an exact algorithm for the pallet loading problem abstract the two dimensional packing problem of finding optimal layouts for identical rectangular boxes on a rectangular pallet has interested or practitioners for many years the problem is np complete and solution methods to date tend to be heuristic this paper discusses the development of an exact tree search algorithm based on a graph theoretic model of the problem

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

on the positional determinacy of edge labeled games it is well known that games with the parity winning condition admit positional determinacy the winner has always a positional memoryless strategy this property continues to hold if edges rather than vertices are labeled we show that in this latter case the converse is also true that is a winning condition over arbitrary set of colors admits positional determinacy in all games if and only if it can be reduced to a parity condition with some finite number of priorities

regular packing of congruent polygons on the rectangular sheet this paper deals with the problem of searching for the fragment of dense lattice dense double lattice packing of congruent polygons on the rectangular sheet which is optimal by the dense factor in doing so we consider only those packings for which the polygons form rows columns parallel to one of the sheet sides the problem arises in many branches of industry in particular in footwear industry the mathematical model of the problem is built simplification of the mathematical model reducing of the problem dimensionality and its decomposition on a number of independent subproblems is done on the basis of analysis of the properties of the mathematical model the approximate method to solve the problem is suggested the example of solving of the real problem is given

an n tet graph approach for non guillotine packings of n dimensional boxes into an n container abstract in this paper we propose a simple recursive uniform algorithm for the problem of packing n dimensional boxes into an n container we are particularly concerned about the special case n 3 where the boxes can be packed in a given subset of their six possible positionings our method studies symmetries in the packings by the use of an ordered set of three directed graphs with the same edges a 3 tet or triad and induced smaller structures of the same kind named minors with the method degeneracy and symmetry issues which curtail the implicit enumeration to practically acceptable running times become transparent in order to illustrate the performance of the algorithm computational results from solving randomly generated 3 d examples are presented and compared with the ones of a layers and knapsack approach the present study has real world applications for the problems of pallet and container loading

an exact algorithm for higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds and other heuristics we develop a two level tree search algorithm for solving higher dimensional packing problems to optimality computational results are reported including optimal solutions for all two dimensional test problems from recent literature this is the third in a series of articles describing new approaches to higher dimensional packing

entanglement and the complexity of directed graphs entanglement is a parameter for the complexity of finite directed graphs that measures to what extent the cycles of the graph are intertwined it is defined by way of a game similar in spirit to the cops and robber games used to describe treewidth directed treewidth and hypertree width nevertheless on many classes of graphs there are significant differences between entanglement and the various incarnations of treewidth entanglement is intimately related with the computational and descriptive complexity of the modal m calculus the number of fixed point variables needed to describe a finite graph up to bisimulation is captured by its entanglement this plays a crucial role in the proof that the variable hierarchy of the m calculus is strict we study complexity issues for entanglement and compare it to other structural parameters of directed graphs one of our main results is that parity games of bounded entanglement can be solved in polynomial time specifically we establish that the complexity of solving a parity game can be parametrised in terms of the minimal entanglement of subgames induced by a winning strategy furthermore we discuss the case of graphs of entanglement two while graphs of entanglement zero and one are very simple graphs of entanglement two allow arbitrary nesting of cycles and they form a sufficiently rich class for modelling relevant classes of structured systems we provide characterisations of this class and propose decomposition notions similar to the ones for treewidth dag width and kelly width

monadic second order logic on tree like structures an operation m which constructs from a given structure m a tree like structure whose domain consists of the finite sequences of elements of m is considered a notion of automata running on such tree like structures is defined it is shown that automata of this kind characterise expressive power of monadic second order logic msol over tree like structures using this characterisation it is proved that msol theory of a tree like structure is effectively reducible to that of the original structure as another application of the characterisation it is shown that msol on trees of arbitrary degree is equivalent to first order logic extended with unary least fixpoint operator

approximation algorithms for the oriented two dimensional bin packing problem given a set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins we consider the problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized the problem is strongly np hard and finds practical applications in cutting and packing we discuss a simple deterministic approximation algorithm which is used in the initialization of a tabu search approach we then present a tabu search algorithm and analyze its average performance through extensive computational experiments

a hybrid genetic algorithm for the two dimensional single large object placement problem in the two dimensional single large object placement problem we are given a rectangular master surface which has to be cut into a set of smaller rectangular items with the aim of maximizing the total value of the pieces cut we consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface we present new greedy algorithms and a hybrid genetic approach with elitist theory immigration rate heuristics on line and tailored crossover operators extensive computational results for a large number of small and large benchmark test problems are presented the results show that our approach outperforms existing heuristic algorithms

an exact algorithm for the two dimensional strip packing problem this paper considers the two dimensional strip packing problem 2sp in which a set of rectangular items have to be orthogonally packed without overlapping into a strip of a given width and infinite height by minimizing the overall height of the packing the 2sp is np hard in the strong sense and finds many practical applications we propose reduction procedures lower and upper bounds and an exact algorithm for the 2sp the new lower bounds are both combinatorial bounds and bounds derived from different relaxations of mathematical formulations of the 2sp the new upper bounds are obtained by constructive heuristics based on different strategies to place the items into the strip the new exact method is based on a branch and bound approach computational results on different sets of test problems derived from the literature show the effectiveness of the new lower and upper bounds and of the new exact algorithm

algorithms for two dimensional cutting stock and strip packing problems using dynamic programming and column generation abstract we investigate several two dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed we first present two dynamic programming based algorithms for the rectangular knapsack rk problem and its variants in which the patterns must be staged the first algorithm solves the recurrence formula proposed by beasley the second algorithm for staged patterns also uses a recurrence formula we show that if the items are not so small compared to the dimensions of the bin then these algorithms require polynomial time using these algorithms we solved all instances of the rk problem found at the or library including one for which no optimal solution was known we also consider the two dimensional cutting stock problem we present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns we propose two strategies to tackle the residual instances we also investigate a variant of this problem where the bins have different sizes at last we study the two dimensional strip packing problem we also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed in this case we solve instances for two three and four staged patterns we report on some computational experiments with the various algorithms we propose in this paper the results indicate that these algorithms seem to be suitable for solving real world instances we give a detailed description a pseudo code of all the algorithms presented here so that the reader may easily implement these algorithms

application of a mixed simulated annealing genetic algorithm heuristic for the two dimensional orthogonal packing problem abstract in this paper a pure meta heuristic genetic algorithm and a mixed meta heuristic simulated annealing genetic algorithm were applied to two dimensional orthogonal packing problems and the results were compared the major motivation for applying a modified genetic algorithm is as an attempt to alleviate the problem of pre mature convergence we found that in the long run the mixed heuristic produces better results while the pure heuristic produces only good results but produces them faster

an initial semantics for the m calculus on trees and rabin s complementation lemma in this paper we show that the function associated with any closed or nonclosed term of the calculus on trees can be represented by a recognizable set of trees whose nodes are labeled by letters and by sets of variables rabin s complementation lemma is an immediate consequence of this result

an integrated cutting stock and sequencing problem in this paper an integrated problem formulated as an integer linear programming problem is presented to find an optimal solution to the cutting stock problem under particular pattern sequencing constraints the solution uses a lagrangian approach the dual problem is solved using a modified subgradient method a heuristic for the integrated problem is also presented the computational results obtained from a set of unidimensional instances that use these procedures are reported

mathematical model and method of searching for a local extremum for the non convex oriented polygons allocation problem abstract this work deals with the problem of optimal allocation of objects of a so called irregular form the objects are allocated on a strip of given width and with defects this problem is insufficiently studied but it is typical for many industries and is also interesting for developing the theory of solving cutting and packing problems an analytical model of the problem using only continuous variables is written in terms of classical mathematical programming and it is constructed on the basis of the original theory of functions and structures of linear inequalities the presented theory allows one to easily describe the conditions of mutual non overlapping of objects and their allocation in the stock region the exact method for searching a local minimum of the problem from any feasible initial point is based on the application of the active set strategy ideas a number of examples of solving practical problems are considered

the finite graph problem for two way alternating automata two way alternating automata on infinite trees were introduced by vardi reasoning about the part with two way automata lecture notes in computer science vol 11 springer berlin 1998 pp 628 641 here we consider alternating two way automata on graphs and show the decidability of the following problem does a given automaton with the buchi condition accept any finite graph using this result we demonstrate the decidability of the finite model problem for a certain fragment of the modal calculus with backward modalities

holomorphic convexity and carleman approximation by entire functions on stein manifolds we give necessary and sufficient conditions for totally real sets in stein manifolds to admit carleman approximation of class mathcal c k k 1 by entire functions

two dimensional packing problems a survey we consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste in two dimensional bin packing problems these units are finite rectangles and the objective is to pack all the items into the minimum number of units while in two dimensional strip packing problems there is a single standardized unit of given width and the objective is to pack all the items within the minimum height we discuss mathematical models and survey lower bounds classical approximation algorithms recent heuristic and metaheuristic methods and exact enumerative approaches the relevant special cases where the items have to be packed into rows forming levels are also discussed in detail

uniform algebras generated by holomorphic and pluriharmonic functions it is shown that if f 1 f n are pluriharmonic on b n the open unit ball in c n and c 1 on b n and the n n matrix f j z k is invertible at every point of b n then the norm closed algebra generated by the ball algebra a b n and f 1 f n is equal to c b n extensions of this result to more general strictly pseudoconvex domains are also presented

on the weak computability of a four dimensional orthogonal packing and time scheduling problem this paper proposes a four dimensional orthogonal packing and time scheduling problem the problem differs from the classical packing problems in that the position and orientation of each item in the container can be changed over time in this way the four dimensional space time problem better uses the container time also we consider a general case that all parameters are real numbers which makes the problems more difficult to solve this paper proposes an algorithm and proves that the algorithm could solve the problem optimally by a finite number of operations we say this problem is weak computational meaning that if there exists a universal machine that could represent real numbers and could do unit arithmetic or logical operation on real numbers in finite time then the algorithm could find optimal solutions in finite time this paper also presents a proof of the weak computability over a general case of the three dimensional orthogonal packing problem where all parameters are positive real numbers

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

a two stage intelligent search algorithm for the two dimensional strip packing problem this paper presents a two stage intelligent search algorithm for a two dimensional strip packing problem without guillotine constraint in the first stage a heuristic algorithm is proposed which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed for a given space in the second stage a local search and a simulated annealing algorithm are combined to improve solutions of the problem in particular a multi start strategy is designed to enhance the search capability of the simulated annealing algorithm extensive computational experiments on a wide range of benchmark problems from zero waste to non zero waste instances are implemented computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently on average it performs particularly better for large instances

the complexity of mean payoff pushdown games two player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems in this work we consider solving recursive game graphs or pushdown game graphs that model the control flow of sequential programs with recursion while pushdown games have been studied before with qualitative objectives such as reachability and regular objectives in this work we study for the first time such games with the most well studied quantitative objective the mean payoff objective in pushdown games two types of strategies are relevant 1 global strategies which depend on the entire global history and 2 modular strategies which have only local memory and thus do not depend on the context of invocation but rather only on the history of the current invocation of the module our main results are as follows 1 one player pushdown games with mean payoff objectives under global strategies are decidable in polynomial time 2 two player pushdown games with mean payoff objectives under global strategies are undecidable 3 one player pushdown games with mean payoff objectives under modular strategies are np hard 4 two player pushdown games with mean payoff objectives under modular strategies can be solved in np i e both one player and two player pushdown games with mean payoff objectives under modular strategies are np complete we also establish the optimal strategy complexity by showing that global strategies for mean payoff objectives require infinite memory even in one player pushdown games and memoryless modular strategies are sufficient in two player pushdown games finally we also show that all the problems have the same complexity if the stack boundedness condition is added where along with the mean payoff objective the player must also ensure that the stack height is bounded

simulating alternating tree automata by nondeterministic automata new results and new proofs of the theorems of rabin mcnaughton and safra abstract we give a proof that alternating tree automata can be simulated by nondeterministic tree automata which yields new complexity results and a unified proof of the theorems of rabin mcnaughton and safra we also give a simple axiomatic framework for uniformizing strategies

a mathematical model and a solution method for the problem of placing various sized circles into a strip abstract a mathematical model of the problem of placing different radii circles into a strip is constructed several peculiarities of the mathematical model are considered on the ground of these peculiarities an original method of transition from one local minimum to another one to provide a decrease of the objective function value is suggested the method is based on the idea of increasing the problem dimension and a reduced gradient method as well as on the concept of active inequalities and the newton method an algorithm to solve the problem is described in detail numerical examples which demonstrate the possibilities of the method are also given

infinite games on finitely coloured graphs with applications to automata on infinite trees we examine a class of infinite two person games on finitely coloured graphs the main aim is to construct finite memory winning strategies for both players this problem is motivated by applications to finite automata on infinite trees a special attention is given to the exact amount of memory needed by the players for their winning strategies based on a previous work of gurevich and harrington and on subsequent improvements of mcnaughton we propose a unique framework that allows to reestablish and to improve various results concerning memoryless strategies due to emerson and jutla mostowski klarlund

a tree search algorithm for the p median problem abstract in this paper we present two lower bounds for the p median problem the problem of locating p facilities medians on a network these bounds are based on two separate lagrangean relaxations of a zero one formulation of the problem with subgradient optimisation being used to maximise these bounds penalty tests based on these lower bounds and a heuristically determined upper bound to the problem are developed and shown to result in a large reduction in problem size the incorporation of the lower bounds and the penalty tests into a tree search procedure is described and computational results are given for problems with an arbitrary number of medians and having up to 200 vertices a comparison is also made between these algorithms and the dual based algorithm of erlenkotter

infinite games with finite knowledge gaps infinite games where several players seek to coordinate under imperfect information are deemed to be undecidable unless the information is hierarchically ordered among the players we identify a class of games for which joint winning strategies can be constructed effectively without restricting the direction of information flow instead our condition requires that the players attain common knowledge about the actual state of the game over and over again along every play we show that it is decidable whether a given game satisfies the condition and prove tight complexity bounds for the strategy synthesis problem under regular winning conditions given by deterministic parity automata

extensions to a lagrangean relaxation approach for the capacitated warehouse location problem abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon the lagrangean relaxation of a mixed integer formulation of the problem where we use subgradient optimisation in an attempt to maximise this lower bound problem reduction tests based upon this lower bound and the original problem are given incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers

uniform approximation on manifolds it is shown that if a is a uniform algebra generated by a family of complex valued c 1 functions on a compact c 1 manifold with boundary m the maximal ideal space of a is m and e is the set of points where the dierentials of the functions in fail to span the complexied cotangent space to m then a contains every continuous function on m that vanishes on e this answers a 45 year old question of michael freeman who proved the special case in which the manifold m is two dimensional more general forms of the theorem are also established the results presented strengthen results due to several mathematicians

the three dimensional knapsack problem with balancing constraints abstract in this paper we introduce a new packing problem the three dimensional knapsack problem with balancing constraints 3bkp the extension of the three dimensional knapsack problem 3kp where additional constraints related to the packing center of mass are given the 3bkp consists in orthogonally packing a subset of three dimensional weighted items into a knapsack in order to maximize the total profit of the loaded items the items must not overlap and the packing center of mass must lie into a predefined boxed domain inside the knapsack we assume that items can be rotated we give a mip model for the problem upper bounds and an efficient heuristic to solve large size instances the computational results show that the mip model cannot find optimal solutions except for small size instances but it can be used to calculate upper and lower bounds it is shown that our heuristic outperforms the solution quality both of the mip model and the heuristics available in the literature explicitly designed to solve the 3kp

memoryless determinacy of parity and mean payoff games a simple proof we give a simple direct and constructive proof of memoryless determinacy for parity and mean payoff games first we prove by induction that the finite duration versions of these games played until some vertex is repeated are determined and both players have memoryless winning strategies in contrast to the proof of ehrenfeucht and mycielski internat j game theory 8 1979 109 113 our proof does not refer to the infinite duration versions second we show that memoryless determinacy straightforwardly generalizes to infinite duration versions of parity and mean payoff games

an effective quasi human based heuristic for solving the rectangle packing problem abstract in this paper we introduce an effective deterministic heuristic less flexibility first for solving the classical np complete rectangle packing problem many effective heuristics implemented for this problem are cpu intensive and non deterministic in nature others including the polynomial approximation methodology j assoc comput mach 32 1 1985 130 are too laborious for practical problem sizes the technique we propose is inspired and developed by enhancing some rule of thumb guidelines resulting from the generation long work experience of human professionals in ancient days although the less flexibility first heuristic is a deterministic algorithm the results are very encouraging this algorithm can consistently produce packing densities of around 99 on most randomly generated large examples as compared with the recent results of a well known simulated annealing based rectangle packing rp algorithm ieee trans computer aided design integrated circuits systems 17 1 1998 60 the results are much better both in less dead space 2 4 vs 6 7 and much less cpu time 9 57 vs 331 78 seconds experimenting our heuristics on a public rectangle packing data set covering instances of 16 97 rectangles the average unpack ratio is quite satisfactory 0 92 for bounding boxes limited to be optimum and 2 68 for the completed packing while most cases spend only a few minutes in cpu time

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

a method for the cutting stock problem with different qualities abstract an operations research project at a swedish company is reported the project deals with the problem of allocating wood products with different quality requirements to be cut from wooden boards which in turn consists of a number of quality zones a major difficulty is the existence of distinct time limits since the cross cutting of the boards is done in a real time process in this paper we derive a mathematical model which includes a set packing problem that accurately describes the real world problem this problem is then solved using a combination of lagrangean relaxation and subgradient optimization numerical results for practical problems are reported the results indicate a valid model and satisfactory efficiency of the proposed solution procedure

uniform strategies rational relations and jumping automata a general concept of uniform strategies has recently been proposed as a relevant notion in game theory for computer science which subsumes various notions from the literature it relies on properties involving sets of plays in two player turn based arenas equipped with arbitrary binary relations between plays these properties are expressed in a language based on ctl with a quantifier over related plays there are two semantics for our quantifier a strict one and a full one that we study separately regarding the strict semantics the existence of a uniform strategy is undecidable for rational binary relations but introducing jumping tree automata and restricting attention to recognizable relations allows us to establish a 2 exptime complete complexity and still capture a class of two player imperfect information games with epistemic temporal objectives regarding the full semantics relying on information set automata we establish that the existence of a uniform strategy is decidable for rational relations and we provide a nonelementary synthesis procedure we also exhibit an essentially optimal subclass of rational relations for which the problem becomes 2 exptime complete considering rich classes of relations makes the theory of uniform strategies powerful it directly entails various results in logics of knowledge and time some of them already known and others new

a bidirectional building approach for the 2d constrained guillotine knapsack packing problem abstract this paper investigates the 2d guillotine knapsack packing problem in which the objective is to select and cut a set of rectangles from a sheet with fixed size and maximize the total profit of the selected rectangles the orientation of the rectangles is fixed and the guillotine cut in which the cut must be parallel to the sides of the sheet to divide it into two completely separated sheets is required two well known methods namely the top down and bottom up approaches are combined into a coherent algorithm to address this problem computational experiments on benchmark test sets show that the approach finds the optimal solution for almost all moderately sized instances and outperforms all existing approaches for larger instances

exact solution of the two dimensional finite bon packing problem given a set of rectangular pieces to be cut from an unlimited number of standardized stock pieces bins the two dimensional finite bin packing problem is to determine the minimum number of stock pieces that provide all the pieces the problem is np hard in the strong sense and finds many practical applications in the cutting and packing area we analyze a well known lower bound and determine its worst case performance we propose not lower bounds which are used within a branch and bound algorithm for the exact solution of the problem extensive computational testing on problem instances from the literature involving up to 120 pieces shows the effectiveness of the proposed approach

a population heuristic for constrained two dimensional non guillotine cutting abstract in this paper we present a heuristic algorithm for the constrained two dimensional non guillotine cutting problem this is the problem of cutting a number of rectangular pieces from a single large rectangle so as to maximise the value of the pieces cut in addition the number of pieces of each type that are cut must lie within prescribed limits our heuristic algorithm is a population heuristic where a population of solutions to the problem are progressively evolved this heuristic is based on a new non linear formulation of the problem computational results are presented for a number of standard test problems taken from the literature and for a number of large randomly generated problems

analysis of upper bounds for the pallet loading problem this paper is concerned with upper bounds for the well known pallet loading problem plp which is the problem of packing identical boxes into a rectangular pallet so as to maximize the number of boxes fitted after giving a comprehensive review of the known upper bounds in the literature we conduct a detailed analysis to determine which bounds dominate which others the result is a ranking of the bounds in a partial order it turns out that two of the bounds dominate all others a bound due to nelissen and a bound obtained from the linear programming relaxation of a set packing formulation experiments show that the latter is almost always optimal and can be computed quickly

best first search methods for constrained two dimensional cutting stock problems best first search is a widely used problem solving technique in the field of artificial intelligence the method has useful applications in operations research as well here we describe an application to constrained two dimensional cutting stock problems of the following type a stock rectangle s of dimensions l w is supplied there are n types of demanded rectangles r1 r2 rn with the ith type having length li width wi value vi and demand constraint bi it is required to produce from the stock rectangle s ai copies of ri 1 i n to maximize a1v1 a2v2 anvn subject to the constraints ai bi only orthogonal guillotine cuts are permitted all parameters are integers a best first tree search algorithm based on wang s bottom up approach is described that guarantees optimal solutions and is more efficient than existing methods

the line segmentation problem this paper describes a line segmentation problem in a multistage multimachine production system the production facility can concurrently produce several types of circuit boards because each production stage consists of multiple machines the items produced are categorized into families and items belonging to the same family share the common major setup while switching over from one family to another requires a major setup the line segmentation problem determines an allocation of machines at each production stage to families so as to minimize the time to complete all jobs as a result of segmenting the line several minilines are formed which are dedicated to the production of items in each family forming dedicated minilines and producing the items in a family on the same line captures the benefits of group technology and focused factory we first formalize the line segmentation problem as a quadratic integer programming problem and establish its np completeness since the problem is np complete we

the complexity of stochastic muller games the theory of graph games with w regular winning conditions is the foundation for modeling and synthesizing reactive processes in the case of stochastic reactive processes the corresponding stochastic graph games have three players two of them system and environment behaving adversarially and the third uncertainty behaving probabilistically we consider two problems for stochastic graph games the qualitative problem asks for the set of states from which a player can win with probability 1 almost sure winning and the quantitative problem asks for the maximal probability of winning optimal winning from each state we consider w regular winning conditions formalized as muller winning conditions we present optimal memory bounds for pure deterministic almost sure winning and optimal winning strategies in stochastic graph games with muller winning conditions we also study the complexity of stochastic muller games and show that both the qualitative and quantitative analysis problems are pspace complete our results are relevant in synthesis of stochastic reactive processes

a cutting plane approach for the two dimensional orthogonal non guillotine cutting problem abstract the two dimensional orthogonal non guillotine cutting problem ngcp appears in many industries like wood and steel industries and consists in cutting a rectangular master surface into a number of rectangular pieces each with a given size and value the pieces must be cut with their edges always parallel or orthogonal to the edges of the master surface orthogonal cuts the objective is to maximize the total value of the pieces cut in this paper we propose a two level approach for solving the ngcp where at the first level we select the subset of pieces to be packed into the master surface without specifying the layout while at a second level we check only if a feasible packing layout exists this approach has been already proposed by fekete and schepers s p fekete j schepers a new exact algorithm for general orthogonal d dimensional knapsack problems esa 97 springer lecture notes in computer science 1284 1997 144 156 s p fekete j schepers on more dimensional packing iii exact algorithms tech rep 97 290 universitat zu koln germany 2000 s p fekete j schepers j c van der veen an exact algorithm for higher dimensional orthogonal packing tech rep under revision on operations research braunschweig university of technology germany 2004 and caprara and monaci a caprara m monaci on the two dimensional knapsack problem operations research letters 32 2004 2 14 we propose improved reduction tests for the ngcp and a cutting plane approach to be used in the first level of the tree search to compute effective upper bounds computational tests on problems derived from the literature show the effectiveness of the proposed approach that is able to reduce the number of nodes generated at the first level of the tree search and the number of times the existence of a feasible packing layout is tested

measure properties of regular sets of trees abstract we investigate measure theoretic properties of regular sets of infinite trees as a first result we prove that every regular set is universally measurable and that every borel measure on the polish space of trees is continuous with respect to a natural transfinite stratification of regular sets into 1 ranks we also expose a connection between regular sets and the algebra of r sets introduced by a kolmogorov in 1928 as a foundation for measure theory we show that the game tree languages w i k are wadge complete for the finite levels of the hierarchy of r sets we apply these results to answer positively an open problem regarding the game interpretation of the probabilistic calculus

graph operations on parity games and polynomial time algorithms in this paper we establish polynomial time algorithms for special classes of parity games in particular we study various constructions for combining graphs that often arise in structural graph theory and show that polynomial time solvability of parity games is preserved under these operations this includes the join of two graphs repeated pasting along vertices and the addition of a vertex as a consequence we obtain polynomial time algorithms for parity games whose underlying graph is an orientation of a complete graph such as tournaments a complete bipartite graph a block graph or a block cactus graph these are classes where the problem was not known to be efficiently solvable before

deciding parity games in quasi polynomial time it is shown that the parity game can be solved in quasi polynomial time the parameterized parity game with n nodes and m distinct values a k a colors or priorities is proven to be in th

a hybrid genetic algorithm heuristic for a two dimensional orthogonal packing problem in this paper we address a two dimensional 2d orthogonal packing problem where a fixed set of small rectangles has to be placed on a larger stock rectangle in such a way that the amount of trim loss is minimized the algorithm we propose hybridizes a placement procedure with a genetic algorithm based on random keys the approach is tested on a set of instances taken from the literature and compared with other approaches the computation results validate the quality of the solutions and the effectiveness of the proposed algorithm

using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems a popular approach when using a genetic algorithm in the solution of constrained problems is to exploit problem specific information by representing individuals as ordered lists a construction heuristic is then often used as a decoder to produce a solution from each ordering in such a situation further information is often available in the form of bounds on the partial solutions this paper uses two two dimensional packing problems to illustrate how this information can be incorporated into the genetic operators to improve the overall performance of the search our objective is to use the packing problems as a vehicle for investigating a series of modifications of the genetic algorithm approach based on information from bounds on partial solutions

a combinatorial characterization of higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling previous efforts for exact algorithms have been unable to avoid structural problems that appear for instance in two or higher dimensional space we present a new approach for modeling packings using a graph theoretical characterization of feasible packings our characterization allows it to deal with classes of packings that share a certain combinatorial structure instead of having to consider one packing at a time in addition we can make use of elegant algorithmic properties of certain classes of graphs this allows our characterization to be the basis for a successful branch and bound framework this is the first in a series of papers describing new approaches to higher dimensional packing

parameterized complexity of strip packing and minimum volume packing abstract we study the parameterized complexity of minimum volume packing and strip packing in the two dimensional version the input consists of a set of rectangles s with integer side lengths in the minimum volume packing problem given a set of rectangles s and a number k the goal is to decide if the rectangles can be packed in a bounding box of volume at most k in the strip packing problem we are given a set of rectangles s numbers w and k the objective is to find if all the rectangles can be packed in a box of dimensions w k we prove that the 2 dimensional volume packing is in fpt by giving an algorithm that runs in 2 2 k k o 1 time we also show that strip packing is w 1 hard even in two dimensions and give an fpt algorithm for a special case of strip packing some of our results hold for the problems defined in higher dimensions as well

energy parity games energy parity games are infinite two player turn based games played on weighted graphs the objective of the game combines a qualitative parity condition with the quantitative requirement that the sum of the weights i e the level of energy in the game must remain positive beside their own interest in the design and synthesis of resource constrained omega regular specifications energy parity games provide one of the simplest model of games with combined qualitative and quantitative objectives our main results are as follows a exponential memory is sufficient and may be necessary for winning strategies in energy parity games b the problem of deciding the winner in energy parity games can be solved in np conp and c we give an algorithm to solve energy parity by reduction to energy games we also show that the problem of deciding the winner in energy parity games is logspace equivalent to the problem of deciding the winner in mean payoff parity games which can thus be solved in np conp as a consequence we also obtain a conceptually simple algorithm to solve mean payoff parity games

alternating traps in muller and parity games muller games are played by two players moving a token along a graph the winner is determined by the set of vertices that occur infinitely often the central algorithmic problem is to compute the winning regions for the players different classes and representations of muller games lead to problems of varying computational complexity one such class are parity games these are of particular significance in computational complexity as they remain one of the few combinatorial problems known to be in np co np but not known to be in p we show that winning regions for a muller game can be determined from the alternating structure of its traps to every muller game we then associate a natural number that we call its trap depth this parameter measures how complicated the trap structure is we present algorithms for parity games that run in polynomial time for graphs of bounded trap depth and in general run in time exponential in the trap depth

presence or absence of analytic structure in maximal ideal spaces we study extensions of wermer s maximality theorem to several complex variables we exhibit various smoothly embedded manifolds in complex euclidean space whose hulls are non trivial but contain no analytic disks we answer a question posed by lee stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold

uniform algebras and approximation on manifolds let cn be a bounded domain and let a c be a uniform algebra generated by a set f of holomorphic and pluriharmonic functions under natural assumptions on and f we show that the only obstruction to a c is that there is a holomorphic disk d such that all functions in f are holomorphic on d i e the only obstruction is the obvious one this generalizes work by a izzo we also have a generalization of wermer s maximality theorem to the distinguished boundary of the bidisk

an algorithm for the two dimensional assortment problem abstract in this paper we consider the two dimensional assortment problem this is the problem of choosing from a set of stock rectangles a subset which can be used for cutting into a number of smaller rectangular pieces constraints are imposed upon the number of such pieces which result from the cutting a heuristic algorithm for the guillotine cutting version of the problem is developed based on a greedy procedure for generating two dimensional cutting patterns a linear program for choosing the cutting patterns to use and an interchange procedure to decide the best subset of stock rectangles to cut computational results are presented for a number of test problems which indicate that the algorithm developed produces good quality results both for assortment problems and for two dimensional cutting problems

strategy construction for parity games with imperfect information we consider two player parity games with imperfect information in which strategies rely on observations that provide imperfect information about the history of a play to solve such games i e to determine the winning regions of players and corresponding winning strategies one can use the subset construction to build an equivalent perfect information game recently an algorithm that avoids the inefficient subset construction has been proposed the algorithm performs a fixed point computation in a lattice of antichains thus maintaining a succinct representation of state sets however this representation does not allow to recover winning strategies in this paper we build on the antichain approach to develop an algorithm for constructing the winning strategies in parity games of imperfect information one major obstacle in adapting the classical procedure is that the complementation of attractor sets would break the invariant of downward closedness on which the antichain representation relies we overcome this difficulty by decomposing problem instances recursively into games with a combination of reachability safety and simpler parity conditions we also report on an experimental implementation of our algorithm to our knowledge this is the first implementation of a procedure for solving imperfect information parity games on graphs

packing different sized circles into a rectangular container abstract this paper is motivated by the problem of fitting pipes of different diameters into a shipping container here we study the subproblem of fitting circles of different sizes into a rectangle since that problem is a central part of the larger problem we formulate this situation as a nonlinear mixed integer programming problem and develop a number of heuristic procedures for approximately solving this problem the heuristics are based on a variety of solution building rules that emulate the process of packing a container some of these methods including a genetic algorithm were based on a more structured design intended to provide solutions which are stable from a stowage viewpoint these heuristics are described in detail and their relative performances are compared for a sample set of 66 randomly generated problems based on this sample the best performing heuristic methods were a quasi random technique and a genetic algorithm of the stable solution structure

an application of simulated annealing to the cutting stock problem we solve a two dimensional cutting stock problem by applying a general global optimization algorithm the simulated annealing our algorithms applied to the cutting problems involving both the guillotine and non guillotine constraints underlying that the latter is to be preferred for a big number of items several tests prove the validity of the algorithms

a recursive exact algorithm for weighted two dimensional cutting gilmore and gomory s algorithm is one of the better actually known exact algorithms for solving unconstrained guillotine two dimensional cutting problems herz s algorithm is more effective but only for the unweighted case we propose a new exact algorithm adequate for both weighted and unweighted cases which is more powerful than both algorithms the algorithm uses dynamic programming procedures and one dimensional knapsack problem to obtain efficient lower and upper bounds and important optimality criteria which permit a significant branching cut in a recursive tree search procedure recursivity computational power adequateness to parallel implementations and generalization for solving constrained two dimensional cutting problems are some important features of the new algorithm

cutting stock problems and solution procedures this paper discusses some of the basic formulation issues and solution procedures for solving one and two dimensional cutting stock problems linear programming sequential heuristic and hybrid solution procedures are described for two dimensional cutting stock problems with rectangular shapes we also propose an approach for solving large problems with limits on the number of times an ordered size may appear in a pattern

efficient and dynamic algorithms for alternating buchi games and maximal end component decomposition the computation of the winning set for buchi objectives in alternating games on graphs is a central problem in computer aided verification with a large number of applications the long standing best known upper bound for solving the problem is o n m where n is the number of vertices and m is the number of edges in the graph we are the first to break the o n m boundary by presenting a new technique that reduces the running time to o n2 this bound also leads to o n2 time algorithms for computing the set of almost sure winning vertices for buchi objectives 1 in alternating games with probabilistic transitions improving an earlier bound of o n m 2 in concurrent graph games with constant actions improving an earlier bound of o n3 and 3 in markov decision processes improving for m n4 3 an earlier bound of o m m we then show how to maintain the winning set for buchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in o n amortized time per operation our algorithms are the first dynamic algorithms for this problem we then consider another core graph theoretic problem in verification of probabilistic systems namely computing the maximal end component decomposition of a graph we present two improved static algorithms for the maximal end component decomposition problem our first algorithm is an o m m time algorithm and our second algorithm is an o n2 time algorithm which is obtained using the same technique as for alternating buchi games thus we obtain an o min m m n2 time algorithm improving the long standing o n m time bound finally we show how to maintain the maximal end component decomposition of a graph under a sequence of edge insertions or a sequence of edge deletions in o n amortized time per edge deletion and o m worst case time per edge insertion again our algorithms are the first dynamic algorithms for this problem

an exact algorithm for general orthogonal two dimensional knapsack problems abstract we present a new exact tree search procedure for solving two dimensional knapsack problems in which a number of small rectangular pieces each of a given size and value are required to be cut from a large rectangular stock plate the objective is to maximise the value of pieces cut or minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern the algorithm limits the size of the tree search by using a bound derived from a langrangean relaxation of a 0 1 integer programming formulaton of the problem subgradient optimisation is used to optimise this bound reduction tests derived from both the original problem and the lagrangean relaxation produce substantial computational gains the computational performance of the algorithm indicates that it is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

characterization and modelling of guillotine constraints this paper focuses on guillotine cuts which often arise in real life cutting stock problems in order to construct a solution verifying guillotine constraints the first step is to know how to determine whether a given cutting pattern is a guillotine pattern for this purpose we first characterize guillotine patterns by proving a necessary and sufficient condition then we propose a polynomial algorithm to check this condition based on this mathematical characterization of guillotine patterns we then show that guillotine constraints can be formulated into linear inequalities the performance of the algorithm to check guillotine cutting patterns is evaluated by means of computational results

a lagrangean relax and cut approach for the two stage capacitated facility location problem abstract the two stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand the optimal assignment of customers to depots and the optimal product flow from plants to depots in order to solve this problem a lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints the resulting lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch and bound methods the best bound available from this relaxation is computed by means of a weighted dantzig wolfe decomposition approach feasible solutions are constructed from those of the lagrangean subproblems by applying simple reassignment procedures furthermore the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program computational results and comparisons with some other bounds and heuristics for this problem are presented

the dh kd algorithm a hybrid approach for unconstrained two dimensional cutting problems abstract we study both weighted and unweighted unconstrained two dimensional guillotine cutting problems we develop a hybrid approach which combines two heuristics from the literature the first one dh uses a tree search procedure introducing two strategies depth first search and hill climbing the second one kd is based on a series of one dimensional knapsack problems using dynamic programming techniques the dh kd algorithm starts with a good initial lower bound obtained by using the kd algorithm at each level of the tree search the proposed algorithm uses also the kd algorithm for constructing new lower bounds and uses another one dimensional knapsack for constructing refinement upper bounds the resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time our algorithm is compared to morabito et al s algorithm the unweighted case and to beasley s 2 approach the weighted case on some examples taken from the literature as well as randomly generated instances

an and or graph approach to the solution of two dimensional non guillotine cutting problems this work is concerned with unconstrained two dimensional non guillotine cutting problems where a rectangular plate is cut into a number of rectangular pieces in such a way as to optimize an objective function we reduce the state space of the problem by considering non guillotine cutting patterns which are combinations of guillotine and simple non guillotine cuts these cutting patterns are represented as complete paths in an and or graph and a branch and bound method is described we provide rules and heuristics that reduce the graph search and present computational results from randomly generated examples as well as from the literature

a genetic algorithm for the two dimensional strip packing problem with rectangular pieces given a set of rectangular pieces and a container of fixed width and variable length the two dimensional strip packing problem 2d spp consists of orthogonally placing all the pieces within the container without overlapping such that the overall length of the layout is minimised until now mainly heuristics for example genetic algorithms ga were proposed for the 2d spp which use encoded solutions that are manipulated by standard operators in this paper a ga for the 2d spp is suggested that works without any encoding of solutions rather fully defined layouts are manipulated as such by means of specific genetic operators two additional constraints namely the orientation constraint and the guillotine constraint can be taken into account the ga is subjected to a comprehensive test using benchmark instances with up to 5000 pieces compared to eleven competing methods from the literature the ga performs best

computational results from a new lagrangean relaxation algorithm for the capacitated plant location problem abstract in this paper we present an algorithm for the capacitated plant location problem based on a reformulation obtained by the introduction of auxiliary variables by relaxation of the coupling constraints two separate subproblems are obtained this is the basis for the so called variable splitting approach we present a complete algorithmic procedure including solution procedures for subproblems heuristics for the generation of feasible solutions and suggestion for branching rules we also present numerical results for capacitated plant location problems of the size x n where m 10 20 n 20 50

a tabu search algorithm for a two dimensional non guillotine cutting problem in this paper we study a two dimensional non guillotine cutting problem the problem of cutting rectangular pieces from a large stock rectangle so as to maximize the total value of the pieces cut the problem has many industrial applications whenever small pieces have to be cut from or packed into a large stock sheet we propose a tabu search algorithm several moves based on reducing and inserting blocks of pieces have been defined intensification and diversification procedures based on long term memory have been included the computational results on large sets of test instances show that the algorithm is very efficient for a wide range of packing and cutting problems

cross decomposition for mixed integer programming many methods for solving mixed integer programming problems are based either on primal or on dual decomposition which yield respectively a benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via lagrangean relaxation these methods exploit either the primal or the dual structure of the problem we propose a new approach cross decomposition which allows exploiting simultaneously both structures the development of the cross decomposition method captures profound relationships between primal and dual decomposition it is shown that the more constraints can be included in the langrangean relaxation provided the duality gap remains zero the fewer the benders cuts one may expect to need if the linear programming relaxation has no duality gap only one benders cut is needed to verify optimality