optimization engineering techniques for the exact solution of np hard combinatorial optimization problems abstract the design of effective exact enumerative algorithms for finding the optimal solution of a given np hard combinatorial optimization problem whose mathematical model is given by an integer linear program is considered the commonly used approaches i e dynamic programming branch and bound branch and cut and branch and price are shortly described since the most effective algorithm to be used for finding the optimal solution of a given problem strongly depends on the specific instance to be solved then the best results are often obtained by using hybrid algorithms i e by combining different approaches according to what can be called an optimization engineering technique to illustrate the above mentioned approach the well known 0 1 knapsack problem asymmetric travelling salesman problem and 2 constraint bin packing problem are considered

fuzzy continuous input output controllers are universal approximators the aim of this paper is twofold i to analyse why the fuzzy input output controllers proposed by buckley and hayashi fuzzy sets and systems 58 1993 273 are hardly continuous and based on this analysis a special necessary and sufficient condition for the controllers being continuous is obtained ii to amend buckley hayashi s controllers so that to make them unconditionally continuous

fuzzy input output controllers are universal approximators abstract in this paper we show that a fuzzy input output controller can approximate to any degree of accuracy a continuous fuzzy system this generalizes the known result that fuzzy controllers crisp input output can approximate continuous functions

cofinality spectrum theorems in model theory set theory and general topology we connect and solve two longstanding open problems in quite different areas the modeltheoretic question of whether sop2 is maximal in keisler s order and the question from general topology set theory of whether p t the oldest problem on cardinal invariants of the continuum we do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely using model theoretic methods by a cofinality spectrum problem s we essentially mean the data of a pair of models m m1 which code sufficient set theory possibly in an expanded language along with a distinguished set of formulas s which define linear orders in m1 let ts the treetops of s be the smallest regular cardinal such that one of a set of derived trees in m1 has a strictly increasing sequence with no upper bound let c s ts be the set of pairs of regular cardinals 1 2 such that 1 2 or what is equivalent such that d contains no cuts for cf is good we obtain several consequences notably existence of a minimum keisler class among the non simple theories

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

implementing an efficient minimum capacity cut algorithm in this paper we present an efficient implementation of theo mn n2 logn time algorithm originally proposed by nagamochi and ibaraki 1992 for computing the minimum capacity cut of an undirected network to enhance computation various ideas are added so that it can contract as many edges as possible in each iteration to evaluate the performance of the resulting implementation we conducted extensive computational experiments and compared the results with that of padberg and rinaldi s algorithm 1990 which is currently known as one of the practically fastest programs for this problem the results indicate that our program is considerably faster than padberg and rinaldi s program and its running time is not significantly affected by the types of the networks being solved

separating clique trees and bipartition inequalities having a fixed number of handles and teeth in polynomial time many important cutting planes have been discovered for the traveling salesman problem among them are the clique tree inequalities and the more general bipartition inequalities little is known in the way of exact algorithms for separating these inequalities in polynomial time in the size of the fractional point x which is being separated an algorithm is presented here that separates bipartition inequalities of a fixed number of handles and teeth which includes clique tree inequalities of a fixed number of handles and teeth in polynomial time

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

formalized theory of general fuzzy reasoning first a rigorous mathematical theory for calculating the conclusion of general fuzzy reasoning is proposed second a formalized non fuzzy version of the theory of general fuzzy reasoning is founded in classical logics

asymptotic cones of finitely presented groups let g be a connected semisimple lie group with at least one absolutely simple factor s such that r rank s 2 and let be a uniform lattice in g a if ch holds then has a unique asymptotic cone up to homeomorphism b if ch fails then has 22 asymptotic cones up to homeomorphism

solution of large scale symmetric travelling salesman problems in this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality our implementation is based on a fast lp solver ibm s mpsx and makes effective use of polyhedral results on the symmetric travelling salesman polytope we describe the important ingredients of our code and give an extensive documentation of its computational performance

minimizing symmetric submodular functions we describe a purely combinatorial algorithm which given a submodular set functionf on a finite setv finds a nontrivial subseta ofv minimizingf a f v a this algorithm an extension of the nagamochi ibaraki minimum cut algorithm as simplified by stoer and wagner m stoer f wagner a simple min cut algorithm proceedings of the european symposium on algorithms esa 94 lncs 855 springer berlin 1994 pp 141 147 and by frank a frank on the edge connectivity algorithm of nagamochi and ibaraki laboratoire artemis imag universite j fourier grenbole 1994 minimizes any symmetric submodular function using o v 3 calls to a function value oracle 1998 the mathematical programming society inc published by elsevier science b v

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

an efficient algorithm for the minimum capacity cut problem given a finite undirected graph with nonnegative edge capacities the minimum capacity cut problem consists of partitioning the graph into two nonempty sets such that the sum of the capacities of edges connecting the two parts is minimum among all possible partitionings the standard algorithm to calculate a minimum capacity cut due to gomory and hu 1961 runs in o n4 time and is difficult to implement we present an alternative algorithm with the same worst case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm we report computational results for graphs with up to 2000 nodes

minimum cuts in near linear time we significantly improve known time bounds for solving the minimum cut problem on undirected graphs we use a semiduality between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques we give a randomized monte carlo algorithm that finds a minimum cut in an m edge n vertex graph with high probability in o m log 3 n time we also give a simpler randomized algorithm that finds all minimum cuts with high probability in o m log 3 n time this variant has an optimal rnc parallelization both variants improve on the previous best time bound of o n 2 log 3 n other applications of the tree packing approach are new nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space efficient manner

efficient separation routines for the symmetric traveling salesman problem i general tools and comb separation this is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the traveling salesman problem via branch and cut in this paper we give the basic ideas behind these heuristics and design heuristics for comb separation the second paper will deal with more complex inequalities such as clique tree star or path and ladder inequalities and give computational results

divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

an nc algorithm for minimum cuts we show that the minimum cut problem for weighted undirected graphs can be solved in nc using three separate and independently interesting results the first is an m 2 n processor nc algorithm for finding a 2 epsilon approximation to the minimum cut the second is a randomized reduction from the minimum cut problem to the problem of obtaining a 2 epsilon approximation to the minimum cut this reduction involves a natural combinatorial set isolation problem that can be solved easily in rnc the third result is a derandomization of this rnc solution that requires a combination of two widely used tools pairwise independence and random walks on expanders we believe that the set isolation approach will prove useful in other derandomization problems the techniques extend to two related problems we describe nc algorithms finding minimum k way cuts for any constant k and finding all cuts of value within any constant factor of the minimum another application of these techniques yields an nc algorithm for finding a em sparse k connectivity certificate for all polynomially bounded values of k previously an nc construction was only known for polylogarithmic values of k