a combined phase i phase ii scaled potential algorithm for linear programming we develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program without the addition of any m terms the method and its lower bounding procedure are particularly simple compared with previous interior algorithms not requiring feasibility

on anstreicher s combined phase i phase ii projective algorithm for linear programming anstreicher has proposed a variant of karmarkar s projective algorithm that handles standard form linear programming problems nicely we suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm much of the analysis depends on a two constraint linear programming problem that is a relaxation of the scaled original problem

a complete algorithm for linear fractional programs abstract the linear fractional programming lfp algorithms attempt to optimize a quotient of two linear functions subject to a set of linear constraints the existing lfp algorithms are problem dependent and none is superior to others in all cases these algorithms explicitly require i the denominator of the objective function does not vanish in the feasible region ii the denominator of the objective function is positive iii the feasible region is bounded moreover some of these algorithms fail whenever iv some constraints are redundant we present a simplex type algorithm which is compact and efficiently detects conditions i iii and relaxes assumption iv the proposed algorithm is evolutionary in the sense that it builds up in a systematic manner to solve any lfp type problems numerical examples illustrate the algorithm

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

strict monotonicity and improved complexity in the standard form projective algorithm for linear programming in a recent paper shaw and goldfarb show that a version of the standard form projective algorithm can achieve o left sqrt nl right step complexity as opposed to the o nl step complexity originally demonstrated for the algorithm the analysis of shaw and goldfarb shows that the algorithm using a constant fixed steplength approximately follows the central trajectory in this paper we show that simple modifications of the projective algorithm obtain the same complexity improvement while permitting a linesearch of the potential function on each step an essential component is the addition of a single constraint motivated by shaw and goldfarb s analysis which makes the standard form algorithm strictly monotone in the true objective

polynomial kernels for proper interval completion and related problems given a graph g v e and a positive integer k the proper interval completion problem asks whether there exists a set f of at most k pairs of vxv e such that the graph h v e f is a proper interval graph the proper interval completion problem finds applications in molecular biology and genomic research this problem is known to be fpt kaplan tarjan and shamir focs 94 but no polynomial kernel was known to exist we settle this question by proving that proper interval completion admits a kernel with o k 3 vertices moreover we prove that a related problem the so called bipartite chain deletion problem admits a kernel with o k 2 vertices completing a previous result of guo isaac 07

conical projection algorithms for linear programming the linear programming problem is manipulated to be stated as a non linear programming problem in which karmarkar s logarithmic potential function is minimized in the positive cone generated by the original feasible set the resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non linear programming algorithm several different procedures for the internal algorithm are proposed giving priority either to the reduction of the potential function or of the actual cost we show that karmarkar s algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration all variants of the new algorithm have the same complexity as karmarkar s method but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm

a survey of search directions in interior point methods for linear programming a basic characteristic of an interior point algorithm for linear programming is the search direction many papers on interior point algorithms only give an implicit description of the search direction in this report we derive explicit expressions for the search directions used in many well known algorithms comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms moreover we give a survey of projected gradient and newton directions for all potential and barrier functions this is done both for the affine and projective variants

interior point algorithms for linear programming with inequality constraints interior methods for linear programming were designed mainly for problems formulated with equality constraints and non negative variables the formulation with inequality constraints has shown to be very convenient for practical implementations and the translation of methods designed for one formulation into the other is not trivial this paper relates the geometric features of both representations shows how to transport data and procedures between them and shows how cones and conical projections can be associated with inequality constraints

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

segmenting multisensor aerial images in class scale space abstract we introduce a class scale space for automated segmentation of aerial images from diverse sensors we describe the performance of a segmentation algorithm operating in this space this algorithm is carried out by three processors in succession multiscale feature extractor multiclass pattern classifier and class scale logic the multiscale feature extractor extracts features at three levels of precision the multiclass pattern classifier is an array of neural classifiers one classifier for each class scale pair each classifier is optimized by genetic feature selectors and genetic weight initiators the array of neural classifiers produces an array of segmented images class scale logic operates on these images to produce a final segmented image that combines coarsely detected regions with finely detected curves and points we describe applications of these techniques to segmenting and matching sar ir and visible light images

the affine scaling direction for linear programming is a limit of projective scaling directions abstract there are two ways to convert a standard form linear programming problem to a form suitable for application of a projective scaling interior point algorithm based on karmarkar s method one of these adds a dummy variable which is identically one we show that as the number of dummy variables all identically one added tends to infinity the resulting direction in all the original variables tends to the direction chosen by dikin s affine scaling algorithm

on lower bound updates in primal potential reduction methods for linear programming we present a procedure for computing lower bounds for the optimal cost in a linear programming problem whenever the procedure succeeds it finds a dual feasible slack and the associated duality gap although no projective transformations or problem restatements are used the method coincides with the procedures by todd and burrell and by de ghellinck and vial when these procedures are applicable the procedure applies directly to affine potential reduction algorithms and improves on existent techniques for finding lower bounds

a potential function reduction algorithm for solving a linear program directly from an infeasible warm start this paper develops an algorithm for solving a standard form linear program directly from an infeasible warm start i e directly from a given infeasible solution hat x that satisfies a hat x b but hat x ngeqslant 0 the algorithm is a potential function reduction algorithm but the potential function is somewhat different than other interior point method potential functions and is given by f x b q ln c t x b sum limits j 1 n ln x j h j c t x b where q n sqrt n is a given constant h is a given strictly positive shift vector used to shift the nonnegativity constaints andb is a lower bound on the optimal value of the linear program the duality gapc t x b is used both in the leading term as well as in the barrier term to help shift the nonnegativity constraints the algorithm is shown under suitable conditions to achieve a constant decrease in the potential function and so achieves a constant decrease in the duality gap and hence also in the infeasibility in o n iterations under more restrictive assumptions regarding the dual feasible region this algorithm is modified by the addition of a dual barrier term and will achieve a constant decrease in the duality gap and in the infeasibility in o sqrt n iterations

an o n l iteration bound primal dual cone affine scaling algorithm for linear programming in this paper we introduce a primal dual affine scaling method the method uses a search direction obtained by minimizing the duality gap over a linearly transformed conic section this direction neither coincides with known primal dual affine scaling directions jansen et al 1993 monteiro et al 1990 nor does it fit in the generic primal dual method kojima et al 1989 the new method requires o sqrt n l main iterations it is shown that the iterates follow the primal dual central path in a neighbourhood larger than the conventional mathcal n 2 neighbourhood the proximity to the primal dual central path is measured by trigonometric functions

global convergence property of the affine scaling methods for primal degenerate linear programming problems in this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy the behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence for dual nondegenerate lp problems the result can be regarded as a counterpart to dikin s global convergence result on the affine scaling method assuming primal nondegeneracy

the interval merging problem abstract a closed interval is an ordered pair of real numbers x y with x y the interval x y represents the set i r x i y given a set of closed intervals i a 1 b 1 a 2 b 2 a k b k the interval merging problem is to find a minimum cardinality set of intervals m i x 1 y 1 x 2 y 2 x j y j j k such that the real numbers represented by i i 1 k a i b i equal those represented by m i i 1 j x i y i in this paper we show the problem can be solved in o d log d sequential time and in o log d parallel time using o d processors on an erew pram where d is the number of the endpoints of i moreover if the input is given as a set of sorted endpoints then the problem can be solved in o d sequential time and in o log d parallel time using o d log d processors on an erew pram

solving combinatorial optimization problems using karmakar s algorithm we describe a cutting plane algorithm for solving combinatorial optimization problems the primal projective standard form variant of karmarkar s algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem

convergence and boundary behavior of the projective scaling trajectories for linear programming we analyze the convergence and boundary behavior of the continuous trajectories of the vector field induced by the projective scaling algorithm as applied to possibly degenerate linear programming problems in karmarkar s standard form we show that a projective scaling trajectory tends to an optimal solution which in general depends on the starting point when the optimal solution is unique we show that all projective scaling trajectories approach the optimal solution through the same asymptotic direction our analysis is based on the affine scaling trajectories for the homogeneous standard form linear program that arises from karmarkar s standard form linear program by removing the unique nonhomogeneous constraint

an extension of the potential reduction algorithm for linear complementarity problems with some priority goals abstract we extend the potential reduction algorithm to solve the restricted convex linear complementarity problem lcp x t s 0 x i 0 s j 0 s mx q and 0 x s r n for some index sets i and j in polynomial time the algorithm will either discover that no solution exists or generate a sequence of pairs x k s k which simultaneously drives x k t s k x k i and s k j to zero in particular we discuss how to apply the algorithm to solving the combined phase i phase ii convex lcp and to identifying a vertex linear programming solution

containing and shrinking ellipsoids in the path following algorithm we describe a new potential function and a sequence of ellipsoids in the path following algorithm for convex quadratic programming each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors furthermore the volumes of the ellipsoids shrink at the ratio 2 omega sqrt n in comparison to 2 1 in karmarkar s algorithm and 2 1 n in the ellipsoid method we also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming

a fully dynamic algorithm for recognizing and representing proper interval graphs in this paper we study the problem of recognizing and representing dynamically changing proper interval graphs the input to the problem consists of a series of modifications to be performed on a graph where a modification can be a deletion or an addition of a vertex or an edge the objective is to maintain a representation of the graph as long as it remains a proper interval graph and to detect when it ceases to be so the representation should enable one to efficiently construct a realization of the graph by an inclusion free family of intervals this problem has important applications in physical mapping of dna we give a near optimal fully dynamic algorithm for this problem it operates in o log n worst case time per edge insertion or deletion we prove a close lower bound of omega log n log log n log b amortized time per operation in the cell probe model with word size b we also construct optimal incremental and decremental algorithms for the problem which handle each edge operation in o 1 time as a byproduct of our algorithm we solve in o log n worst case time the problem of maintaining connectivity in a dynamically changing proper interval graph

improved bounds and containing ellipsoids in karmarkar s linear programming algorithm karmarkar s projective algorithm for linear programming provides not only primal solutions but dual solutions giving bounds on the optimal value here we show how improved bounds can be obtained at the expense of solving a two dimensional linear programming problem at every iteration and also how an ellipsoid containing all dual optimal solutions can be generated from available information we also give the results of limited computational experiments related to these topics

on monotonicity in the scaled potential algorithm for linear programming in this note we show that a simple modification of ye s affinely scaled potential reduction algorithm makes the method monotone in the true objective on primal steps based on computational experience with the standard form projective algorithm the monotonicity modification should substantially improve the performance of the algorithm when it is initialized with a lower bound much less than the optimal objective value imposing monotonicity on primal steps also results in stronger lower bound updates which is not the case with the standard form projective algorithm

complexity models for incremental computation def nrp hbox rm protect small nrp def poly hbox rm protect small p def nick hbox rm protect small nc def incrplt hbox rm it incr allowbreak protect small polylogtime def logsp hbox rm protect small logspace def incrls hbox rm it incr allowbreak protect small logspace def rincrls hbox rm it rincr allowbreak protect small logspace def incrpls hbox rm it incr allowbreak protect small polylogspace def nlgsp hbox rm protect small nlogspace we present a new complexity theoretic approach to incremental computation we define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes we show that problems that have small sequential space complexity also have small incremental time complexity we show that all common logsp complete problems for poly are also incrplt complete for poly we introduce a restricted notion of completeness called nrp completeness and show that problems which are nrp complete for poly are also incrplt complete for poly we also give incrementally complete problems for nlgsp logsp and non uniform nick 1 we show that under certain restrictions problems which have efficient dynamic solutions also have efficient parallel solutions we also consider a non uniform model of incremental computation and show that in this model most problems have almost linear complexity in addition we present some techniques for lower bounding the complexity of explicitly defined problems we also look at the time complexity of circuit value and network stability problems restricted to comparator gates we show that the comparator circuit value problem and the lex first maximal matching problem are in incrls while the comparator network stability and the man optimal stable marriage problem are in rincrls nlgsp this shows that the dynamic versions of these problems are solvable quickly in parallel even though there are no known nick algorithms to solve them from scratch

a polynomial solution to the k fixed endpoint path cover problem on proper interval graphs we study a variant of the path cover problem namely the k fixed endpoint path cover problem or kpc for short given a graph g and a subset t of k vertices of v g a k fixed endpoint path cover of g with respect to t is a set of vertex disjoint paths p that covers the vertices of g such that the k vertices of t are all endpoints of the paths in p the kpc problem is to find a k fixed endpoint path cover of g of minimum cardinality note that if t is empty or equivalently k 0 the stated problem coincides with the classical path cover problem the kpc problem generalizes some path cover related problems such as the 1hp and 2hp problems which have been proved to be np complete note that the complexity status for both 1hp and 2hp problems on interval graphs remains an open question damaschke 1993 9 in this paper we show that the kpc problem can be solved in linear time on the class of proper interval graphs that is in o n m time on a proper interval graph on n vertices and m edges the proposed algorithm is simple requires linear space and also enables us to solve the 1hp and 2hp problems on proper interval graphs within the same time and space complexity

a relaxed revision of karmarkar s method a relaxed version of karmarkar s method is developed this method is proved to have the same polynomial time complexity as karmarkar s method and its efficient implementation using inexact projections is discussed computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable

following a balanced trajectory from an infeasible point to an optimal linear programming solution with a polynomial time algorithm this paper is concerned with the problem of following a trajectory from an infeasible starting point directly to an optimal solution of the linear programming problem a class of trajectories for the problem is defined based on the notion of a balanced solution to the problem given a prespecified positive balancing constant an infeasible solution x is said to be balanced if the optimal value gap is less than or equal to times the infeasibility gap mathematically this can be written as ctx z tx where the linear form tx is the phase i objective function and z is the optimal objective value of the linear program the concept of a balanced solution is used to define a class of trajectories from an infeasible point to an optimal solution of a given linear program each trajectory has the property that all points on or near the trajectory in a suitable metric are balanced the main thrust of the paper is the development of an algorithm that traces a given balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial time more specifically the algorithm allows for fixed improvement in the bound on the phase i and the phase ii objectives in on newton steps

fully dynamic representations of interval graphs abstract we present a fully dynamic algorithm that maintains three different representations of an interval graph a minimal interval model of the graph the pq tree of its maximal cliques and its modular decomposition after each vertex or edge modification insertion or deletion the algorithm determines whether the new graph is an interval graph in o n time and in the positive updates the three representations within the same complexity furthermore we state that the modular decomposition tree of an interval graph and the pq tree of its maximal cliques are algorithmically equivalent and we completely explicit the relationship between the two structures

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

should tables be sorted we examine optimality questions in the following information retrieval problem given a set s of n keys store them so that queries of the form is x in s can be answered quickly it is shown that in a rather general model including al1 the commonly used schemes lceil lg n l rceil probes to the table are needed in the worst case provided the key space is sufficiently large the effects of smaller key space and arbitrary encoding are also explored

global convergence of the affine scaling methods for degenerate linear programming problems in this paper we show the global convergence of the affine scaling methods without assuming any condition on degeneracy the behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence of the affine scaling methods

a dantzig wolfe like variant of karmarkar s interior point linear programming algorithm we show that a variant of karmarkar s projective algorithm for linear programming can be viewed as following the approach of dantzig wolfe decomposition at each iteration the current primal feasible solution generates prices which are used to form a simple subproblem the solution to the subproblem is then incorporated into the current feasible solution with a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration we also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by e klotz and l schrage

sparsification a technique for speeding up dynamic graph algorithms we provide data strutures that maintain a graph as edges are inserted and deleted and keep track of the following properties with the following times minimum spanning forests graph connectivity graph 2 edge connectivity and bipartiteness in time o n 1 2 per change 3 edge connectivity in time o n 2 3 per change 4 edge connectivity in time o n n per change k edge connectivity for constant k in time o n log n per change 2 vertex connectivity and 3 vertex connectivity in the o n per change and 4 vertex connectivity in time o n n per change further results speed up the insertion times to match the bounds of known partially dynamic algorithms all our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph which we call sparsification

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

a fully dynamic algorithm for the recognition of p4 sparse graphs in this paper we solve the dynamic recognition problem for the class of p 4 sparse graphs the objective is to handle edge vertex additions and deletions to recognize if each such modification yields a p 4 sparse graph and if yes to update a representation of the graph our approach relies on maintaining the modular decomposition tree of the graph which we use for solving the recognition problem we establish properties for each modification to yield a p 4 sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in o 1 time and vertex modifications in o d time for a vertex of degree d thus our algorithm implies an optimal edges only dynamic algorithm and a new optimal incremental algorithm for p 4 sparse graphs

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

a combined phase i phase ii projective algorithm for linear programming we devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution inclusion of such a constraint allows the algorithm to be applied to a possibly infeasible standard form linear program without the addition of any bigm terms or conversion to a primal dual problem

convergence of interior point algorithms for the monotone linear complementarity problem the literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values but very little is known about the convergence of the iterate sequences this paper studies the horizontal linear complementarity problem and derives general convergence properties for algorithms based on newton iterations this problem provides a simple and general framework for most existing primal dual interior point methods the conclusion is that most of the published algorithms of this kind generate convergent sequences in many cases whenever the convergence is not too fast in a certain sense the sequences converge to the analytic center of the optimal face

a potential reduction algorithm with user specified phase i phase ii balance for solving a linear program from an infeasible warm start this paper develops a potential reduction algorithm for solving a linear programming problem directly from a warm start initial point that is neither feasible nor optimal the algorithm is an interior point variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement phase i and objective value improvement phase ii the key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense given a prespecified balancing parameter beta 0 the algorithm maintains the following phase i phase ii beta balancing constraint throughout left c t x z right beta xi t x where c t x is the objective function z is the unknown optimal objective value of the linear program and xi t x measures the infeasibility of the current iterate x this balancing constraint can be used to either emphasize rapid attainment of feasibility set beta large

convergence analysis of the projective scaling algorithm based on a long step homogeneous affine scaling algorithm in this paper we give a new convergence analysis of a projective scaling algorithm we consider a long step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem this algorithm takes a fixed fraction 2 3 of the way towards the boundary of the nonnegative orthant at each iteration the iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection which generates the same search direction as the original projective scaling algorithm at each iterate the recent convergence results for the long step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm specifically we will show i polynomiality of the algorithm with complexities of o nl and o n2l iterations for 2 3 and 2 3 respectively ii global covnergence of the algorithm when the optimal face is unbounded iii convergence of the primal iterates to a relative interior point of the optimal face iv convergence of the dual estimates to the analytic center of the dual optimal face and v convergence of the reduction rate of the objective function value to 1

theoretical efficiency of a shifted barrier function algorithm for linear programming abstract this paper examines the theoretical efficiency of solving a standard form linear program by solving a sequence of shifted barrier problems of the form minimize x c t x e j 1 n ln x j eh j s t a x b x eh 0 for a given and fixed shift vector h 0 and for a sequence of values of 0 that converges to zero the resulting sequence of solutions to the shifted barrier problems will converge to a solution to the standard form linear program the advantage of using the shifted barrier approach is that a starting feasible solution is unnecessary and there is no need for a phase i phase ii approach to solving the linear program either directly or through the addition of an artificial variable furthermore the algorithm can be initiated with a warm start i e an initial guess of a primal solution x that need not be feasible the number of iterations needed to solve the linear program to a desired level of accuracy will depend on a measure of how close the initial solution x is to being feasible and optimal the number of iterations will also depend on the judicious choice of the shift vector h if an approximate center of the dual feasible region is known in advance then h can be chosen so that the guaranteed fractional decrease in at each iteration is 1 1 6 n which contributes afactor of 6 n to the number of iterations needed to solve the problem

average case complexity results for a centering algorithm for linear programming problems under gaussian distributions abstract to solve linear programming problems by interior point methods an approximately centered interior point has to be known such a point can be found by an algorithmic approach a so called phase 1 algorithm or centering algorithm for random linear programming problems distributed according to the rotation symmetry model especially with normal distribution we present probabilistic results on the quality of the origin as starting point and the average number of steps of a centering algorithm

optimal greedy algorithms for indifference graphs abstract a fundamental problem in social sciences and management is understanding and predicting decisions made by individuals various groups or the society as a whole in this context one important concept is the notion of indifference we characterize the class of indifference graphs that is graphs which arise in the process of quantifying indifference relations in particular we show that these graphs are characterized by the existence of a special ordering of their vertices as it turns out this ordering leads naturally to optimal greedy algorithms for a number of computational problems including coloring finding a shortest path between two vertices computing a maximum matching the center and a hamiltonian path

interval graphs canonical representations in logspace we present a logspace algorithm for computing a canonical labeling in fact a canonical interval representation for interval graphs to achieve this we compute canonical interval representations of interval hypergraphs this approach also yields a canonical labeling of convex graphs as a consequence the isomorphism and automorphism problems for these graph classes are solvable in logspace for proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems