combinatorial interior point methods for generalized network flow problems we present combinatorial interior point methods for the generalized minimum cost flow and the generalized circulation problems based on wallacher and zimmermann s combinatorial interior point method for the minimum cost network flow problem the algorithms have features of both a combinatorial algorithm and an interior point method they work towards optimality by iteratively reducing the value of a potential function while maintaining interior point solutions at each iteration flow is augmented along a generalized circulation which is computed by solving a tvpi two variables per inequality system the algorithms run in time where m and n are respectively the number of arcs and nodes in the graph and l is the length of the input data

refined proximity and sensitivity results in linearly constrained convex separable integer programming abstract we define a new measure of the structure of a linear constraint matrix and establish some propertie we then use the measure to refine proximity and sensitivity results in the literature in particular we refine bounds on the distance between continuous and discrete optima of linearly constrained convex separable programming problems and on the size of the feasible region for a linear program we apply the results to improve the computational complexity of hochbaum and shanthikumar s algorithm for integer nonlinear programming reducing the computational effort for certain problems from exponential to polynomial

convex separable optimization is not much harder than linear optimization the polynomiality of nonlinear separable convex concave optimization problems on linear constraints with a matrix with small subdeterminants and the polynomiality of such integer problems provided the inteter linear version of such problems ins polynomial is proven this paper presents a general purpose algorithm for converting procedures that solves linear programming problems the conversion is polynomial for constraint matrices with polynomially bounded subdeterminants among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix to integer separable convex programming an algorithm for finding a e accurate optimal continuous solution to the nonlinear problem that is polynomial in log 1 e and the input size and the largest subdeterminant of the constraint matrix is also presented these developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function the practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function only a polynomial number of function evaluations on a prespecified grid

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

computational complexity of a cost allocation approach to a fixed cost spanning forest problem we present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem referred to as the fixed cost spanning forest fcsf problem the customers in the fcsf problem represented by nodes in a graph g are in need of service that can be produced at some facilities yet to be constructed the cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers we formulate this cost allocation problem as a cooperative game referred to as the fcsf game in general the core of a fcsf game may be empty however for the case when g is a tree it is shown that the core is not empty moreover we prove that in this case core points can be generated in strongly polynomial time we further provide a nonredundant characterization of the core of the fcsf game defined over a tree in the special case when all nodes are communities this is shown to lead in some instances to a strongly polynomial algorithm for computing the nucleolus

linear programming relaxations and belief propagation an empirical study the problem of finding the most probable map configuration in graphical models comes up in a wide range of applications in a general graphical model this problem is np hard but various approximate algorithms have been developed linear programming lp relaxations are a standard method in computer science for approximating combinatorial problems and have been used for finding the most probable assignment in small graphical models however applying this powerful method to real world problems is extremely challenging due to the large numbers of variables and constraints in the linear program tree reweighted belief propagation is a promising recent algorithm for solving lp relaxations but little is known about its running time on large problems in this paper we compare tree reweighted belief propagation trbp and powerful general purpose lp solvers cplex on relaxations of real world graphical models from the fields of computer vision and computational biology we find that trbp almost always finds the solution significantly faster than all the solvers in cplex and more importantly trbp can be applied to large scale problems for which the solvers in cplex cannot be applied using trbp we can find the map configurations in a matter of minutes for a large range of real world problems

a simple variant of the mizuno todd ye predictor corrector algorithm and its objective function free complexity in this paper we propose a simple variant of the mizuno todd ye predictor corrector algorithm for linear programming our variant employs a simple and a natural scaling invariant search directio

bimonotone linear inequalities and sublattices of rn abstract a bimonotone linear inequality is a linear inequality with at most two nonzero coefficients that are of opposite signs if both different from zero a linear inequality defines a halfspace that is a sublattice of r n a subset closed with respect to componentwise maximum and minimum if and only if it is bimonotone veinott has shown that a polyhedron is a sublattice if and only if it can be defined by a finite system of bimonotone linear inequalities whereas topkis has shown that every sublattice of r n and of more general product lattices is the solution set of a system of nonlinear bimonotone inequalities in this paper we prove that a subset of r n is the solution set of a countable system of bimonotone linear inequalities if and only if it is a closed convex sublattice similarly we note that a subset of r n is closed and convex if and only if it is the solution set of a countable system of linear inequalities we also present necessary and or sufficient conditions for a sublattice to be the intersection of the cartesian product of its projections on the coordinate axes with the solution set of a possibly infinite system of bimonotone linear inequalities we provide explicit constructions of such systems of bimonotone linear inequalities under certain assumptions on the sublattice we obtain veinott s polyhedral representation theorem and a 0 1 version of birkhoff s representation theorem as corollaries we also point out a few potential pitfalls regarding properties of sublattices of r n

temporal constraint networks abstract this paper extends network based methods of constraint satisfaction to include continuous variables thus providing a framework for processing temporal constraints in this framework called temporal constraint satisfaction problem tcsp variables represent time points and temporal information is represented by a set of unary and binary constraints each specifying a set of permitted intervals the unique feature of this framework lies in permitting the processing of metric information namely assessments of time differences between events we present algorithms for performing the following reasoning tasks finding all feasible times that a given event can occur finding all possible relationships between two given events and generating one or more scenarios consistent with the information provided we distinguish between simple temporal problems stps and general temporal problems the former admitting at most one interval constraint on any pair of time points we show that the stp which subsumes the major part of vilain and kautz s point algebra can be solved in polynomial time for general tcsps we present a decomposition scheme that performs the three reasoning tasks considered and introduce a variety of techniques for improving its efficiency we also study the applicability of path consistency algorithms as preprocessing of temporal problems demonstrate their termination and bound their complexities

integer programs for logic constraint satisfaction abstract logic constraint satisfaction problems are in general np hard and a general deterministic polynomial time algorithm is not known since several logic constraint problems can be reduced in polynomial time to the satisfaction of a conjunctive normal form cnf sat this case is very important we present here a technique to transform a cnf sat problem in an integer optimization problem that can be solved by linear programming the size of the obtained integer program has a polynomial growth in comparison with the original problem size

a two resource allocation problem solvable in linear time the allocation problem discussed here is as follows the processing time of a task is a linear function t ax by of amounts x y of resources allocated to it where t a b are positive constants given the amounts x and y of available resources we wish to allocate them among n independent tasks so as to minimize the maximal completion time we solve this problem in linear time

thermodynamic cost of acquiring information connections between information theory and thermodynamics have proven to be very useful to establish bounding limits for physical processes ideas such as landauer s erasure principle and information assisted work extraction have greatly contributed not only to broadening our understanding about the fundamental limits imposed by nature but also paving the way for practical implementations of information processing devices the intricate information thermodynamics relation also entails a fundamental limit on parameter estimation establishing a thermodynamic cost for information acquisition we show that the amount of information that can be encoded in a physical system by means of a unitary process is limited by the dissipated work during the implementation of the process this includes a thermodynamic tradeoff for information acquisition likewise the information acquisition process is ultimately limited by the second law of thermodynamics this tradeoff for information acquisition may find applications in several areas of knowledge

fluctuation theorems work is not an observable the characteristic function of the work performed by an external time dependent force on a hamiltonian quantum system is identified with the time ordered correlation function of the exponentiated system s hamiltonian a similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the jarzynski work theorem

an algorithm of internal feasible directions for linear integer programming abstract an approximate algorithm for solving pure integer problems is examined the algorithm belongs to the class of component algorithms using feasible integer directions the basic characteristics of the algorithm are analyzed testing on an ibm pc at personal computer is carried out the proposed algorithm is compared with other well known approximate algorithms

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 simpler and faster strongly polynomial algorithm for generalized flow maximization we present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm 34 for the uncapacitated problem formulation the complexity bound o mn m n log n log n2 m improves on the previous estimate by almost a factor o n2 even for small numerical parameter values our running time bound is comparable to the best weakly polynomial algorithms the key new technical idea is relaxing the primal feasibility conditions this allows us to work almost exclusively with integral flows in contrast to all previous algorithms for the problem

locating tree shaped facilities using the ordered median objective in this paper we consider the location of a tree shaped facility s on a tree network using the ordered median function of the weighted distances to represent the total transportation cost objective this function unifies and generalizes the most common criteria used in location modeling e g median and center if there are n demand points at the nodes of the tree t v e this function is characterized by a sequence of reals 1 n satisfying 1 2 n 0 for a given subtree s let x s x1 xn be the set of weighted distances of the n demand points from s the value of the ordered median objective at s is obtained as follows sort the n elements in x s in nonincreasing order and then compute the scalar product of the sorted list with the sequence two models are discussed in the tactical model there is an explicit bound l on the length of the subtree and the goal is to select a subtree of size l which minimizes the above transportation cost function in the strategic model the length of the subtree is variable and the objective is to minimize the sum of the transportation cost and the length of the subtree we consider both discrete and continuous versions of the tactical and the strategic models we note that the discrete tactical problem is np hard and we solve the continuous tactical problem in polynomial time using a linear programming lp approach we also prove submodularity properties for the strategic problem these properties allow us to solve the discrete strategic version in strongly polynomial time moreover the continuous version is also solved via lp for the special case of the k centrum objective we obtain improved algorithmic results using a dynamic programming dp algorithm and discretization and nestedness results

querying temporal and spatial constraint networks in ptime we start with the assumption that temporal and spatial knowledge usually captured by constraint networks can be represented and queried more effectively by using the scheme of indefinite constraint databases because query evaluation in this scheme is in general a hard computational problem we seek tractable instances of query evaluation we assume that we have a class of constraints c with some reasonable computational and closure properties the computational properties of interest are that the satisfiability problem and an appropriate version of the variable elimination problem for c should be solvable in ptime under this assumption we exhibit general classes of indefinite constraint databases and first order modal queries for which query evaluation can be done with ptime data complexity we then search for tractable instances of c among the subclasses of horn disjunctive linear constraints over the rationals from previous research we know that the satisfiability problem for horn disjunctive linear constraints is solvable in ptime but not the variable elimination problem thus we try to discover subclasses of horn disjunctive linear constraints with tractable variable elimination problems the class of utvpi constraints is the largest class that we show to have this property finally we restate our general tractability results with c ranging over the newly discovered tractable classes interesting tractable query answering problems for indefinite temporal and spatial constraint databases are identified in this way we close our complexity analysis by precisely outlining the frontier between tractable and possibly intractable query answering problems

testing the necklace condition for shortest tours and optimal factors in the plane a tour of a finite set p of points is a necklace tour if there are disks with the points in p as centers such that two disks intersect if and only if their centers are adjacent in it has been observed by sanders that a necklace tour is an optimal traveling salesman tour

a submodular optimization problem with side constraints in this paper we consider the general problem of optimizing over the intersection of a submodular base polyhedron and an affine space an example is the following flow problem defined on a capacitated network we wish to send a commodity from locations in a producing country to locations in a number of client countries so as to simultaneously maximize profits and send to each client country a prescribed proportion of the total sent we present an algorithm for the general problem and analyze its complexity

a polynomial combinatorial algorithm for generalized minimum cost flow we propose the first combinatorial solution to the generalized minimum cost flow problem flow with losses and gains despite a rich history dating back to kantorovich and dantzig until now the only known way to solve the problem in polynomial time was via general purpose linear programming techniques polynomial combinatorial algorithms were previously known only for the version of our problem without costs we design the first such algorithms for the version with costs our algorithms also find provably good solutions faster than optimal ones providing the first strongly polynomial approximation schemes for the problem our techniques extend to optimize linear programs with two variables per inequality polynomial combinatorial algorithms were previously developed for testing the feasibility of such linear programs we propose the first such methods for the optimization version

on some balanced totally balanced and submodular delivery games this paper studies a class of delivery problems associated with the chinese postman problem and a corresponding class of delivery games a delivery problem in this class is determined by a connected graph a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office it is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server who starts at the post office and is expected to traverse all edges in the graph before returning to the post office a graph g is called chinese postman submodular or for short cp submodular cp totally balanced cp balanced respectively if for each delivery problem in which g is the underlying graph the associated delivery game is submodular totally balanced balanced respectively for undirected graphs we prove that cp submodular graphs and cp totally balanced graphs are weakly cyclic graphs and conversely an undirected graph is shown to be cp balanced if and only if it is a weakly euler graph for directed graphs cp submodular graphs can be characterized by directed weakly cyclic graphs further it is proven that any strongly connected directed graph is cp balanced for mixed graphs it is shown that a graph is cp submodular if and only if it is a mixed weakly cyclic graph finally we note that undirected directed and mixed weakly cyclic graphs can be recognized in linear time

an iterative algorithm for two level hierarchical time minimization transportation problem abstract this paper discusses a two level hierarchical time minimization transportation problem in which the whole set of source destination links consists of two disjoint partitions namely level i and level ii links some quantity of a homogeneous product is first shipped from sources to destinations by level i decision makers using only level i links and on its completion the level ii decision maker transports the remaining quantity of the product in an optimal fashion using only level ii links the objective is to find that feasible solution for level i decision corresponding to which the optimal feasible solution for level ii decision maker is such that the sum of shipment times in level i and level ii is minimum a polynomial time iterative algorithm is proposed to solve the two level hierarchical time minimization transportation problem at each iteration a lexicographic optimal solution of a restricted version of a related standard time minimization transportation problem is examined to generate a pair of level i and level ii shipment times and finally the global optimal solution is obtained by selecting the best out of these generated pairs numerical illustration is included in support of theory

approximation algorithms for the assembly line crew scheduling problem given an m n nonnegative matrix we study the problem of independently permuting the elements in each column so as to minimize the maximum row sum this problem is np hard even when we restrict it to the m 3 case a special case of our problem is the multiprocessor scheduling problem without precedence constraints in this paper we give a 2 1 m algorithm for the general m n case we also design a 3 2 algorithm for the m 3 case which applies a restricted longest processing time first heuristic finally using an edge matching algorithm we produce a more elaborate algorithm that guarantees a bound of 4 3 for the m 3 case

fast and simple approximation schemes for generalized flow we present fast and simple fully polynomial time approximation schemes fptas for generalized versions of maximum flow multicommodity flow minimum cost maximum flow and minimum cost multicommodity flow we extend and refine fractional packing frameworks introduced in fptas s for traditional multicommodity flow and packing linear programs our fptas s dominate the previous best known complexity bounds for all of these problems some by more than a factor of n2 where n is the number of nodes this is accomplished in part by introducing an efficient method of solving a sequence of generalized shortest path problems our generalized multicommodity fptas s are now as fast as the best non generalized ones we believe our improvements make it practical to solve generalized multicommodity flow problems via combinatorial methods

parallel linear programming in fixed dimension almost surely in constant time for any fixed dimension d thelinear programming problem with n inequality constraints can be solved on a probabilistic crcw pram with o n processors almost surely in constant time the algorithm always findsthe correct solution with nd log 2 d processors the probability that the algorithm will not finish within o d 2 log 2 d time tends to zero exponentially with n authors abstract

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

on the complexity of quadratic programming in real number models of computation abstract the complexity of linearly constrained nonconvex quadratic programming is analyzed within the framework of real number models namely the one of blum shub and smale and its modification recently introduced by koiran weak bss model in particular we show that this problem is not np complete in the koiran setting applications to the full bss model are discussed

half integrality of node capacitated multiflows and tree shaped facility locations on trees in this paper we establish a novel duality relationship between node capacitated multiflows and tree shaped facility locations we prove that the maximum value of a tree distance weighted maximum node capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree and the maximum is attained by a half integral multiflow utilizing this duality we show that a half integral optimal multiflow and an optimal location can be found in strongly polynomial time these extend previously known results in the maximum free multiflow problems we also show that the set of tree distance weights is the only class having bounded fractionality in maximum node capacitated multiflow problems

the maximum concurrent flow problem the maximum concurrent flow problem mcfp is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently the ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow the mcfp objective is to maximize the throughput subject to the capacity constraints we develop a fully polynomial time approximation scheme for the mcfp for the case of arbitrary demands and uniform capacity computational results are presented it is shown that the problem of associating costs distances to the edges so as to maximize the minimum cost of routing the concurrent flow is the dual of the mcfp a path cut type duality theorem to expose the combinatorial structure of the mcfp is also derived our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm applications include packet switched networks 1 4 8 and cluster analysis 16

minimal energy cost for thermodynamic information processing measurement and information erasure the fundamental lower bounds on the thermodynamic energy cost of measurement and information erasure are determined the lower bound on the erasure validates landauer s principle for a symmetric memory for other cases the bound indicates the breakdown of the principle our results constitute the second law of information thermodynamics in which information content and thermodynamic variables are treated on an equal footing

alternating two way ac tree automata we explore the notion of alternating two way tree automata modulo the theory of finitely many associative commutative ac symbols this was prompted by questions arising in cryptographic protocol verification in particular in modeling group key agreement schemes based on diffie hellman like functions where the emptiness question for intersections of such automata is fundamental this also has independent interest we show that the use of general push clauses or of alternation leads to undecidability already in the case of one ac symbol with only functions of arity zero on the other hand emptiness is decidable in the general case of several function symbols including several ac symbols provided push clauses are unconditional and intersection clauses are final this class of automata is also shown to be closed under intersection

range assignment of base stations maximizing coverage area without interference abstract we study the problem of assigning non overlapping geometric objects centered at a given set of points such that the sum of area covered by them is maximized the problem remains open since 2002 as mentioned in a lecture slide of david eppstein in this paper we have performed an exhaustive study on the problem we show that if the points are placed in r 2 then the problem is np hard even for simplest type of covering objects like disks or squares in contrast eppstein 2017 10 proposed a polynomial time algorithm for maximizing the sum of radii or perimeter of non overlapping disks when the points are arbitrarily placed in r 2 we show that eppstein s algorithm for maximizing sum of perimeter of the disks in r 2 gives a 2 approximation solution for the sum of area maximization problem we also propose a ptas for the same problem our results can be extended in higher dimensions as well as for a class of centrally symmetric convex objects

inapproximability and a polynomially solvable special case of a network improvement problem abstract we consider a network improvement problem in which we wish to spend as little as possible to ensure that the distances from a given vertex to all other vertices are within given bounds we first prove that achieving an approximation ratio 1 is np hard for some constant 0 then we show that the problem can be solved by a combinatorial algorithm running in o v log v time when the network is restricted to a tree

low rank matrix approximation in the infinity norm abstract the low rank matrix approximation problem with respect to the entry wise l norm is the following given a matrix m and a factorization rank r find a matrix x whose rank is at most r and that minimizes max i j m i j x i j in this paper we prove that the decision variant of this problem for r 1 is np complete using a reduction from the problem not all equal 3sat we also analyze several cases when the problem can be solved in polynomial time and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low rank matrix that is to the problem of recovering a low rank matrix whose entries have been rounded up to some accuracy

all linear and integer programs are slim 3 way transportation programs we show that any rational convex polytope is polynomial time representable as a 3 way line sum transportation polytope of slim r c 3 format this universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure we provide a polynomial time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs our construction resolves several standing problems on 3 way transportation polytopes for example it demonstrates that unlike the case of 2 way contingency tables the range of values an entry can attain in any slim 3 way contingency table with specified 2 margins can contain arbitrary gaps our smallest such example has format 6 4 3 our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables for example it automatically provides new proofs for some classical results including a well known real feasible but integer infeasible 6 4 3 transportation polytope of m vlach and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator

permuting elements within columns of a matrix in order to minimize maximum row sum we study the problem of permuting the elements within columns of a given m n matrix a so as to minimize its maximum row sum sum of the elements in a row we introduce and analyze an approximation algorithm of greedy type for this np complete problem we prove that our algorithm produces matrices with maximum row sums no more than 3 2 1 2m times greater than those found by an optimization rule moreover examples are presented which achieve this relative performance thus our algorithm represents a substantial improvement in that all earlier algorithms have a worst case performance that is asymptotically twice that of an optimization rule we verify that our algorithm requires at most o m2n time which is a modest increase over the earlier algorithms requiring mn log n time in the worst case

incremental closure for systems of two variables per inequality subclasses of linear inequalities where each inequality has at most two vari ables are popular in abstract interpretation and model checking because they strike a balance between what can be described and what can be efficiently computed this paper focuses on the tvpi class of inequalities for which each coefficient of each two variable inequality is unrestricted an implied tvpi in equality can be generated from a pair of tvpi inequalities by eliminating a given common variable echoing resolution on clauses this operation called result can be applied to derive tvpi inequalities which are entailed implied by a given tvpi system the key operation on tvpi is calculating closure satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations a closed system can be derived by repeatedly applying the result operator the process of adding a single tvpi inequality to an already closed input tvpi system and then finding the closure of this augmented system is called incremental closure this too can be calcu lated by the repeated application of the result operator this paper studies the calculus defined by result the structure of result derivations and how deriva tions can be combined and controlled a series of lemmata on derivations are presented that collectively provide a pathway for synthesising an algorithm for incremental closure the complexity of the incremental closure algorithm is analysed and found to be o n 2 m 2 lg m where n is the number of variables and m the number of inequalities of the input tvpi system

optimization over the efficient set of a parametric multiple objective linear programming problem abstract in this paper we consider the problem p for optimizing a function over the efficient set of a multiple objective linear programming molp problem with parameters in the right hand side vector three solution algorithms in a most general case of problem p and improvements of them in some special cases are presented in the case when the objective function of problem p is linear it can be solved based on t 2 linear programming problems with mixed one zero integer variables if the parametric set is finite and based on t 2 linear programming problems if the right hand side vector of the molp problem is a linear function of the parameters and the parametric set is a polyhedron where t 2 is in general the number of maximal efficient faces of the molp problem corresponding to a value of the parameters a numerical example is given to illustrate the working of the algorithms

workforce planning in mixed model assembly systems serial assembly systems are formed by arranging several production cells or stations in series we study a popular class of serial assembly lines where all stations have the same production cycle we address a workforce planning problem for such lines which finds applications in labor intensive operations in automobile fire engine aircraft and pc board assembly the problem presented can be applied to lines that produce several variations of a basic stable design i e mixed model transfer lines given a set of n jobs we want to find a sequence that minimizes the maximum workforce requirements over all production cycles an optimal polynomial algorithm for the two station line is presented and the three station case is proved to be strongly x1d4a9 x1d4ab complete several heuristic algorithms that produce upper and lower bounds are developed for the general problem worst case behavior of the upper bounds as well as average performance of lower and upper bounds are reported computational results show that s

new algorithms for generalized network flows this paper of which a preliminary version appeared in istcs 92 is concerned with generalized network flow problems in a generalized network each edgee u v has a positive flow multiplier a e associated with it the interpretation is that if a flow ofx e enters the edge at nodeu then a flow ofa e x e exits the edge atv the uncapacitated generalized transshipment problem ugt is defined on a generalized network where demands and supplies real numbers are associated with the vertices and costs real numbers are associated with the edges the goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand and the sum over the edges of the product of the cost and the flow is minimized adler and cosares operations research 39 1991 955 960 reduced the restricted uncapacitated generalized transshipment problem where only demand nodes are present to a system of linear inequalities with two variables per inequality the algorithms presented by the authors in siam journal on computing to appear result in a faster algorithm for restricted ugt generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges i e upper bounds on the amount of flow the goal is to find a flow such that the excesses at the nodes are proportional to the demands and maximized we present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of ugt the algorithm can be used to find an optimal flow or an approximation thereof when used to find a constant factor approximation the algorithm is not only more efficient than previous algorithms but also strongly polynomial it is believed to be the first strongly polynomial approximation algorithm for generalized circulation the existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm

polynomial algorithms for lp over a subring of the algebraic integers with applications to lp with circulant matrices we show that a modified variant of the interior point method can solve linear programs lps whose coefficients are real numbers from a subring of the algebraic integers by defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring we prove the modified algorithm runs in time polynomial in the encoding size of the input coefficients the dimension of the problem and the order of the subring we then extend the tardos scheme to our case obtaining a running time which is independent of the objective and right hand side data as a consequence of these results we are able to show that lps with real circulant coefficient matrices can be solved in strongly polynomial time finally we show how the algorithm can be applied to lps whose coefficients belong to the extension of the integers by a fixed set of square roots

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

deciding linear inequalities by computing loop residues v r pratt has shown that the real and integer feastbdlty of sets of linear mequallues of the form x y c can be decided quickly by examining the loops m certain graphs pratt s method is generahzed first to real feaslbdlty of mequahues m two variables and arbitrary coefficients and ultimately to real feaslbdlty of arbitrary sets of hnear mequahtles the method is well suited to apphcatlons m program verification

a strongly polynomial algorithm for a special class of linear programs we extend the list of linear programming problems that are known to be solvable in strongly polynomial time to include a class of lps which contains special cases of the generalized transshipment problem the result is facilitated by exploiting some special properties associated with leontief substitution systems and observing that a feasible solution to the system ax b x 0 in which no variable appears in more than two equations can be found in strongly polynomial time for b belonging to some set

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

a strongly polynomial algorithm for generalized flow maximization a strongly polynomial algorithm is given for the generalized flow maximization problem it uses a new variant of the scaling technique called continuous scaling the main measure of progress is that within a strongly polynomial number of steps an arc can be identified that must be tight in every dual optimal solution and thus can be contracted as a consequence of the result we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix

isotonic median regression a linear programming approach the isotonic median regression problem arises in statistics it is known that the isotonic median regression problem with respect to a complete order may be solved by a pool adjacent violators algorithm in this paper we show that this algorithm is a dual method for solving a linear programming formulation of the problem the linear programming approach provides additional insight into the algorithm as well as a simple proof of its validity we also analyze the computational complexity of the algorithm and discuss its significance from the standpoint of linear programming theory

tight bounds and 2 approximation algorithms for integer programs with two variables per inequality the problem of integer programming in bounded variables over constraints with no more than two variables in each constraint is np complete even when all variables are binary this paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality and with all variables bounded between 0 andu for such systems a 2 approximation algorithm is presented that runs in time o mnu2 log un2m so it is polynomial in the input size if the upper boundu is polynomially bounded the algorithm works by finding first a super optimal feasible solution that consists of integer multiples of 1 2 that solution gives a tight bound on the value of the minimum it furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem these properties are a generalization of the properties of the vertex cover problem the algorithm described is in particular a 2 approximation algorithm for the problem of minimizing the total weight of true variables among all truth assignments to the 2 satisfiability problem

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

linear programming with two variables per inequality in poly log time the parallel time complexity of the linear programming problem with at most two variables per inequality is discussed let n and m denote the number of variables and the number of inequalities respectively in a linear programming problem it is assumed that all inequalities are weak under the concurrent read exclusive write pram model an o log m log 2 n log 2 n time parallel algorithm for deciding feasibility is described it requires mn o log n processors in the worst case though it is not known whether this bound is tight when the problem is feasible a solution can be computed within the same complexity moreover linear programming problems with at most two nonzero coefficients in the objective function can be solved in poly log time on a similar number of processors consequently all these problems can be solved sequentially with only o log m log 2 n 2 log 2 n space these bounds assume that numbers take o 1 space and arithmetic on them takes o 1 time the p

the linear complementarity problems with a few variables per constraint in this paper we consider the sparse linear complementarity problem denoted by k lcp the coefficient matrices are restricted to have at most k nonzero entries per row it is known that the 1 lcp is solvable in linear time and the 3 lcp is strongly np hard we show that the 2 lcp is strongly np hard and it can be solved in polynomial time if it is sign balanced i e each row of the matrix has at most one positive and one negative entry our second result matches the currently best known complexity bound for the corresponding sparse linear feasibility problem in addition we show that an integer variant of the sign balanced 2 lcp is weakly np hard and pseudo polynomially solvable and the generalized 1 lcp is strongly np hard

descent direction algorithm with multicommodity flow problem for signal optimization and traffic assignment jointly network managers wish to optimize control parameters such as signal setting which are very related to the traffic assignment models on the other hand traffic assignment patterns as an important instrument for predicting the amount of flow on network links are dependent to control decisions according to the significance of this concept some important papers about this mutually relation are reviewed in this paper then we implement a nonlinear algorithm on a minimal cost multicommodity flow mcmf problem to optimize some control policies subject to optimal flows although we take signal times into account but this approach has a far more reaching application in urban network control and design we employ a hybrid intelligent algorithm integrating decent direction algorithm and an interior point algorithm in a mutually consistent scheme for obtaining optimal signals and equilibrium flows an example is given to illustrate the effectiveness of our scheme

a signal dependent time frequency representation fast algorithm for optimal kernel design a time frequency representation based on an optimal signal dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time frequency distributions that the best kernel and distribution depend on the signal to be analyzed the optimization formulation for the signal dependent kernel results in a linear program with a unique feature a tree structure that summarizes a set of constraints on the kernel the authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest for a kernel with q variables the running time of the algorithm is o q log q which is several orders of magnitude less than any other known method for solving this class of linear program this efficiency enables the computation of the signal dependent optimal kernel time frequency representation at a cost that is on the same order as a fixed kernel distribution an important property of the optimal kernel is that it takes on essentially only the values of 1 and 0

a cooperative location game based on the 1 center location problem abstract in this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces the players are the customers demand points in the location problem and the characteristic value of a coalition is the cost of serving its members specifically the cost in our games is the service radius of the coalition we call these games the minimum radius location games mrlg we study the existence of core allocations and the existence of polynomial representations of the cores of these games focusing on network spaces i e finite metric spaces induced by undirected graphs and positive edge lengths and on the lp metric spaces defined over r d

minimum cost multiflows in undirected networks letn g t c a be a network whereg is an undirected graph t is a distinguished subset of its vertices calledterminals and each edgee ofg has nonnegative integer valuedcapacity c e andcost a e theminimum cost maximum multi commodity flow problem studied in this paper is to find ac admissible multiflowf ing such that i f is allowed to contain partial flows connecting any pairs of terminals ii the total value off is as large as possible and iii the total cost off is as small as possible subject to ii this generalizes on one hand the undirected version of the classical minimum cost maximum flow problem when t 2 and on the other hand the problem of finding a maximum fractional packing oft paths whena 0 lovasz and cherkassky independently proved that the latter has a half integral optimal solution

a polynomial predictor corrector trust region algorithm for linear programming in this paper we present a scaling invariant interior point predictor corrector type algorithm for linear programming lp whose iteration complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the lp constraint matrix at the predictor stage the algorithm either takes the step along the standard affine scaling as direction or a new trust region type direction whose construction depends on a scaling invariant bipartition of the variables determined by the as direction this contrasts with the layered least squares direction introduced in s vavasis and y ye math program 74 1996 pp 79 120 whose construction depends on multiple layered partitions of the variables that are not scaling invariant moreover it is shown that the overall arithmetic complexity of the algorithm weakly depends on the right hand side and the cost of the lp in view of the work involved in the computation of the trust region steps

a strongly polynomial algorithm for linear systems having a binary solution this paper proposes a strongly polynomial algorithm which either finds a solution of a linear system ax b 0 x 1 or correctly decides that the system has no 0 1 solutions the algorithm can be used as the basis for the construction of a polynomial algorithm for linear programming

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

on some network flow games we analyze three subclasses of cooperative games arising from network optimization problems in which the resources such as arcs or nodes in the network are controlled by individuals who have conflicting objectives the first subclass of cooperative games is induced by network optimization problems over directed augmented trees we show that for this subclass of games the kernel coincides with the nucleolus and that the nucleolus can be characterized as the unique revenue allocation vector in which every pair of arc owners who are adjacent in the tree are located symmetrically with respect to their bargaining range we further give a linear characterization of the core of this subclass of games which is then used to provide a more explicit representation of the nucleolus and to construct a strongly polynomial algorithm for generating it the second subclass of cooperative games is induced by maximum flow problems in simple undirected networks we provide a useful parametric representation of the core of this subclass of games which is used to characterize the nucleolus and the intersection of the core and the kernel explicitly we show that the intersection of the core and the kernel consists of all revenue allocations in the core which assign equal payoffs to any pair of unseparated arc owners we further demonstrate that among the core vectors the nucleolus is the unique revenue allocation vector in which the smallest allocations are maximized in a lexicographical sense the third cooperative game that we study is the assignment game introduced by shapley and shubik 1972 this game is induced by the assignment problem which can be cast as a network optimization problem we investigate the relationship between the kernel and the core of the assignment game and provide a necessary and sufficient condition for the core to be contained in the kernel we further show that in general the intersection of the kernel and the core is not a convex set we also exhibit that under certain conditions the nucleolus has a simple characterization as the unique vector in the core in which the smallest revenue allocations are maximized in a lexicographical sense finally we consider the horse market example of bohm bawerk 1923 for which it is shown that the core is contained in the kernel and that the nucleolus is the midpoint of the core

about strongly polynomial time algorithms for quadratic optimization over submodular constraints we present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints the main results are an o nm log n 2 m algorithm for the problemnetwork defined on a network onm arcs andn nodes an o n logn algorithm for thetree problem onn variables an o n logn algorithm for thenested problem and a linear time algorithm for thegeneralized upper bound problem these algorithms are the best known so far for these problems the status of the general problem and open questions are presented as well

linear programming the simplex algorithm and simple polytopes in the first part of the paper we survey some far reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes in the second part we discuss some recent developments concerning the simplex algorithm we describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes 1997 the mathematical programming society inc published by elsevier science b v

on the algorithmic inversion of the discrete radon transform the present paper deals with the computational complexity of the discrete inverse problem of reconstructing finite point sets and more general functionals with finite support that are accessible only through some of the values of their discrete radon transform it turns out that this task behaves quite differently from its well studied companion problem involving 1 dimensional x rays concentrating on the case of coordinate hyperplanes in rd and on functionals zd d with d 0 1 r n0 for some arbitrary but fixed r we show in particular that the problem can be solved in polynomial time if information is available for m such hyperplanes when m d 1 but is np hard for m d and d 0 1 r however for d n0 a case that is relevant in the context of contingency tables the problem is still in p similar results are given for the task of determining the uniqueness of a given solution and for a related counting problem

improved algorithms for linear inequalities with two variables per inequality the authors show that a system of m linear inequalities with n variables where each inequality involves at most two variables can be solved in tilde o mn 2 time we denote tilde o f o f polylog n polylog m deterministically and in tilde o n 3 mn expected time using randomization parallel implementations of these algorithms run in tilde o n time where the deterministic algorithm uses tilde o mn processors and the randomized algorithm uses tilde o n 2 m processors the bounds significantly improve over previous algorithms the randomized algorithm is based on novel insights into the structure of the problem

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

generalized jarzynski equality under nonequilibrium feedback control the jarzynski equality is generalized to situations in which nonequilibrium systems are subject to a feedback control the new terms that arise as a consequence of the feedback describe the mutual information content obtained by measurement and the efficacy of the feedback control our results lead to a generalized fluctuation dissipation theorem that reflects the readout information and they can be experimentally tested using small thermodynamic systems we illustrate our general results by introducing an information ratchet which can transport a brownian particle in one direction and extract positive work from the particle

constraint propagation with interval labels abstract constraint propagation is often used in ai systems to perform inference about quantities this paper studies one particular kind of constraint propagation where quantities are labelled with signs or with intervals and these labels are propagated through recorded constraints we review the uses of such inference schemes in ai systems of various kinds and evaluate their strengths and weaknesses in particular we determine the completeness and running time of constraint propagation for various kinds of labels and constraints

strongly polynomial time and nc algorithms for detecting cycles in periodic graphs this paper is concerned with the problem of recognizing in a graph with rational vector weights associates with the edges the existence of a cycle whose total weight is the zero vector this problem is known to be equivalent to the problem of recognizing the existence of cycles in periodic dynamic graphs and to the validity of systems of recursive formulas it was previously conjectured that combinatorial algorithms exist for the cases of two and three dimensional vector weights it is shown that strongly polynomial algorithms exist for any fixed dimension d moreover these algorithms also establish membership in the class nc on the other hand it is shown that when the dimension of the weights is not fixed the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

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

on the inverse problem of linear programming and its application to minimum weight perfect k matching we first introduce the inverse problem of linear programming we then apply it to the inverse problem of minimum weight perfect k matching of bipartite graphs we show that there is a strongly polynomial algorithm for solving the problem