an o n log log n time algorithm for triangulating a simple polygon given a simple n vertex polygon the triangulation problem is to partition the interior of the polygon into n 2 triangles by adding n 3 nonintersecting diagonals we propose an o n log log n time algorithm for this problem improving on the previously best bound of o n log n and showing that triangulation is not as hard as sorting improved algorithms for several other computational geometry problems including testing whether a polygon is simple follow from our result

the minimum cost flow problem a unifying approach to dual algorithms and a new tree search algorithm this paper is concerned with the minimum cost flow problem it is shown that the class of dual algorithms which solve this problem consists of different variants of a common general algorithm we develop a new variant which is in fact a new form of the primal dual algorithm and which has several interesting properties it uses explicitly only dual variables the slope of the change in the dual objective is monotone the bound on the maximum number of iterations to solve a problem with integral bounds on the flow is better than bounds for other algorithms

on the construction of good fully symmetric integration rules in this paper we review some classical theory of moments for polynomials this theory turns out to be a useful tool when we want to construct good fully symmetric integration rules fsg rules for fully symmetric regions we also show how to apply this theory by constructing several new fsg rules of degree 7 9 and 11 for the 3 cube

on the structure of all minimum cuts in a network and applications this paper presents a characterization of all minimum cuts separating a source from a sink in a network a binary relation is associated with any maximum flow in this network and minimum cuts are identified with closures for this relation as a consequence finding all minimum cuts reduces to a straightforward enumeration applications of this results arise in sensitivity and parametric analyses of networks the vertex packing and maximum closure problems in unconstrained pseudo boolean optimization and project selection as well as in other areas of application of minimum cuts

an efficient approximation algorithm for the survivable network design problem the survivable network design problem sndp is to construct a minimum cost subgraph satisfying certain given edge connectivity requirements the first polynomial time approximation algorithm was given by williamson et al combinatorica 15 1995 435 454 this paper gives an improved version that is more efficient consider a graph ofn vertices and connectivity requirements that are at mostk both algorithms find a solution that is within a factor 2k 1 of optimal fork 2 and a factor 2 of optimal fork 1 our algorithm improves the time from o k3n4 to o k 2 n 2 kn 2 sqrt log log n our algorithm shares features with those of williamson et al combinatorica 15 1995 435 454 but also differs from it at a high level necessitating a different analysis of correctness and accuracy our analysis is based on a combinatorial characterization of the redundant edges several other ideas are introduced to gain efficiency these include a generalization of padberg and rao s characterization of minimum odd cuts use of a representation of all minimum s t cuts in a network and a new priority queue system the latter also improves the efficiency of the approximation algorithm of goemans and williamson siam journal on computing 24 1995 296 317 for constrained forest problems such as minimum weight matching generalized steiner trees and others 1998 the mathematical programming society inc published by elsevier science b v

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

approximate minimum cost multicommodity flows in o e 2 knm time we show that an e approximate solution of the cost constrainedk commodity flow problem on ann nodem arc network g can be computed by sequentially solving o k 2 loggk loggm log g 1 gk single commodity minimum cost flow problems on the same network in particular an approximate minimum cost multicommodity flow can be computed in tilde o g 2 gknm running time where the notation o means up to logarithmic factors this result improves the time bound mentioned by grigoriadis and khachiyan 4 by a factor ofm n and that developed more recently by karger and plotkin 8 by a factor of 1 we also provide a simple tilde o nm time algorithm for single commodity budget constrained minimum cost flows which is tilde o 3 times faster than the algorithm developed in the latter paper

enhancing multiple point geostatistical modeling 1 graph theory and pattern adjustment in recent years higher order geostatistical methods have been used for modeling of a wide variety of large scale porous media such as groundwater aquifers and oil reservoirs their popularity stems from their ability to account for qualitative data and the great flexibility that they offer for conditioning the models to hard quantitative data which endow them with the capability for generating realistic realizations of porous formations with very complex channels as well as features that are mainly a barrier to fluid flow one group of such models consists of pattern based methods that use a set of data points for generating stochastic realizations by which the large scale structure and highly connected features are reproduced accurately the cross correlation based simulation ccsim algorithm proposed previously by the authors is a member of this group that has been shown to be capable of simulating multimillion cell models in a matter of a few cpu seconds the method is however sensitive to pattern s specifications such as boundaries and the number of replicates in this paper the original ccsim algorithm is reconsidered and two significant improvements are proposed for accurately reproducing large scale patterns of heterogeneities in porous media first an effective boundary correction method based on the graph theory is presented by which one identifies the optimal cutting path surface for removing the patchiness and discontinuities in the realization of a porous medium next a new pattern adjustment method is proposed that automatically transfers the features in a pattern to one that seamlessly matches the surrounding patterns the original ccsim algorithm is then combined with the two methods and is tested using various complex two and three dimensional examples it should however be emphasized that the methods that we propose in this paper are applicable to other pattern based geostatistical simulation methods

beyond the flow decomposition barrier we introduce a new approach to the maximum flow problem this approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities our approach leads to an o min n 2 3 m 1 2 m log n 2 m log u time bound for a network with n vertices m arcs and integral arc capacities in the range 1 u this is a fundamental improvement over the previous time bounds we also improve bounds for the gomory hu tree problem the parametric flow problem and the approximate s t cut problem

constructing fully symmetric cubature formulae for the sphere we construct symmetric cubature formulae of degrees in the 13 39 range for the surface measure on the unit sphere we exploit a recently published correspondence between cubature formulae on the sphere and on the triangle specifically a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function u 1 u 2 u 3 1 2 where u 1 u 2 and u 3 are homogeneous coordinates

generalized goal programming polynomial methods and applications in this paper we address a general goal programming problem with linear objectives convex constraints and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets in particular classical linear goal programming problems as well as several models in location and regression analysis are modeled within this framework

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

a capacity rounding algorithm for the minimum cost circulation problem a dual framework of the tardos algorithm recently e tardos gave a strongly polynomial algorithm for the minimum cost circulation problem and solved the open problem posed in 1972 by j edmonds and r m karp her algorithm runs in o m2t m n logm time wherem is the number of arcs n is the number of vertices andt m n is the time required for solving a maximum flow problem in a network withm arcs andn vertices in the present paper taking an approach that is a dual of tardos s we also give a strongly polynomial algorithm for the minimum cost circulation problem our algorithm runs in o m2s m n logm time and reduces the computational complexity wheres m n is the time required for solving a shortest path problem with a fixed origin in a network withm arcs n vertices and a nonnegative arc length function the complexity is the same as that of orlin s algorithm recently developed by efficiently implementing the edmonds karp scaling algorithm

integer programming formulations for the elementary shortest path problem abstract given a directed graph g v a with arbitrary arc costs the elementary shortest path problem espp consists of finding a minimum cost path between two nodes s and t such that each node of g is visited at most once if negative costs are allowed the problem is np hard in this paper several integer programming formulations for the espp are compared we present analytical results based on a polyhedral study of the formulations and computational experiments where we compare their linear programming relaxation bounds and their behavior within a branch and cut framework the computational results show that a formulation with dynamically generated cutset inequalities is the most effective

a new algorithm for solving the feasibility problem of a network flow abstract a network d with the constraints of conservation and boundedness is given the feasibility problem is defined as is there a flow x satisfying the constraints in this paper we suppose that the lower and upper bounds are integers we first present a new theorem characterizing the conditions for feasibility and then describe a scaling algorithm to solve the problem our algorithm relaxes the capacity bounds and then successively reduces the value of relaxation if the network is infeasible then our algorithm not only diagnoses it and finds a positive cut but also gives a geometrical description to feasibility concept our algorithm also helps modeler to repair an infeasible network or to estimate the maximum cost of repairing for a network with n nodes and m arcs the algorithm runs in o m 2 log nu time where u is the largest absolute arc bounds

relaxed most negative cycle and most positive cut canceling algorithms for minimum cost flow this paper presents two new scaling algorithms for the minimum cost network flow problem one a primal cycle canceling algorithm the other a dual cut canceling algorithm both algorithms scale a relaxed optimality parameter and create a second inner relaxation the primal algorithm uses the inner relaxation to cancel a most negative node disjoint family of cycles w r t the scaled parameter the dual algorithm uses it to cancel most positive cuts w r t the scaled parameter we show that in a network with n nodes and m arcs both algorithms need to cancel only omn objects per scaling phase furthermore we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms our algorithms have potential practical advantages compared to some other canceling algorithms as well along the way we give a comprehensive survey of cycle and cut canceling algorithms for min cost flow we also clarify the formal duality between cycles and cuts

fully symmetric integration formulas for the surface of the sere in s dimensions methods of polynomial degree d to approximate integrals over the surface of the s dimensional sphere are discussed such methods are constructed by choosing the abscissas and weights so that the formulas integrate a certain subset of the polynomials of maximum degree d exactly a simple set of monomials is described the exact integration of which will ensure that the method has the required polynomial degree results are obtained which enable the derivation of consistency conditions on the rule structures an example of a new s dimensional degree 11 rule is given

faster scaling algorithms for general graph matching problems an algorithm for minimum cost matching on a general graph with integral edge costs is presented the algorithm runs in time close to the fastest known bound for maximum cardinality matching specifically let n m and n denote the number of vertices number of edges and largest magnitude of a cost respectively the best known time bound for maximum cardinal ity matching m 0 am the new algorithm for minimum cost matching has time bound 0 in a m n iog n m log nn a slight modification of the new algorithm finds a maximum cardinality matching in 0 fire time other applications of the new algorlthm are given mchrding an efficient implementa tion of christofides traveling salesman approximation algorithm and efficient solutions to update problems that require the linear programming duals for matching

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

a new approach for computing a most positive cut using the minimum flow algorithms abstract the minimum flow problem is one of the network flow problems in which computes minimum flow from a source node s to a sink node t to this point the algorithms of the minimum flow problem have been only referred to solve certain examples like the machine setup problem in this paper we describe correspondence between the minimum flow problem and the most positive cut problem we present a new algorithm to solve the most positive cut problem using the minimum flow algorithms the running time of the algorithm is equal to one minimum flow computation also we prove that the infeasibility of a network can be distinguished by solving a minimum flow problem

computing maximal polymatroidal network flows in the classical network flow model flows are constrained by the capacities of individual arcs in the polymatroidal network flow model introduced in this paper flows are constrained by the capacities of sets of arcs yet the essential features of the classical model are retained the augmenting path theorem the integral flow theorem and the max flow min cut theorem arc all shown to yield to straightforward generalization we describe a maximal flow algorithm which finds augmenting paths by labeling arcs instead of nodes as in the case of the classical model as a counterpart of a known result for the classical model we prove that the number of augmentations required to achieve a maximal value flow is bounded by the cube of the number of arcs in the network provided each successive augmentation is made along a shortest augmenting path with ties between shortest paths broken by lexicography

faster scaling algorithms for network problems this paper presents algorithms for the assignment problem the transportation problem and the minimum cost flow problem of operations research the algorithms find a minimum cost solution yet run in time close to the best known bounds for the corresponding problems without costs for example the assignment problem equivalently minimum cost matching in a bipartite graph can be solved in o sqrt nm log nn time where n m and n denote the number of vertices number of edges and largest magnitude of a cost costs are assumed to be integral the algorithms work by scaling as in the work of goldberg and tarjan in each scaled problem an approximate optimum solution is found rather than an exact optimum

single machine scheduling with precedence constraints we discuss the problem of sequencing precedence constrained jobs on a single machine to minimize the average weighted completion time this problem has attracted much attention in the mathematical programming community since sidney s pioneering work in 1975 sidney j b 1975 decomposition algorithms for single machine scheduling with precedence relations and deferral costs operations research23 283 298 we look at the problem from a polyhedral perspective and uncover a relation between sidney s decomposition theorem and different linear programming relaxations more specifically we present a generalization of sidney s result which particularly allows us to reason that virtually all known 2 approximation algorithms are consistent with his decomposition moreover we establish a connection between the single machine scheduling problem and the vertex cover problem indeed in the special case of series parallel precedence constraints we prove that the sequencing problem can be seen as a special case of the vertex cover problem we also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem finally we give a 3 2 approximation algorithm for two dimensional partial orders and we also provide a characterization of the active inequalities of a linear programming relaxation in completion time variables

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

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

an epsilon relaxation method for separable convex cost network flow problems we propose a new method for the solution of the single commodity separable convex cost network flow problem the method generalizes the epsilon relaxation method developed for linear cost problems and reduces to that method when applied to linear cost problems we show that the method terminates with a near optimal solution and we provide an associated complexity analysis we also present computational results showing that the method is much faster than earlier relaxation methods particularly for ill conditioned problems

a unified framework for primal dual methods in minimum cost network flow problems we introduce a broad class of algorithms for finding a minimum cost flow in a capacitated network the algorithms are of the primal dual type they maintain primal feasibility with respect to capacity constraints while trying to satisfy the conservation of flow equation at each node by means of a wide variety of procedures based on flow augmentation price adjustment and ascent of a dual functional the manner in which these procedures are combined is flexible thereby allowing the construction of algorithms that can be tailored to the problem at hand for maximum effectiveness particular attention is given to methods that incorporate features from classical relaxation procedures experimental codes based on these methods outperform by a substantial margin the fastest available primal dual and primal simplex codes on standard benchmark problems

m convex function minimization by continuous relaxation approach proximity theorem and algorithm the concept of m convexity for functions in integer variables introduced by murota adv math 124 1996 pp 272 311 plays a primary role in the theory of discrete convex analysis in this paper we consider the problem of minimizing an m convex function which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points we propose a new approach for m convex function minimization based on continuous relaxation by establishing proximity theorems we develop a new algorithm based on continuous relaxation we apply the approach to some special cases of the separable convex quadratic resource allocation problem and the convex quadratic tree resource allocation problem to obtain faster algorithms

efficiency of the primal network simplex algorithm for the minimum cost circulation problem we study the number of pivots required by the primal network simplex algorithm to solve the minimum cost circulation problem we propose a pivot selection rule with a bound of n logn 2 o 1 on the number of pivots for an n vertex network this is the first known subexponential bound the network simplex algorithm with this rule can be implemented to run in n logn 2 o 1 time in the special case of planar graphs we obtain a polynomial bound on the number of pivots and the running time we also consider the relaxation of the network simplex algorithm in which cost increasing pivots are allowed as well as cost decreasing ones for this algorithm we propose a pivot selection rule with a bound of o nm min log nc m log n on the number of pivots for a network with n vertices m arcs and integer arc costs bounded in magnitude by c the total running time is o nm log n min log nc m log n this bound is within a logarithmic factor of those of the best previously known algorithms for the minimum

algorithms for network piecewise linear programs a comparative study abstract piecewise linear programming plp is an important area of mathematical programming and concerns the minimisation of a convex separable piecewise linear objective function subject to linear constraints in this paper a subarea of plp called network piecewise linear programming nplp is explored the paper presents four specialised algorithms for nplp strongly feasible primal simplex dual method out of kilter and strongly polynomial cost scaling and their relative efficiency is studied a statistically designed experiment is used to perform a computational comparison of the algorithms the response variable observed in the experiment is the cpu time to solve randomly generated network piecewise linear problems classified according to problem class transportation transshipment and circulation problem size extent of capacitation and number of breakpoints per arc results and conclusions on performance of the algorithms are reported

diagnosing infeasibilities in network flow problems we consider the problem of finding a feasible flow in a directed networkg n a in which each nodei n has a supplyb i and each arc i j a has a zero lower bound on flow and an upper bounduij it is well known that this feasibility problem can be transformed into a maximum flow problem it is also well known that there is no feasible flow in the networkg if and only if there is a subsets of nodes such that the net supplies of the nodes ins exceeds the capacity of the arcs emanating froms such a sets is called a witness of infeasibility or simply awitness of the network flow problem in the case that there are many different witnesses for an infeasible problem a small cardinality witness may be preferable in practice because it is generally easier for the user to assimilate and may provide more guidance to the user on how to identify the cause of the infeasibility here we show that the problem of finding a minimum cardinality witness is np hard we also consider the problem of determining aminimal witness that is a witnesss such that no proper subset ofs is also a witness in this paper we show that we can determine a minimal witness by solving a sequence of at mostn maximum flow problems moreover if we use the preflow push algorithm to solve the resulting maximum flow problems and organize computations properly then the total time taken by the algorithm is comparable to that of solving a single maximum flow problem this approach determines a minimal cardinality witness in o n2m1 2 time using simple data structures and in o nm logn time using the standard implementation of the dynamic tree data structures we also show that the recognition version of the minimal witness problem is equivalent to a recognition version of a related problem known as theminimum rooted cut problem 1998 the mathematical programming society inc published by elsevier science b v

consistent structures of invariant quadrature rules for the n simplex in this paper we develop a technique to obtain in a systematic way the consistency conditions for the n dimensional simplex t n for any dimension n and degree of precision d the introduction of a convenient basis of invariant polynomials provides a powerful tool to analyze and obtain consistent structures we also present tables listing the optimal consistent structures for dimensions n 2 8 and degree of precision up to d 23 this paper is devoted only to structures no quadrature rules are presented here

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

separation dimension and facet algorithms for node flow polyhedra production planning problems such as an available to promise atp model of chen et al 2002 can involve material compatibility constraints that specify when components from various suppliers can be feasibly assembled into a final product in a companion paper to chen et al 2002 ball et al 2003 showed that in many cases such constraints can be modeled as the set of feasible source sink flows through an acyclic network the flow through a node is the sum of the flows on all paths containing it the number of paths is often exponential in the number of nodes and so it is more computationally tractable to consider the set of node flows in place of that of path flows here nodes represent components and paths represent product configurations in the context of np hard mixed integer programming models such as the atp model when the description of the set of node flows is too complicated to be explicitly written out the material compatibility constraints can be handled in a cutting plane framework by using a separation algorithm ball et al characterized the polyhedron of node flows for some special cases we extend this work in various practical and theoretical directions we allow arbitrary directed networks we allow both upper and lower bounds on flows we characterize which valid inequalities are facets we give fast algorithms for separation violation and dimension and we put all the pieces together into an algorithm for facet separation all algorithms are very efficient as they are based on max flow and min cost flow subroutines

an efficient cost scaling algorithm for the assignment problem the cost scaling push relabel method has been shown to be efficient for solving minimum cost flow problems in this paper we apply the method to the assignment problem and investigate implementations of the method that take advantage of assignment s special structure the results show that the method is very promising for practical use

approximation algorithms for the discrete time cost tradeoff problem due to its obvious practical relevance the time cost tradeoff problem has attracted the attention of many researchers over the last forty years while the linear time cost tradeoff problem can be solved in polynomial time its discrete variant is known to be np hard we present the first approximation algorithms for the discrete time cost tradeoff problem specifically given a fixed budget we consider the problem of finding a shortest schedule for a project we give an approximation algorithm with performance ratio 3 2 for the class of projects where all feasible durations of activities are either 0 1 or 2 we extend our result by giving approximation algorithms with performance guarantee olog l where l is the ratio of the maximum duration of any activity to the minimum nonzero duration of any activity finally we discuss bicriteria approximation algorithms which compute schedules for a given deadline or budget such that both project duration and cost are within a constant factor of the duration and cost of an optimum schedule for the given deadline or budget

on the computational behavior of a polynomial time network flow algorithm a variation on the edmonds karp scaling approach to the minimum cost network flow problem is examined this algorithm which scales costs rather than right hand sides also runs in polynomial time large scale computational experiments indicate that the computational behavior of such scaling algorithms may be much better than had been presumed within several distributions of square dense capacitated transportation problems a cost scaling code scale exhibits linear growth in average execution time with the number of edges while two network simplex codes rnet and gnet exhibit greater than linear growth our experiments reveal that median and mean execution times are predictable with surprising accuracy for all of the three codes and all three distributions from which test problems were generated moreover for fixed problem size individual execution times appear to behave as though they are approximately lognormally distributed with constant variance the experiments also reveal sensitivity of the parameters in the models and in the models themselves to variations in the distribution of problems this argues for caution in the interpretation of such computational studies beyond the realm in which the computations were performed

finding minimum cost circulations by canceling negative cycles a classical algorithm for finding a minimum cost circulation consists of repeatedly finding a residual cycle of negative cost and canceling it by pushing enough flow around the cycle to saturate an arc we show that a judicious choice of cycles for canceling leads to a polynomial bound on the number of iterations in this algorithm this gives a very simple strongly polynomial algorithm that uses no scaling a variant of the algorithm that uses dynamic trees runs in o nm log n min log nc m log n time on a network of n vertices m arcs and arc costs of maximum absolute value c this bound is comparable to those of the fastest previously known algorithms

the general structure of edge connectivity of a vertex subset in a graph and its incremental maintenance odd case let g v e be an undirected graph s be a subset of its vertices frak c s be the set of minimum edge cuts partitioning s and lambda s be the cardinality of such a cut we suggest a graph structure called the connectivity carcass of s that represents both cuts in frak c s and the partition of v by all these cuts its size is o min e lambda s v in this paper we present general constructions and study in detail the case lambda s odd the specifics of the case lambda s even are considered elsewhere for an adequate description of the connectivity carcass we introduce a new type of graph locally orientable graphs which generalize digraphs the connectivity carcass consists of a locally orientable quotient graph of g a cactus tree in case lambda s odd just a tree representing all distinct partitions of s by cuts in frak c s and a mapping connecting them one can build it in o s max flow computations in g for an arbitrary sequence of u edge insertions not changing lambda s the connectivity carcass can be maintained in time o v min e lambda s v u for two vertices of g queries asking whether they are separated by a cut in frak c s are answered in o 1 worst case time per query another possibility is to maintain the carcass in o s min e lambda s v u time but to answer the queries in o 1 time only if at least one of the vertices belongs to s

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

dynamic trees as search trees via euler tours applied to the network simplex algorithm thedynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges linking and splitting by deleting edges cutting while at the same time allowing reporting of certain combinations of vertex or edge values for many applications of dynamic trees values must be combined along paths for other applications values must be combined over entire trees for the latter situation an idea used originally in parallel graph algorithms to represent trees by euler tours leads to a simple implementation with a time of o logn per tree operation wheren is the number of tree vertices we apply this representation to the implementation of two versions of the network simplex algorithm resulting in a time of o logn per pivot wheren is the number of vertices in the problem network

scaling methods for finding a maximum free multiflow of minimum cost suppose we are given an undirected graph with nonnegative integer valued edge capacities and costs in which a subset of nodes is specified we consider the problem of finding a collection of flows between arbitrary pairs of specified nodes such that the capacity constraints are satisfied and the sum of costs of flows is minimum provided that the sum of values of flows is maximum it is known that this problem has a half integer optimal solution and such a solution can be found in strongly polynomial time using the ellipsoid method in this paper we give two purely combinatorial polynomial algorithms for finding a half integer optimal solution these are based on capacity and cost scaling techniques and use the double covering method earlier worked out for the problem

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

multiflows and disjoint paths of minimum total cost in this paper we discuss a number of recent and earlier results in the field of combinatorial optimization that concerns problems on minimum cost multiflows multicommodity flows and edge disjoint paths more precisely we deal with an undirected networkn consisting of a supply graphg a commodity graphh and nonnegative integer valued functions of capacities and costs of edges ofg and consider the problems of minimizing the total cost among i all maximum multiflows and ii all maximuminteger multiflows

parallel and fast sequential algorithms for undirected edge connectivity augmentation e2 such that i ei optmially increases the connectivity by for any integer the main result of the paper is that this sequence of edge sets can be divided into o n groups such that within one group all ei are basically the same using this result we improve on the running time of edge connectivity augmentation as well as we give the first parallel rnc augmentation algorithm we also present new efficient subroutines for finding the so called extreme sets and the cactus representation of min cuts required in our algorithms augmenting the connectivity of hypergraphs with ordinary edges is known to be structurally harder than that of ordinary graphs in a weaker version when one exceptional hyperedge is allowed in the augmenting edge set we derive similar results as for ordinary graphs

near optimal solutions and large integrality gaps for almost all instances of single machine precedence constrained scheduling we consider the problem of minimizing the weighted sum of completion times on a single machine subject to bipartite precedence constraints in which all minimal jobs have unit processing time and zero weight and all maximal jobs have zero processing time and unit weight for various probability distributions over these instances including the uniform distribution we show several almost all type results first we show that almost all instances are prime with respect to a well studied decomposition for this scheduling problem second we show that for almost all instances every feasible schedule is arbitrarily close to optimal finally for almost all instances we give a lower bound on the integrality gap of various linear programming relaxations of this problem

finding minimum cost circulations by successive approximation we develop a new approach to solving minimum cost circulation problems our approach combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling we measure the accuracy of a solution by the amount that the complementary slackness conditions are violated we propose a simple minimum cost circulation algorithm one version of which runs in on3lognc time on an n vertex network with integer arc costs of absolute value at most c by incorporating sophisticated data structures into the algorithm we obtain a time bound of onm logn2 mlognc on a network with m arcs a slightly different use of our approach shows that a minimum cost circulation can be computed by solving a sequence of on lognc blocking flow problems a corollary of this result is an on2log nlognc time m processor parallel minimum cost circulation algorithm our approach also yields strongly polynomial minimum cost circulation algorithms our results provide evidence that the minimum cost circulation problem is not much harder than the maximum flow problem we believe that a suitable implementation of our method will perform extremely well in practice

construction of fully symmetric cubature formulae of degree 4 k 3 for fully symmetric planar regions abstract a new method is described for the construction of cubature formulae of degree 4k 3 for two dimensional symmetric regions this method is a generalisation of the t method some formulae of degree 5 9 13 and 17 for the square the circle and the entire plane are constructed

new algorithms for maximum weight matching and a decomposition theorem we revisit the classical maximum weight matching problem in general graphs with nonnegative integral edge weights we present an algorithm that operates by decomposing the problem into w unweighted versions of the problem where w is the largest edge weight our algorithm has running time as good as the current fastest algorithms for the maximum weight matching problem when w is small one of the highlights of our algorithm is that it also produces an integral optimal dual solution thus our algorithm also returns an integral certificate corresponding to the maximum weight matching that was computed our algorithm yields a new proof to the total dual integrality of edmonds matching polytope and it also gives rise to a decomposition theorem for the maximum weight of a matching in terms of the maximum size of a matching in certain subgraphs we also consider the maximum weight capacitated b matching problem in bipartite graphs with nonnegative integral edge weights and show that it can also be decomposed i

relaxation methods for minimum cost ordinary and generalized network flow problems we propose a new class of algorithms for linear cost network flow problems with and without gains these algorithms are based on iterative improvement of a dual cost and operate in a manner that is reminiscent of coordinate ascent and gauss seidel relaxation methods we compare our coded implementations of these methods with mature state of the art primal simplex and primal dual codes and find them to be several times faster on standard benchmark problems and faster by an order of magnitude on large randomly generated problems our experiments indicate that the speedup factor increases with problem dimension

note on weintraub s minimum cost circulation algorithm in 1974 weintraub management sci 21 1974 pp 87 97 published an algorithm for the minimum cost circulation problems with convex cost function in this note weintraub s algorithm is considered when applied to a minimum cost circulation problem with linear objective function it is shown that a minor variation of the algorithm runs in polynomial time the resulting algorithm although it is not strongly polynomial does not rely on scaling it is a generalization of the maximum flow algorithm due to edmonds and karp j assoc comput mach 19 1972 pp 248 264 that augments along the fattest augmenting path in the residual graph the algorithm described here is slower than the fastest minimum cost circulation algorithms known the authors interest in the algorithm is partially historical a scaling free almost polynomial time algorithm was published in 1974 and partially due to the different ideas involved

tight mip formulations for bounded up down times and interval dependent start ups switching machines on and off is an important aspect of unit commitment problems and production planning problems among others here we study tight mixed integer programming formulations for two aspects of such problems bounded length on and off intervals and interval dependent start ups the problem with both these aspects admits a general dynamic programming shortest path formulation which leads to a tight extended formulation with a number of binary variables that is quadratic in the number n of time periods we are thus interested in tight formulations with a linear number of binary variables for the bounded interval problem we present a tight network dual formulation based on new integer cumulative variables that allows us to simultaneously treat lower and upper bounds on the interval lengths and also to handle time varying bounds this in turn leads to more general results including simpler proofs of known tight formulations for problems with just lower bounds for the interval dependent start up problem we develop a path formulation that allows us to describe the convex hull of solutions in the space of machine state variables and interval dependent start up variables

a polynomial time algorithm for rayleigh ratio on discrete variables replacing spectral techniques for expander ratio normalized cut and cheeger constant a general form of minimizing the rayleigh ratio on discrete variables is shown here for the first time to be polynomial time solvable this is significant because major problems in clustering partitioning and imaging can be presented as the rayleigh ratio minimization on discrete variables and an orthogonality constraint these challenging problems are modeled as the normalized cut problem the graph expander ratio problem the cheeger constant problem or the conductance problem all of which are np hard these problems have traditionally been solved heuristically using the spectral technique a unified framework is provided here whereby all these problems are formulated as a constrained minimization form of a quadratic ratio referred to here as the rayleigh ratio the quadratic ratio is to be minimized on discrete variables and a single sum constraint that we call the balance or orthogonality constraint when the discreteness constraints on the variables are disregarded the resulting continuous relaxation is solved by the spectral method it is shown here that the rayleigh ratio minimization subject to the discreteness constraints requiring each variable to assume one of two values in b 1 is solvable in strongly polynomial time equivalent to a single minimum s t cut algorithm on a graph of same size as the input graph for any nonnegative value of b this discrete form for the rayleigh ratio problem was often assumed to be np hard not only is it shown here that the discrete rayleigh ratio problem is polynomial time solvable but also the algorithm is more efficient than the spectral algorithm furthermore an experimental study demonstrates that the new algorithm provides in practice an improvement often dramatic on the quality of the results of the spectral method both in terms of approximating the true optimum of the rayleigh ratio problem on both the discrete variables and the balance constraint and in terms of the subjective partition quality a further contribution here is the introduction of a problem the quantity normalized cut generalizing all the rayleigh ratio problems the discrete version of that problem is also solved with the efficient algorithm presented this problem is shown in a companion paper to enable the modeling of features essential to clustering that are valuable in practical applications

approximating the single source unsplittable min cost flow problem in the single source unsplittable min cost flow problem commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity moreover the total cost must not exceed a given budget this problem has been introduced by kleinberg 7 and generalizes several np complete problems from various areas in combinatorial optimization such as packing partitioning scheduling load balancing and virtual circuit routing kolliopoulos and stein 9 and dinitz garg and goemans 4 developed algorithms improving the first approximation results of kleinberg for the problem of minimizing the violation of edge capacities and for other variants however known techniques do not seem to be capable of providing solutions without also violating the cost constraint we give the first approximation results with hard cost constraints moreover all our results dominate the best known bicriteria approximations finally we provide results on the hardness of approximation for several variants of the problem

distributed asynchronous relaxation methods for convex network flow problems we consider the solution of the single commodity strictly convex network flow problem in a distributed asynchronous computation environment the dual of this problem is unconstrained differentiable and well suited for solution via gauss seidel relaxation we show that the structure of the dual allows the successful application of a distributed asynchronous method whereby relaxation iterations are carried out in parallel by several processors in arbitrary order and with arbitrarily large interprocessor communication delays

computing minimum length representations of sets of words of uniform length motivated by text compression the problem of representing sets of words of uniform length by partial words i e sequences that may have some wildcard characters or holes was recently considered and shown to be in p polynomial time algorithms that construct representations were described using graph theoretical approaches as more holes are allowed representations shrink and if representation is given the set can be reconstructed we further study this problem by determining for a binary alphabet the largest possible value of the size of a set of partial words that is important in deciding the representability of a given set s of words of uniform length this largest value surprisingly is i 0 s 1 2 i where i is the number of ones in the binary representation of i a well studied digital sum and it is achieved when the cardinality of s is a power of two we show that circular representability is in p and that unlike non circular representability it is easy to decide we also consider the problem of computing minimum length representation circular total words those without holes and reduce it to a cost flow network problem

an o m m nlogn log nc time algorithm to solve the minimum cost tension problem this paper presents a new polynomial time algorithm to solve the minimum cost tension problem it runs in o m m nlogn log nc time where n m c denote the number of nodes number of arcs and maximum arc capacity value of an arc cost respectively the algorithm improves the o m 2nlogc time algorithm of maurras 1994 20 also our algorithm under the similarity assumption gabow 1985 12 runs in o m m nlogn logn time which improves the o n 4m 3logn time algorithm of hadjiat and maurras 1997 18

a fast algorithm for computing minimum 3 way and 4 way cuts for an edge weighted graph g with n vertices and m edges we present a new deterministic algorithm for computing a minimum k way cut for k 3 4 the algorithm runs in o n k 1 f n m o mn k log n 2 m time and o n 2 space for k 3 4 where f n m denotes the time bound required to solve the maximum flow problem in g the bound for k 3 matches the current best deterministic bound o mn 3 for weighted graphs but improves the bound o mn 3 to o n 2 f n m o min mn 8 3 m 3 2 n 2 for unweighted graphs the bound o mn 4 for k 4 improves the previous best randomized bound o n 6 for m o n 2 the algorithm is then generalized to the problem of finding a minimum 3 way cut in a symmetric submodular system

how to compute least infeasible flows it is well known how to use maximum flow to decide when a flow problem with demands lower bounds and upper bounds is infeasible less well known is how to compute a flow that is least infeasible this paper considers many possible ways to define least infeasible and shows how to compute optimal flows for each definition for each definition it also gives a dual characterization in terms of cuts a polynomial routine for recognizing that type of least infeasible flow and relates that definition to dual cut canceling min cost flow network algorithms

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

computation of moderate degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving a novel method is presented for expressing the moment equations involved in computing fully symmetric cubature rules on the triangle by using symmetric polynomials to represent the two kinds of invariance inherent in these rules this method results in a system of polynomial equations that is amenable to solution using algebraic solving techniques using grobner bases rules of degree up to 15 are computed and presented some of them new and with all their points inside the triangle since all solutions to the polynomial system are computed it is for the first time possible to prove whether a given rule type results in specific rules of a given quality it is thus proved that for degrees up to 14 there are no non fortuitous rules that can improve on the presented results for degree 10 an example is also provided showing how the proposed method can be used to exclude the existence of better fortuitous rules as well

on augmentation algorithms for linear and integer linear programming from edmonds karp to bland and beyond motivated by bland s linear programming lp generalization of the renowned edmonds karp efficient refinement of the ford fulkerson maximum flow algorithm we analyze three closely related natural augmentation rules for lp and integer linear programming ilp augmentation algorithms for all three rules and in both contexts lp and ilp we bound the number of augmentations extending bland s discrete steepest descent augmentation rule i e choosing directions with the best ratio of cost improvement per unit 1 norm length and then making maximal augmentations in such directions from lp to ilp we i show that the number of discrete steepest descent augmentations is bounded by the number of elements in the graver basis of the problem matrix and ii give the first strongly polynomial time algorithm for n fold ilp for lp two of the rules can suffer from a zig zagging phenomenon and so in those cases we apply the rules more subtly to achieve good bounds our results improve on what is known fo

a randomized maximum flow algorithm a randomized algorithm for computing a maximum flow is presented for an n vertex m edge network the running time is o nm n 2 log n 2 with probability at least 1 2 sqrt nm the algorithm is always correct and in the worst case runs in o nm log n time the only use of randomization is to randomly permute the adjacency lists of the network vertices at the start of the execution the analysis introduces the notion of premature target relabeling ptr events and shows that each ptr event contributes o log n amortized time to the overall running time the number of ptr events is always o nm however it is shown that when the adjacency lists are randomly permuted then this quantity is o n 3 2 m 1 2 n 2 log n with high probability

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations

orthogonal polynomials and cubature formulae on balls simplices and spheres we report on recent developments on orthogonal polynomials and cubature formulae on the unit ball bd the standard simplex td and the unit sphere sd the main result shows that orthogonal structures and cubature formulae for these three regions are closely related this provides a way to study the structure of orthogonal polynomials for example it allows us to use the theory of h harmonics to study orthogonal polynomials on bd and on td it also provides a way to construct new cubature formulae on these regions

an o n 2 m n log n log n min cost flow algorithm the minimum cost flow problem is given a network with n vertices and m edges find a maximum flow of minimum cost many network problems are easily reducible to this problem a polynomial time algorithm for the problem has been known for some time but only recently a strongly polynomial algorithm was discovered in this paper an o n 2 m n log n log n algorithm is designed the previous best algorithm due to fujishige and orlin had an o m 2 m n log n log n time bound thus for dense graphs an improvement of two orders of magnitude is obtained the algorithm in this paper is based on fujishige s algorithm which is based on tardos s algorithm fujishige s algorithm consists of up to m iterations each consisting of o m log n steps each step solves a single source shortest path problem with nonnegative edge lengths this algorithm is modified in order to make an improved analysis possible the new algorithm may still consist of up to m iterations and an iteration may still consist of up to o m log n steps but it can still be shown that the total number of steps is bounded by o n 2 log n the improvement is due to a new technique that relates the time spent to the progress achieved

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs