores the uncertain nature of the demands thus typically providing recommendations that may result in overproduction or in profit loss this paper proposes a stochastic version of the csp which explicitly takes into account uncertainty using a scenario based approach we develop a two stage stochastic programming formulation the highly non convex nature of the model together with its huge size prevent the application of standard software we use a solution approach designed to exploit the specific problem structure encouraging preliminary computational results are provided

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

labeled random finite sets and the bayes multi target tracking filter an analytic solution to the multi target bayes recursion known as the delta generalized labeled multi bernoulli delta glmb filter has been recently proposed by vo and vo in labeled random finite sets and multi object conjugate priors ieee trans signal process vol 61 no 13 pp 3460 3475 2014 as a sequel to that paper the present paper details efficient implementations of the delta glmb multi target tracking filter each iteration of this filter involves an update operation and a prediction operation both of which result in weighted sums of multi target exponentials with intractably large number of terms to truncate these sums the ranked assignment and k th shortest path algorithms are used in the update and prediction respectively to determine the most significant terms without exhaustively computing all of the terms in addition using tools derived from the same framework such as probability hypothesis density filtering we present inexpensive relative to the delta glmb filter look ahead strategies to reduce the number of computations characterization of the l 1 error in the multi target density arising from the truncation is presented

a cutting plane algorithm for the one dimensional cutting stock problem with multiple stock lengths abstract a cutting plane approach combining chvatal gomory cutting planes with column generation is generalized for the case of multiple stock lengths in the one dimensional cutting stock problem appropriate modifications of the column generation procedure and the rounding heuristic are proposed a comparison with the branch and price method for the problem with one stock type and representative test results are reported

algorithms for finding the weight constrained k longest paths in a tree and the length constrained k maximum sum segments of a sequence in this work we obtain the following new results given a tree t v e with a length function e r and a weight function w e r a positive integer k and an interval l u the weight constrainedklongest paths problem is to find the k longest paths among all paths in t with weights in the interval l u we show that the weight constrainedklongest paths problem has a lower bound w vlogv k in the algebraic computation tree model and give an o vlogv k time algorithm for it given a sequence a a 1 a 2 a n of numbers and an interval l u we define the sum and length of a segment a i j to be a i a i 1 a j and j i 1 respectively the length constrainedkmaximum sum segments problem is to find the k maximum sum segments among all segments of a with lengths in the interval l u we show that the length constrainedkmaximum sum segments problem can be solved in o n k time