active constraint variable ordering for faster feasibility of mixed integer linear programs the selection of the branching variable can greatly affect the speed of the branch and bound solution of a mixed integer or integer linear program traditional approaches to branching variable selection rely on estimating the effect of the candidate variables on the objective function we present a new approach that relies on estimating the impact of the candidate variables on the active constraints in the current lp relaxation we apply this method to the problem of finding the first feasible solution as quickly as possible empirical experiments demonstrate a significant improvement compared to a state of the art commercial mip solver

the symmetric traveling salesman polytope and its graphical relaxation composition of valid inequalities the graphical relaxation of the traveling salesman problem is the relaxation obtained by requiring that the salesman visit each city at least once instead of exactly once this relaxation has already led to a better understanding of the traveling salesman polytope in cornuejols fonlupt and naddef 1985 we show here how one can compose facet inducing inequalities for the graphical traveling salesman polyhedron and obtain other facet inducing inequalities this leads to new valid inequalities for the symmetric traveling salesman polytope this paper is the first of a series of three papers on the symmetric traveling salesman polytope the next one studies the strong relationship between that polytope and its graphical relaxation and the last one applies all the theoretical developments of the two first papers to prove some new facet inducing results

an integer programming approach to the bandwidth packing problem we consider the bandwidth packing problem arising from telecommunication networks the problem is to determine the set of calls and an assignment of them to the paths in an arc capacitated network to maximize profit we propose an algorithm to solve the integer programming formulation of the problem an efficient column generation technique to solve the linear programming relaxation is proposed and a modified cover inequality is used to strengthen the ip formulation the algorithm incorporates the column generation technique and the strong cutting plane approach into a branch and bound scheme we test the proposed algorithm on some random problems the results show that the algorithm can be used to solve the problems within reasonably small time limits

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

a cutting plane algorithm for the unrelated parallel machine scheduling problem this paper deals with the polyhedral structure of the scheduling problem r cmax strong valid inequalities are identified for fixed values of the maximum completion time and are used to build a cutting plane scheme from which an exact algorithm and an approximation algorithm are developed the main algorithm includes a preprocessing phase to compute an upper bound with the list scheduling algorithm of davis and jaffe and a lower bound from the preemptive version of the problem computational results show that the proposed exact algorithm gives an optimal solution for almost all tested cases within the fixed time and memory limits for the few cases where the limit has been exceeded rather good solutions are obtained computational requirements of both algorithms are reported for a collection of test problems and the quality of the schedules provided by the approximation algorithm is measured

polynomial time separation of a superclass of simple comb inequalities the comb inequalities are a well known class of facet inducing inequalities for the traveling salesman problem defined in terms of certain vertex sets called the handle and the teeth we say that a comb inequality is simple if the following holds for each tooth either the intersection of the tooth with the handle has cardinality one or the part of the tooth outside the handle has cardinality one or both the simple comb inequalities generalize the classical 2 matching inequalities of edmonds edmonds j 1965 maximum matching and a polyhedron with 0 1 vertices j res nat bur standards69b 125 130 and also the so called chvatal comb inequalities in 1982 padberg and rao padberg m w m r rao 1982 odd minimum cut sets and b matchings math oper res 7 67 80 gave a polynomial time separation algorithm for the 2 matching inequalities i e an algorithm for testing if a given fractional solution to an lp relaxation violates a 2 matching inequality we extend this significantly by giving a polynomial time separation algorithm for a class of valid inequalities which includes all simple comb inequalities

the crown inequalities for the symmetric traveling salesman polytope we define a new family of valid inequalities for the symmetric traveling salesman polytope stspn that is the convex hull of the incidence vectors of all hamiltonian cycles of a complete graph with n nodes the smallest value of n for which the inequalities are defined is 8 we call these inequalities crown inequalities and we prove that they are facet inducing for stspn for n 8 we describe some extensions of the crown inequalities and we prove that they induce facets of stspn as well finally we discuss some issues on the possible applications of these inequalities in a polyhedral cutting plane algorithm

a branch and bound algorithm for the mini max spanning forest problem the mini max spanning forest problem requires to find a spanning forest of an undirected graph that minimizes the maximum of the costs of constituent trees in a previous work we proved this problem np hard in the current paper we present three lower bounds for this problem and develop a branch and bound algorithm to solve the problem exactly the algorithm is implemented and numerical experiments are conducted on a series of test problems the experiments compare the performances of the proposed bounds and search strategies in the algorithm as well as investigate the effects of instance characteristics on the behavior of the algorithm also extension of the problem to the case of more than two root vertices as well as to the problem of determining the root locations are discussed

the existence of periodic solutions of the equation x t ƒ x t x t τ abstract in this paper we study the existence of periodic solutions of the equation x t x t x t where gt 0 x y is continuous on r2 we obtain a sufficient condition in which 1 has at least k non constant oscillatory periodic solutions

numerical experience with lower bounds for miqp branch and bound the solution of convex mixed integer quadratic programming miqp problems with a general branch and bound framework is considered it is shown how lower bounds can be computed efficiently during the branch and bound process improved lower bounds such as the ones derived in this paper can reduce the number of quadratic programming qp problems that have to be solved the branch and bound approach is also shown to be superior to other approaches in solving miqp problems numerical experience is presented which supports these conclusions

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

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

the pickup and delivery problem faces and branch and cut algorithm abstract this paper formulates the pickup and delivery problem also known as the dial a ride problem as an integer program its polyhedral structure is explored and four classes of valid inequalities developed the results of a branch and cut algorithm based on these constraints are presented

octane a new heuristic for pure 0 1 programs we propose a new heuristic for pure 0 1 programs which finds feasible integer points by enumerating extended facets of the octahedron the outer polar of the unit hypercube we give efficient algorithms to carry out the enumeration and we explain how our heuristic can be embedded in a branch and cut framework finally we present computational results on a set of pure 0 1 programs taken from miplib and other sources

fenchel cutting planes for integer programs a technique for generating cutting planes for integer programs is introduced that is based on the ability to optimize a linear function on a polyhedron rather than explicit knowledge of the underlying polyhedral structure of the integer program the theoretical properties of the cuts and their relationship to lagrangian relaxation are discussed the cut generation procedure is described and computational results are presented

separating capacity constraints in the cvrp using tabu search abstract branch and cut methods have shown to be very successful in the resolution of some hard combinatorial optimization problems the success has been remarkable for the symmetric traveling salesman problem tsp the crucial part in the method is the cutting plane algorithm the algorithm that looks for valid inequalities that cut off the current nonfeasible linear program lp solution in turn this part relies on a good knowledge of the corresponding polyhedron and our ability to design algorithms that can identify violated valid inequalities this paper deals with the separation of the capacity constraints for the capacitated vehicle routing problem cvrp three algorithms are presented a constructive algorithm a randomized greedy algorithm and a very simple tabu search procedure as far as we know this is the first time a metaheuristic is used in a separation procedure the aim of this paper is to present this application no advanced tabu technique is used we report computational results with these heuristics on difficult instances taken from the literature as well as on some randomly generated instances these algorithms were used in a branch and cut procedure that successfully solved to optimality large cvrp instances

extended formulations for the a cut problem letg v e be an undirected graph anda v we consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes ina we consider two extended formulations using both node and edge variables an edge variable formulation has previously been considered for this problem chopra and rao 1991 cunningham 1991 we show that the lp relaxations of the extended formulations are stronger than the lp relaxation of the edge variable formulation even with an extra class of valid inequalities added this is interesting because while the lp relaxations of the extended formulations can be solved in polynomial time the lp relaxation of the edge variable formulation cannot we also give a class of valid inequalities for one of the extended formulations computational results using the extended formulations are performed

combinatorial optimization current successes and directions for the future our ability to solve large important combinatorial optimization problems has improved dramatically in the past decade the availability of reliable software extremely fast and inexpensive hardware and high level languages that make the modeling of complex problems much faster have led to a much greater demand for optimization tools this paper highlights the major breakthroughs and then describes some very exciting future opportunities previously large research projects required major data collection efforts expensive mainframes and substantial analyst manpower now we can solve much larger problems on personal computers much of the necessary data is routinely collected and tools exist to speed up both the modeling and the post optimality analysis with the information technology revolution taking place currently we now have the opportunity to have our tools embedded into supply chain systems that determine production and distribution schedules process design and location allocation decisions these tools can be used industry wide with only minor modifications being done by each user

a cutting plane algorithm for the general routing problem the general routing problem grp is the problem of finding a minimum cost route for a single vehicle subject to the condition that the vehicle visits certain vertices and edges of a network it contains the rural postman problem chinese postman problem and graphical travelling salesman problem as special cases we describe a cutting plane algorithm for the grp based on facet inducing inequalities and show that it is capable of providing very strong lower bounds and in most cases optimal solutions

exact value for the average optimal cost of the bipartite traveling salesman and two factor problems in two dimensions we show that the average cost for the traveling salesman problem in two dimensions which is the archetypal problem in combinatorial optimization in the bipartite case is simply related to the average cost of the assignment problem with the same euclidean increasing convex weights in this way we extend a result already known in one dimension where exact solutions are avalaible the recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction overline e n frac 1 pi log n for large number of points 2n as a byproduct of our analysis also the loop covering problem has the same optimal average cost we also explain why this result cannot be extended at higher dimensions we numerically check the exact predictions

simulated annealing with an optimal fixed temperature contrary to conventional belief it turns out that in some problem instances of moderate size fixed temperature simulated annealing algorithms based on a heuristic formula for determining the optimal temperature can be superior to algorithms based on cooling such a heuristic formula however often seems elusive in practical cases considered we include instances of traveling salesman quadratic assignment and graph partitioning problems where we obtain results that compare favorably to the ones known in the literature

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

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

toward solving the steiner travelling salesman problem on urban road maps using the branch decomposition of graphs the steiner travelling salesman problem stsp is an important issue in intelligent transportation systems and has various practical applications such as travelling and parcel delivery in this study we consider the stsp in real world road maps i e given a road map with a set of service points or intersections selection of the representative vertices of a graph g to determine a closed route so a vehicle can visit each service point at least once with the minimum cost it has been theoretically proved that the stsp and its ancestor travelling salesman problem are np hard however it has been empirically demonstrated that they can be solved efficiently on planar graphs with small branchwidths therefore we propose a branch decomposition based extraction algorithm for the stsp in planar graphs our algorithm can solve the stsp in a planar graph g with a time complexity of 2o bw g no 1 where bw g is the branchwidth of g in the case of planar graphs with a small branchwidth our algorithm performs efficiently we evaluated our algorithm by applying it to real world urban road maps which demonstrated the suitability of our method for real world applications

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

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

parallel branch cut and price for large scale discrete optimization in discrete optimization most exact solution approaches are based on branch and bound which is conceptually easy to parallelize in its simplest forms more sophisticated variants such as the so called branch cut and price algorithms are more difficult to parallelize because of the need to share large amounts of knowledge discovered during the search process in the first part of the paper we survey the issues involved in parallelizing such algorithms we then review the implementation of symphony and coin bcp two existing frameworks for implementing parallel branch cut and price these frameworks have limited scalability but are effective on small numbers of processors finally we briefly describe our next generation framework which improves scalability and further abstracts many of the notions inherent in parallel bcp making it possible to implement and parallelize more general classes of algorithms

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

solving a truck dispatching scheduling problem using branch and cut a branch and cut ip solver is developed for a class of structured 0 1 integer programs arising from a truck dispatching scheduling problem this problem involves a special class of knapsack equality constraints families of facets for the polytopes associated with individual knapsack constraints are identified in addition a notion of conflict graph is utilized to obtain an approximating node packing polytope for the convex hull of all 0 1 solutions the branch and cut solver generates cuts based on both the knapsack equality constraints and the approximating node packing polytope and incorporates these cuts into a tree search algorithm that uses problem reformulation and linear programming based heuristics at each node in the search tree to assist in the solution process numerical experiments are performed on large scale real instances supplied by texaco trading transportation inc the optimal schedules correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution

clique relaxations in social network analysis the maximum k plex problem this paper introduces and studies the maximum k plex problem which arises in social network analysis and has wider applicability in several important areas employing graph based data mining after establishing np completeness of the decision version of the problem on arbitrary graphs an integer programming formulation is presented followed by a polyhedral study to identify combinatorial valid inequalities and facets a branch and cut algorithm is implemented and tested on proposed benchmark instances an algorithmic approach is developed exploiting the graph theoretic properties of a k plex that is effective in solving the problem to optimality on very large sparse graphs such as the power law graphs frequently encountered in the applications of interest

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

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

an improved column generation algorithm for minimum sum of squares clustering given a set of entities associated with points in euclidean space minimum sum of squares clustering mssc consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized a column generation algorithm for mssc was given by du merle et al in siam journal scientific computing 21 1485 1505 the bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost we propose a new way to solve this auxiliary problem based on geometric arguments this greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2 300 entities i e more than 10 times as much as previously done

some thoughts on combinatorial optimisation abstract a group of young researchers from the esi x summer school hec jouy en josas 1994 give their personal views on the current status of and prospects for combinatorial optimisation several issues are considered and discussed with emphasis on a selected number of techniques heuristics and polyhedral approaches and problems knapsack quadratic 0 1 programming machine scheduling routing and network design

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

nonlinear chance constrained problems with applications to hydro scheduling we present a branch and cut algorithm for a class of nonlinear chance constrained mathematical optimization problems with a finite number of scenarios unsatisfied scenarios can enter a recovery mode this class corresponds to problems that can be reformulated as deterministic convex mixed integer nonlinear programming problems with indicator variables and continuous scenario variables but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows the branch and cut algorithm is based on an implicit benders decomposition scheme where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces the size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance constrained problem we apply the branch and cut algorithm to the mid term hydro scheduling problem for which we propose a chance constrained formulation a computational study using data from ten hydroplants in greece shows that the proposed methodology solves instances faster than applying a general purpose solver for convex mixed integer nonlinear programming problems to the deterministic reformulation and scales much better with the number of scenarios

a branch and bound algorithm with fuzzy inference for a permutation flowshop scheduling problem abstract this paper considers an m machine permutation flowshop scheduling problem of minimizing the makespan this classical scheduling problem is still important in modern manufacturing systems and is well known to be intractable i e np hard in fact branch and bound algorithms developed so far for this problem have not come to solve large scale problem instances with over a hundred jobs in order to improve the performance of branch and bound algorithms this paper proposes a new dominance relation by which the search load could be reduced and notices that it is based on a sufficient precondition this suggests that the dominance relation holds with high possibility even if the precondition approximately holds thus being more realistic the branch and bound algorithm proposed here takes advantage of this possibility for obtaining an optimal solution as early as possible in the branch and bound search for this purpose this paper utilizes membership functions in the context of the fuzzy inference extensive numerical experiments that were executed through monte carlo simulations and benchmark tests show that the developed branch and bound algorithm can solve 3 machine problem instances with up to 1000 jobs with probability of over 99 and 4 machine ones with up to 900 jobs with over 97

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

relaxed tours and path ejections for the traveling salesman problem abstract we describe an edge based ejection chain method to generate compound neighborhood structures for the traveling salesman problem tsp these neighborhood structures enclose a special substructure which is not necessarily a hamiltonian tour instead the neighborhood components are linked together to compose successive levels of an ejection chain and coordinated by a suitable reference structure to generate compound moves with outstanding proprieties more precisely such a substructure can be viewed as a relaxed tour which allows solution transformations to be obtained without preserving the hamiltonian property at each step furthermore in the ejection chain process the generation of substructures produces a variety of alternating paths for the selection of subsequent ejection moves as well as for the choice of the corresponding trial moves finally we propose two algorithmic variants a preliminary and a full subpath ejection chain method p sec and f sec based on this type of compound neighborhood design both variants are guided by a simple tabu search mechanism computational results on 67 tsplib problems show that both the preliminary and the full methods find optimal and near optimal solutions very quickly frequently outperforming the best heuristic procedures for the tsp in addition the full method generally dominates the best alternative tsp procedures

extended formulations and branch and cut algorithms for the black and white traveling salesman problem in this paper we study integer linear programming models and develop branch and cut algorithms to solve the black and white traveling salesman problem bwtsp bourgeois laporte semet 2003 which is a variant of the well known traveling salesman problem tsp two strategies to model the bwtsp have been used in the literature the problem is either modeled on the original graph as tsp using a single set of binary edge variables and with additional non trivial hop and distance constraints between every pair of black nodes see ghiani laporte semat 2006 or as a sequence of constrained paths composed of white nodes connecting pairs of black nodes see muter 2015 in this paper we study and develop an intermediate approach based on the observation that it is sufficient to guarantee the required distance and hop limit of the path from a given black node to the next black node without explicitly stating which one it is thus instead of stating the two constraints after adding appropriately defined variables for each pair of black nodes they are stated for each black node only that represents the source of each path based on this idea we develop several variants of position and distance dependent reformulations together with corresponding layered graph representations branch and cut algorithms are developed for all proposed formulations and empirically compared by an extensive computational study the obtained results allow us to provide insights into individual advantages and disadvantages of the different models

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

compact formulations of the steiner traveling salesman problem and related problems the steiner traveling salesman problem stsp is a variant of the tsp that is particularly suitable when routing on real life road networks the standard integer programming formulations of both the tsp and stsp have an exponential number of constraints on the other hand several compact formulations of the tsp i e formulations of polynomial size are known in this paper we adapt some of them to the stsp and compare them both theoretically and computationally it turns out that just by putting the best of the formulations into the cplex branch and bound solver one can solve instances with over 200 nodes we also briefly discuss the adaptation of our formulations to some related problems

a new branch and cut algorithm for the capacitated vehicle routing problem we present a new branch and cut algorithm for the capacitated vehicle routing problem cvrp the algorithm uses a variety of cutting planes including capacity framed capacity generalized capacity strengthened comb multistar partial multistar extended hypotour inequalities and classical gomory mixed integer cuts for each of these classes of inequalities we describe our separation algorithms in detail also we describe the other important ingredients of our branch and cut algorithm such as the branching rules the node selection strategy and the cut pool management computational results for a large number of instances show that the new algorithm is competitive in particular we solve three instances b n50 k8 b n66 k9 and b n78 k10 of augerat to optimality for the first time

fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems abstract branch and cut methods for integer programming problems solve a sequence of linear programming problems traditionally these linear programming relaxations have been solved using the simplex method the reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one if the solution to a relaxation is fractional additional constraints can be generated which cut off the solution to the relaxation but donot cut off any feasible integer points gomory cutting planes and other classes of cutting planes are generated from the final tableau in this paper we consider using an interior point method to solve the linear programming relaxations we show that it is still possible to generate gomory cuts and other cuts without having to recreate a tableau and we also show how variables can be fixed without using the optimal reduced costs the procedures we develop do not require that the current relaxation be solved to optimality this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation

on tightening cover induced inequalities abstract here we describe computationally efficient procedures for tightening cover induced inequalities by using 0 1 knapsack constraints and if available cliques whose variables are included in the cover an interesting application is the case where the cover is implied by the knapsack constraint the tightening is achieved by increasing the coefficients of the cover inequality the new constraint is 0 1 equivalent to and lp tighter than the original one the computational complexity of the procedures is o n log n where n is the number of variables in the cover

separating a superclass of comb inequalities in planar graphs many classes of valid and facet inducing inequalities are known for the family of polytopes associated with thesymmetric travelling salesman problem stsp includingsubtour elimination 2 matching andcomb inequalities for a given class of inequalities anexact separation algorithm is a procedure which given an lp relaxation vectorx finds one or more inequalities in the class which are violated byx or proves that none exist such algorithms are at the core of the highly successfulbranch and cut algorithms for the stsp however whereas polynomial time exact separation algorithms are known for subtour elimination and 2 matching inequalities the complexity of comb separation is unknown apartial answer to the comb problem is provided in this paper we define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx e 0 is planar the separation algorithm runs in o n3 time wheren is the number of vertices in the graph

a polyhedral approach to single machine scheduling problems we report new results for a time indexed formulation of nonpreemptive single machine scheduling problems we give complete characterizations of all facet inducing inequalities with integral coefficients and right hand side 1 or 2 for the convex hull of the set of feasible partial schedules i e schedules in which not all jobs have to be started furthermore we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope which is the convex hull of the set of feasible complete schedules i e schedules in which all jobs have to be started to obtain insight in the effectiveness of these classes of facet inducing inequalities we develop a branch and cut algorithm based on them we evaluate its performance on the strongly np hard single machine scheduling problem of minimizing the weighted sum of the job completion times subject to release dates

playing billiards in version space a ray tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a given set of linear separable examples for randomly distributed examples the billiard estimate of the single perceptron with best average generalization probability agrees with known analytic results while for real life classification problems the generalization probability is consistently enhanced when compared to the maximal stability perceptron

path optimization with limited sensing ability we propose a computational strategy to find the optimal path for a mobile sensor with limited coverage to traverse a cluttered region the goal is to find one of the shortest feasible paths to achieve the complete scan of the environment we pose the problem in the level set framework and first consider a related question of placing multiple stationary sensors to obtain the full surveillance of the environment by connecting the stationary locations using the nearest neighbor strategy we form the initial guess for the path planning problem of the mobile sensor then the path is optimized by reducing its length via solving a system of ordinary differential equations odes while maintaining the complete scan of the environment furthermore we use intermittent diffusion which converts the odes into stochastic differential equations sdes to find an optimal path whose length is globally minimal to improve the computation efficiency we introduce two techniques one to remove redundant connecting points to reduce the dimension of the system and the other to deal with the entangled path so the solution can escape the local traps numerical examples are shown to illustrate the effectiveness of the proposed method

the transmission of fluctuation among price indices based on granger causality network abstract in this paper we provide a method of statistical physics to analyze the fluctuation of transmission by constructing granger causality network among price indices pigcn from a systematical perspective using complex network theory combined with granger causality method in economic system there are numerous price indices of which the relationships are extreme complicated thus time series data of 6 types of price indices of china including 113 kinds of sub price indices are selected as example of empirical study through the analysis of the structure of pigcn we identify important price indices with high transmission range high intermediation capacity high cohesion and the fluctuation transmission path of price indices respectively furthermore dynamic relationships among price indices are revealed based on these results we provide several policy implications for monitoring the diffusion of risk of price fluctuation our method can also be used to study the price indices of other countries which is generally applicable