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Showing papers in "Annals of discrete mathematics in 1977"


Book ChapterDOI
TL;DR: In this paper, the authors survey and extend the results on the complexity of machine scheduling problems and give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity.
Abstract: We survey and extend the results on the complexity of machine scheduling problems. After a brief review of the central concept of NP-completeness we give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity. The problems for which a polynomial-bounded algorithm is available are listed and NP-completeness is established for a large number of other machine scheduling problems. We finally discuss some questions that remain unanswered.

1,881 citations


Book ChapterDOI
TL;DR: In this paper, the authors consider the case where n jobs are to be processed by a single machine, and assume that the weighting of jobs is "agreeable", in the sense that the tardiness of a job in a given sequence is wj, max (O, Ci, - di), where Ci is the completion time of job j.
Abstract: Suppose n jobs are to be processed by a single machine. Associated with each job j are a fixed integer processing time pi, a due date di, and a positive weight wj. The weighted tardiness of job j in a given sequence is wj, max (O, Ci, - di), where Ci is the completion time of job j. Assume that the weighting of jobs is “agreeable”, in the sense that pi

623 citations


Book ChapterDOI
TL;DR: In this paper, a min-max relation for submodular functions on graphs is described, and a new combinatorial minmax equality that unifies and extends results including the matroid intersection theorem and the theorem of Lucchesi and Younger on the minimum number of edges which meet every directed cut in a graph is presented.
Abstract: Publisher Summary This chapter describes a min-max relation for submodular functions on graphs. It proves a new combinatorial min-max equality that unifies and extends results including the matroid intersection theorem and the theorem of Lucchesi and Younger on the minimum number of edges, which meet every directed cut in a graph. The method of proof used in the chapter generalizes the method used to prove the polymatroid intersection theorem and the method used to prove the Lucchesi-Younger Theorem including an idea that Lovasz attributes to Neil Robertson.

523 citations


Book ChapterDOI
TL;DR: A conceptually straightforward method for generating sharp lower bounds constitutes the basic element in a family of efficient branch and bound algorithms for solving simple (uncapacitated) plant location problems and special versions hereof including set covering and set partitioning.
Abstract: A conceptually straightforward method for generating sharp lower bounds constitutes the basic element in a family of efficient branch and bound algorithms for solving simple (uncapacitated) plant location problems and special versions hereof including set covering and set partitioning. After an introductory discussion of the problem formulation, a theorem on lower bounds is established and exploited in a heuristic procedure for maximizing lower bounds. For cases where an optimal solution cannot be derived directly from the final tableau upon determination of the first lower bound, a branch and bound algorithm is presented together with a report on computational experience. The lower bound generation procedure was originally developed by the authors in 1967. In the period 1967–69 experiments were performed with various algorithms for solving both plant location and set covering problems. All results appeared in a series of research reports in Danish and attracted accordingly limited attention outside Scandinavia. However, due to their simplicity and high standard of performance, the algorithms are still competitive with more recent approaches. Furthermore, they have appeared to be quite powerful for solving problems of moderate size by hand.

229 citations


Book ChapterDOI
TL;DR: In this paper, a Lagrangian dual for obtaining an upper bound and heuristics for obtaining a lower bound on the value of an optimal solution are introduced, and the main results are analytical worst case analyses of these bounds.
Abstract: The problem of optimally locating bank accounts to maximize clearing times in discused. The importance of this problem depends in part on its mathematical relationship to the well-known uncapacitated plant location problem. A Lagrangian dual for obtaining an upper bound and heuristics for obtaining a lower bound on the value of an optimal solution are introduced. The main results are analytical worst case analyses of these bounds. In particular it is shown that the relative error of the dual bound and a “greedy” heuristic never exceeds [( K – 1)/ K ] K e for a problem in which at most K locations are to be chosen. An interchange heuristic is shown to have a worst case relative error of ( K – 1)/(2 K – 1)

140 citations


Book ChapterDOI
TL;DR: The basic disjunctive cut principle is derived, its interrelations with the other cut-producing procedures are discussed, and applications of it are given.
Abstract: This paper is a survey, with new results, of the disjunctive methods of cutting-plane theory, which were devised by Balas, Glover, Owen, Young, and other researchers, over the past half decade. The basic disjunctive cut principle is derived, its interrelations with the other cut-producing procedures are discussed, and applications of it are given. Many theorems from the literature are concisely proven, and a fairly complete bibliography is provided. In addition, several new results are presented, and finitely convergent disjunctive cutting-plane algorithms are given for a wide class of programs.

68 citations


Book ChapterDOI
TL;DR: In this article, a method for solving multiple criteria integer linear programming problems is presented, and two types of algorithms for extending the Zionts-Wallenius algorithm are briefly presented.
Abstract: Although it may seem counterintuitive, a method for solving multiple criteria integer linear programming problems is not an obvious extension of methods that solve multiple criteria linear programming problems The main difficulty is illustrated by means of an example Then a way of extending the Zionts-Wallenius algorithm [6] for solving integer problems is given, and two types of algorithms for extending it are briefly presented An example is presented for one of the two types Computational considerations are also discussed

56 citations


Book ChapterDOI
TL;DR: This work considers an undirected network where each node and each arc may be in one of two states: operative or inoperative, and considers more global measures such as the probability that all nodes can communicate and all operative nodes can communication.
Abstract: Backtracking algorithms are applied to determine various reliability measures for networks These algorithms are useful in analyzing the reliability of many data communication networks We consider an undirected network where each node and each arc may be in one of two states: operative or inoperative These states are independent random events In addition to the more usual measure of network reliability, the probability that a specified pair of nodes can communicate, we consider more global measures such as the probability that all nodes can communicate and all operative nodes can communicate

55 citations


Book ChapterDOI
TL;DR: Three implicit enumeration algorithms for solving the asymmetric traveling salesman problem with subtour elimination using the assignment problem relaxation similar to the previous approaches by Eastman, Shapiro and Bellmore and Malone are developed and computationally test.
Abstract: In this paper we develop and computationally test three implicit enumeration algorithms for solving the asymmetric traveling salesman problem. All three algorithms use the assignment problem relaxation of the traveling salesman problem with subtour elimination similar to the previous approaches by Eastman, Shapiro and Bellmore and Malone. The present algorithms, however, differ from the previous approaches in two important respects: (i) lower bounds on the objective function for the descendants of a node in the implicit enumeration tree are computed without altering the assignment solution corresponding to the parent node – this is accomplished using a result based on “cost operators”, (ii) a LIFO ( Last In , First Out) depth first branching strategy is used which considerably reduces the storage requirements for the implicit enumeration approach. The three algorithms differ from each other in the details of implementing the implicit enumeration approach and in terms of the type of constraint used for eliminating subtours. Computational experience with randomly generated test problems indicates that the present algorithms are more efficient and can solve larger problems compared to (i) previous subtour elimination algorithms and (ii) the 1-arborescence approach of Held and Karp (as implemented by T. H. C. Smith) for the asymmetric traveling salesman problem. Computational experience is reported for up to 180 node problems with costs (distances) in the interval (1,1000) and up to 200 node problems with bivalent costs.

52 citations


Book ChapterDOI
TL;DR: In this paper, a method for solving pure integer programs by a subadditive method was presented, which uses some elements of both enumeration and cutting plane theory in a unified setting.
Abstract: A method is presented for solving pure integer programs by a subadditive method. This work extends to the integer linear problem a method for solving the group problem. It uses some elements of both enumeration and cutting plane theory in a unified setting. The method generates a subadditive function and solves the original integer linear program.

50 citations


Book ChapterDOI
TL;DR: The proposed LIFO implicit enumeration search algorithm for the symmetric traveling salesman problem which uses the 1-tree relaxation of Held and Karp is proposed and on the basis of the sample it can be stated that the proposed algorithm is faster and generates many fewer subproblems than Held andKarp's algorithm.
Abstract: We propose here a LIFO implicit enumeration search algorithm for the symmetric traveling salesman problem which uses the 1-tree relaxation of Held and Karp. The proposed algorithm has significantly smaller memory requirements than Held and Karp's branch-and-bound algorithm. Computational experience with this algorithm and an improved version of Held and Karp's algorithm is reported and on the basis of the sample it can be stated that the proposed algorithm is faster and generates many fewer subproblems than Held and Karp's algorithm.

Book ChapterDOI
TL;DR: In this article, the facial structure of set packing polyhedra has been studied and a method that can be used repeatedly to construct (arbitrarily) complex facet-producing graphs has been proposed.
Abstract: We review some of the more recent results concerning the facial structure of set packing polyhedra. Utilizing the concept of a facet-producing graph we give a method that can be used repeatedly to construct (arbitrarily) complex facet-producing graphs. A second method, edge-division, is used to further enlarge the class of facet-defining subgraphs.

Book ChapterDOI
TL;DR: In this article, a simple algorithm is presented to determine for a given threshold graph its unique integral separator which minimizes the number of vertices in the threshold graph, where the vertices of a threshold graph are assumed to be independent.
Abstract: A graph is called threshold if there exists a real number b and real numbers a j associated with its vertices w j such that Σ a j ≤ b holds iff S is a stable (independent) set of vertices. The vector 〈 a 1 ,…, a n ; b 〉 associated to a threshold graph is called an integral separator if a i + a j ≥ b + 1 for every edge ( w i w j ). A simple algorithm is presented to determine for a given threshold graph its (unique) integral separator which minimizes b .

Book ChapterDOI
TL;DR: In this article, a primal algorithm for solving (IFP) problems was presented, which can be applied to solve the mixed integer fractional programming problem in finitely many iterations.
Abstract: We construct in this paper new cutting plane algorithms for solving the Integer Fractional Programming (IFP) and the Mixed Integer Fractional Programming (MIFP) problems. By using Charnes and Cooper's approach for solving continuous fractional programs we develop two types of cutting planes, which can be systematically generated and applied while solving (IFP) problems. Similar results are obtained for the (MIFP) problem. By employing Martos' approach for solving continuous fractional programs together with Young's primal algorithm for solving Integer Programming problems, we are able to construct a primal algorithm for solving (IFP) problems in finitely many iterations.

Book ChapterDOI
TL;DR: A number of approaches to the important simple plant location problem are described, including modified simplex methods which exploit triangular bases and several decomposition approaches.
Abstract: The paper is concerned with a number of approaches to the important simple plant location problem. In addition to describing several decomposition approaches, the paper focuses on modified simplex methods which exploit triangular bases.

Book ChapterDOI
TL;DR: In this article, it was shown that given any path of length r there is a cycle of length at least m ≥ r + 3 containing this path, which implies the well-known theorem of Chvatal [4] on hamiltonian graphs and the theorem of Posa [7] which gives sufficient conditions for a graph to contain cycles of a certain length.
Abstract: In this note we establish a sufficient condition for the following property of a graph: given any path of length r there is a cycle of length at least m ≥ r + 3 containing this path. The theorem implies the well-known theorem of Chvatal [4] on hamiltonian graphs and the theorem of Posa [7] which gives sufficient conditions for a graph to contain cycles of a certain length. It is shown that the theorem is neither stronger nor weaker than the theorem of Bondy [3] and the still unsettled conjecture of Woodall [8].

Book ChapterDOI
TL;DR: It is shown how representation of discrete optimization problems as discrete dynamic programming, or network problems, lead naturally to a characterization of the valid inequalities for the feasible solution sets Q of such problems, which allows a reformulation of the “minimum equivalent knapsack inequality” problem, and the ‘cutting stock’ problem.
Abstract: Various discrete optimization problems such as the integer and 0–1 programming problems, and the travelling salesman problem have been represented as discrete dynamic programming, or network problems. We show how such representations lead naturally to a characterization of the valid inequalities for the feasible solution sets Q of such problems. In particular we obtain polytopes Γ of valid inequalities having the facets of Q among their extreme points. In addition the problems of “packing” or “covering” with feasible solutions to the discrete problem have natural network representations, which are the duals of problems over Γ. Reversing the approach, any special properties of the valid inequalities can in turn be used to give new formulations of the corresponding network problems. In particular this allows a reformulation of the “minimum equivalent knapsack inequality” problem, and the “cutting stock” problem.

Book ChapterDOI
TL;DR: Baranyai as discussed by the authors used a linear programming technique of Gomory and Gilmore to extend his result to two other cases: the hereditary closure of the complete h -uniform hypergraphs K h n, for h ≤ 4.
Abstract: A theorem of Baranyai reduces the problem of finding the chromatic index of certain hypergraphs to a cutting stock integer programming problem. Baranyai used this result to establish the chromatic index for the complete h -uniform hypergraphs. We use a linear programming technique of Gomory and Gilmore to extend his result to two other cases: the hereditary closure of the complete h -uniform hypergraphs K h n , for h ≤ 4; and of the complete h -partite hypergraphs.

Book ChapterDOI
TL;DR: In this article, the authors studied properties of facets of full-dimensional polytopes P with binary vertices and showed that the faces of the convex hull of all n -argument parameter vectors are characterized.
Abstract: Properties of facets of full-dimensional polytopes P with binary vertices are studied. If Q is obtained from P by fixing some of the binary variables, then the facets of P that reduce to a given facet of Q are determined by the vertices of a certain polyhedron V . The case where V has a unique vertex is characterized. If P is completely monotonic and the facet of O has 0–1 coefficients, then the vertices of V lie in a hypercube of side 1, and the integer vertices correspond to the sequential lifts or extensions. The self facets, i. e. hyperplanes spanned by binary points, are connected to the hyperplanes spanned by non-negative integral points. Every threshold function can be labelled by its Chow parameter vector. The faces of the convex hull of all n -argument parameter vectors are characterized. This leads to a necessary and sufficient condition for a parameter vector to label a self dual threshold function having a self facet separator.

Book ChapterDOI
TL;DR: In this article, a family of inequalities derived from the logical implications of set partitioning constraints and investigated their properties and potential uses are introduced. But these inequalities do not cover the set covering problem.
Abstract: We introduce a family of inequalities derived from the logical implications of set partitioning constraints and investigate their properties and potential uses. We start with a class of homogeneous canonical inequalities that we call elementary, and discuss conditions under which they are (a) valid, (b) cutting planes, (c) maximal, and (d) facets or improper faces of the set partitioning polytope. We give two procedures for strengthening nonmaximal valid elementary inequalities. Next we derive two nonhomogeneous equivalents of the elementary inequalities, which are of the set packing and set covering types respectively. Using the first of these equivalents, we introduce a “strong” intersection graph, a supergraph of the (common) intersection graph, whose facet generating subgraphs (cliques, odd holes, etc.) give rise to valid inequalities for the set partitioning problem. These inequalities subsume or dominate the similar inequalities that one can derive for the associated set packing problem. One subclass can be used to enhance orthogonality tests in implicit enumeration or column generating algorithms. Further, we introduce two types of composite inequalities, obtainable by combining elementary inequalities according to specific rules, and some related inequalities obtainable directly from the set partitioning constraints. These inequalities provide convenient primal all-integer cutting planes that offer a greater flexibility and are usually stronger than the earlier cuts which do not use the special structure of the set partitioning problem. In the final section we discuss a primal algorithm which uses these cuts in conjunction with implicit enumeration.

Book ChapterDOI
TL;DR: In this article, the authors used an IP duality theory to derive sensitivity analysis tests for IP problems, and obtained results for cost, right hand side and matrix coefficient variation, respectively.
Abstract: This paper uses an IP duality Theory recently developed by the authors and others to derive sensitivity analysis tests for IP problems. Results are obtained for cost, right hand side and matrix coefficient variation.

Book ChapterDOI
TL;DR: In this paper, the authors present an example of dual polytopes in the unit hypercube which are invariant under permutations of the indices of the variables of the polytope.
Abstract: Publisher Summary This chapter presents an example of dual polytopes in the unit hypercube. These polytopes are invariant under permutations of the indices of the variables.

Book ChapterDOI
TL;DR: In this article, a survey of results of the following type is presented: if a linear program and some derived programs have integral solutions, so does its dual, and if integrality is replaced by a condition of the least common denominator of the entries of a solution, then the integrality does not exist.
Abstract: This paper surveys some results of the following type: “If a linear program and some derived programs have integral solutions, so does its dual.” Several well-known minimax theorems in combinatorics can be derived from such general principles. Similar principles can be proved if integrality is replaced by a condition of the least common denominator of the entries of a solution. An analogy between Tutte's 1-factor-theorem and the Lucchesi-Younger Theorem on disjoint directed cuts is pointed out.

Book ChapterDOI
TL;DR: In this paper, the dual greedy algorithm and a modified form of it were introduced to solve the dual partial order problem, where the objective is replaced by a preorder, and the problem is to determine a maximum of a given preorder on Bn in S ⊆Bn.
Abstract: For B = {0,1} and ordered sets (H, ⩽) the objective f : Bn →H shall be maximized under the restriction x ɛS ⊆Bn, The Greedy algorithm can be formulated for this problem without difficulties. The question is for which objectives f and which restrictions S one can use the algorithm to solve the above defined Boolean optimization problem. Dealing with this question, it turned out to be useful to replace the objective by a preorder. The problem then is to determine a maximum of a given preorder on Bn in S ⊆Bn. Concerning some partial orders on Bn problems are characterized for which the optimal solution does not depend on the special choice of the objective. Assumptions with regard to S are closely related to matroid theory; in view of the preorder a certain monotonicity condition is important. The dual greedy algorithm and a modified form of it leads us to the definition of dual partial orders. Herewith it is possible to characterize those S ⊆Bn for which the greedy algorithm and its dual determine the same vector.

Book ChapterDOI
TL;DR: An implicit enumeration algorithm was developed and implemented for this class of problems where only non-zero elements of the large but sparse constraint matrix are stored explicitly and chained row-wise and column-wise.
Abstract: A number of planning problems can be formulated as (0–1)-programs where all variables can be grouped into special ordered sets or generalized upper bounds. An implicit enumeration algorithm was developed and implemented for this class of problems. The generalized upper bounds are handled implicitly. Only non-zero elements of the large but sparse constraint matrix are stored explicitly and chained row-wise and column-wise. The storage structure allows for very efficient testing of partial solutions. Preliminary numerical results indicate that even large-scale problems can be solved efficiently.

Book ChapterDOI
TL;DR: The question of how the two specialized types of algorithms can be married to provide an effective overall approach to the problem is raised.
Abstract: Numerous practical problems involve both logical design choices and continuous-valued decision variables which are predicated in some manner on the logical design. For instance: industrial scheduling problems usually involve both sequencing and the determination of how continuously divisible resources should be applied for the chosen sequence, and network synthesis problems involve both the logical design of the network and the programming of flows for the chosen design. Many such problems which are difficult to solve directly as a whole have the tantalizing properties that (a) specialized algorithms (discrete or combinatorial) are available for close relatives of the logical design aspect of the problem, and (b) for any particular logical design the resulting continuous optimization problem can be solved by an available convex programming method (usually by LP or a network flow technique). This raises the question of how the two specialized types of algorithms can be married to provide an effective overall approach to the problem. Several possible kinds of marriages are surveyed and attractive opportunities for further research are pointed out.

Book ChapterDOI
TL;DR: In this article, the authors considered the problem of minimizing a separable strictly convex function with nonnegative integer variables when the sum of variables is constrained and provided the condition for the optimum and properties of the optimal solution.
Abstract: Minimization of separable strictly convex function is considered with nonnegative integer variables when the sum of variables is constrained. Theorems concerning the condition for the optimum and properties of the optimal solution are presented. For a few types of functions this problem displays “periodic” properties similar to those in linear integer programming: The difference between the noninteger and integer solution is a function depending solely on the position of the noninteger solution inside a hypercube formed by the neighbouring integer points. Utilization of this property shortens drastically the search for the integer solution, in many cases the problem reduces to nonlinear 0/1 problem.

Book ChapterDOI
TL;DR: In this paper, the extension of this technique to other linear programs with logical constraints was examined, and its use as a solution procedure for the 0-1 integer programming problem was discussed.
Abstract: Recent work has shown how to use vertex generation methods to solve linear complementarity problems and cardinality constrained linear programs. These problems can be characterized as linear programs with additional logical constraints. These logical constraints can be incorporated into Chernikova's vertex generating algorithm in a natural and straightforward fashion. This study examines the extension of this technique to other linear programs with logical constraints, and discusses its use as a solution procedure for the 0–1 integer programming problem.

Book ChapterDOI
Jørgen Tind1
TL;DR: In this paper, an economic interpretation of the duality correspondence for antiblocking sets and polyhedra is given, which at least in the polyhedral case play an important role in the study of integer programming problems.
Abstract: This paper first gives an economic interpretation of the duality correspondence for antiblocking sets and polyhedra, which at least in the polyhedral case play an important role in the study of certain integer programming problems, e. g. covering problems. We then discuss, in view of the duality correspondence, how bounds for such problems may be obtained by relatively simple network flow methods.

Book ChapterDOI
TL;DR: In this article, a state-enumeration procedure with reduction methods is presented, which features the consistent use of logical inequalities derived during the computation, especially for influencing the choice of appropriate directions for the search effort.
Abstract: Integer programs with small bound intervals can often be dealt with effectively, by a state-enumeration procedure with reduction methods Our approach features the consistent use of logical inequalities, derived during the computation, especially for influencing the choice of appropriate directions for the search effort