Showing papers in "Annals of discrete mathematics in 1979"
TL;DR: In this article, the authors survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory, and indicate some problems for future research and include a selective bibliography.
Abstract: The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.
5,030 citations
TL;DR: This paper gives an informal introduction to the theory of NP-completeness and derives some fundamental results, in the hope of stimulating further use of this valuable analytical tool.
Abstract: Recent developments in the theory of computational complexity as applied to combinatorial problems have revealed the existence of a large class of so-called NP-complete problems, either all or none of which are solvable in polynomial time. Since many infamous combinatorial problems have been proved to be NP-complete, the latter alternative seems far more likely. In that sense, NP-completeness of a problem justifies the use of enumerative optimization methods and of approximation algorithms. In this paper we give an informal introduction to the theory of NP-completeness and derive some fundamental results, in the hope of stimulating further use of this valuable analytical tool.
260 citations
TL;DR: This chapter discusses a number of questions about this method for trying to solve zero–one integer programming (IP) problems and its relevance for optimizing the original IP problem.
Abstract: Publisher Summary This chapter proposes Lagrangean techniques for discrete optimization problems A simple method for trying to solve zero–one integer programming (IP) problems is discussed This method is used as a starting point for discussing many of the developments since then The behavior of Lagrangean techniques in analyzing and solving zero–one IP problems is typical of their use on other discrete optimization problems The chapter discusses a number of questions about this method and its relevance for optimizing the original IP problem The goal of Lagrangean techniques is to try to establish sufficient optimality conditions: Lagrangean techniques are useful in computing zero–one solutions to IP problems with soft constraints or in parametric analysis of an IP problem over a family of right hand sides Parametric analysis of discrete optimization problems is also discussed The use of Lagrangean techniques as a distinct approach to discrete optimization has proven theoretically and computationally important for three reasons First, dual problems derived from more complex discrete optimization problems can be represented as linear programming (LP) problems, but ones of immense size, which cannot be explicitly constructed and then solved by the simplex algorithm Second, reason for considering the application of Lagrangean techniques to dual problems, in addition to the simplex algorithm, is that the simplex algorithm is exact and the dual problems are relaxation approximations Lagrangean techniques as a distinct approach to discrete optimization problems emphasize the need they satisfy for exploiting special structures, which arise in various models
179 citations
TL;DR: The main results of various approaches to the solution of nonlinear programs in 0-1 variables, as well as some of the many applications, are summarized.
Abstract: Much work has been recently devoted to the solution of nonlinear programs in 0-1 variables Many methods of reduction to equivalent forms and of linearization have been proposed Particular cases solvable in polynomial time by network flows have been detected Algebraic and enumerative algorithms have been devised for the general case This paper summarizes the main results of these various approaches, as well as some of the many applications
170 citations
TL;DR: In this paper, the authors survey some of the recent results that have been obtained in connection with covering, packing, and knapsack problems formulated as linear programming problems in zero-one variables.
Abstract: We survey some of the recent results that have been obtained in connection with covering, packing, and knapsack problems formulated as linear programming problems in zero-one variables.
105 citations
TL;DR: A new technique, known as Linked Ordered Sets, is introduced to handle sums and products of functions of nonlinear variables in either the coefficients or the right hand sides of an otherwise linear, or integer, programming problem.
Abstract: Branch and Bound algorithms have been incorporated in many mathematical programming systems, enabling them to solve large nonconvex programming problems. These are usually formulated as linear programming problems with some variables being required to take integer values. But it is sometimes better to formulate problems in terms of Special Ordered Sets of variables of which either only one, or else only an adjacent pair, may take nonzero values. Algorithms for both types of formulation are reviewed. And a new technique, known as Linked Ordered Sets, is introduced to handle sums and products of functions of nonlinear variables in either the coefficients or the right hand sides of an otherwise linear, or integer, programming problem.
98 citations
TL;DR: In this article, the authors present computer codes for the problems of integer programming, which can be characterized by mixed integer programming (MIP) at one end and combinatorial programming at the other end.
Abstract: Publisher Summary This chapter presents computer codes for the problems of integer programming. The term “integer programming” covers a wide spectrum of models, which can be characterized by mixed integer programming (MIP) at one end and combinatorial programming at the other end. The interest of those working in commercial organizations is currently focused at the MIP end of the spectrum—indeed on problems, which are basically large linear programming (LP) systems with relatively few integer variables. The chapter presents a “consumer research” report on the different products and also the methods for solving pure integer problems—frequently with special combinatorial structures. Thus, in a consumer report, one has to bear in mind, which consumers are intended for each code. The code should be capable of obtaining a guaranteed optimum solution. A large and complex problem may not be capable of yielding an optimum integer solution within feasible cost and time limits on any code so that the user has in fact to be content with a significant solution obtained by heuristic methods.
69 citations
TL;DR: An attempt is made to evaluate the methods for solving multiple criteria problems involving integer programming from a user orientation as well as from the perspective of a researcher trying to develop a workable user-oriented method.
Abstract: Several methods have been proposed for solving multiple criteria problems involving integer programming. This paper contains a brief survey as well as a typology of several such methods. Although computational date is scanty to date, an attempt is made to evaluate the methods from a user orientation as well as from the perspective of a researcher trying to develop a workable user-oriented method.
51 citations
TL;DR: This chapter discusses several most important results in integer programming that have been successfully applied to graph theory and then discusses those fields of graph theory where an integer-programming approach has been most effective.
Abstract: Publisher Summary A very large part of combinatorics deals or can be formulated as to deal with optimization problems in discrete structures. Generally, the constraints and the objective function are linear forms of certain variables that are restricted to integers or, mostly, to 0 and 1. Thus, the combinatorial problem is translated to a linear integer-programming problem. The value of such a translation depends on whether it provides new insight or new methods for the solution. This chapter discusses several most important results in integer programming that have been successfully applied to graph theory and then discusses those fields of graph theory where an integer-programming approach has been most effective. The chapter also discusses many graph theoretical results that have a linear programming flavor but no explicit treatment.
45 citations
TL;DR: In this paper, the authors introduce two important families of static, deterministic, single criterion, one product location problems: center problems and median problems, and present an overview of the families of problems considered.
Abstract: This exposition is concerned with two important families of static, deterministic, single criterion, one product location problems: Center problems and median problems. A similar treatise covering additional families is at present under preparation and will be published elsewhere. Reference is made to the introduction for an overview of the families of problems considered. Each part, which is introduced by its own abstract and concluded by its own bibliography, can be read independently of the others as cross references are virtually non-existent.
45 citations
TL;DR: In this paper, the main results found in the last few years on assignment and traveling salesman problems are surveyed and a survey of algorithms, algorithms and worst case analysis for TSP-heuristics is presented.
Abstract: This paper surveys the main results found in the last years on assignment and travelling salesman problems. Algorithmic aspects as well as theoretical questions are considered. In particular algebraic assignment problems, the permanent conjecture, algorithms for quadratic assignment problems, travelling salesman polytopes and algorithms and worst case analysis for TSP-heuristics are treated.
IBM1
TL;DR: This paper traces the development of Gomory's group problem using the cyclic group problem and knapsack problem as prototypes as prototypes, and presents an algorithm giving a constructive proof of a subadditive dual problem for the knapsacks problem.
Abstract: The study of Gomory's group problem has led to a subadditive approach to integer programming. In this paper, we trace that development using the cyclic group problem and knapsack problem as prototypes. The asymptotic theorem of Gomory is also discussed. Finally, an algorithm giving a constructive proof of a subadditive dual problem for the knapsack problem is presented.
IBM1
TL;DR: This paper relates some of the connections arising from the one dimensional cutting stock to Linear, Dynamic and Integer Programming.
Abstract: Cutting stock problems have many connections with Linear, Dynamic and Integer Programming, many of the connections being through the Knapsack Problem This paper relates some of the connections arising from the one dimensional cutting stock Little background in mathematical programming is presumed
TL;DR: The general discrete optimization problem is defined over an arbitrary finite set S on which a mapping into the real numbers is defined, called the set of feasible solutions or the feasible set, which is a subset of the power set of a given finite set.
Abstract: Publisher Summary The general discrete optimization problem is defined over an arbitrary finite set S on which a mapping into the real numbers is defined. S will be called the set of feasible solutions or the feasible set. In order to provide problem with more algebraic structure, one assumes for combinatorial optimization problems that the set of feasible solutions is a subset of the power set of a given finite set. This enables us to define intersections and unions of elements of S and to identify these elements by a (0, 1)-incidence vector of length |E|. In case of combinatorial optimization problems, only separable objective functions are considered. Most well known combinatorial optimization problems and the classical integer programming problem can be stated in the general form with additional assumptions. One advantage of decision problems (or “yes-no-problems”) of type is that they are canonically related to language recognition problems on turing machines, in that the set P corresponds to the set of languages accepted by the turing machine.
TL;DR: This chapter focuses on a particular type of problems that arise in graph theory and in integer programming; though some of the results can be proved by purely algebraic methods and polytope properties, the “hypergraph” context permits a significant presentation that does not require a particular algebraic knowledge.
Abstract: Publisher Summary Graph theory has appeared as a tool to solve a large class of combinatorial problems Hypergraph theory started to generalize and to simplify the classical theorems about graphs This chapter focuses on a particular type of problems that arise in graph theory and in integer programming; though some of the results can be proved by purely algebraic methods and polytope properties, the “hypergraph” context permits a significant presentation that does not require a particular algebraic knowledge Most of the significant concepts about graphs are in fact (0, 1) solutions of a linear program; if instead one consider the solutions with fractional coordinates, one can get simpler results This is called the “fractional” graph theory The chapter provides a brief overview of classical hypergraph theory and fractional hypergraph theory
TL;DR: In this paper, the authors present an introduction to cutting-plane theory and several examples of how to use the principles and metaprinciples for producing cutting-planes for a wide variety of situations.
Abstract: Publisher Summary This chapter presents an introduction to cutting-plane theory. Two principles for obtaining cutting-planes are discussed, one for S=S1= {x ≥ 0 | Ax ≥ b} and one for S=S2= {x ≥ 0 | x integer and Ax=b}. The principle for S1 is a restatement of the duality theorem of Linear Programming. The chapter presents two metaprinciples for combining cuts obtained by the two principles, under various special circumstances. By using the principle for S=S1 together with one of the metaprinciples, the basic principle of disjunctive constraints, which has an interesting converse, is obtained. Several examples of how to use the principles and metaprinciples for producing cutting-planes for a wide variety of situations are presented. The chapter also presents several additional results of cutting-plane theory, which give a general idea of what kinds of theorems are proved by developing the basic principles further.
IBM1
TL;DR: In this article, enumerative methods in integer programming are discussed and a proper utilization of the logical inequalities is the best way for replacing a host of feasibility tests, which are proposed for enumeration.
Abstract: Publisher Summary This chapter presents enumerative methods in integer programming. The problem areas that require enumeration are discussed. The chapter focuses on integer vector as an entity (partial solution or state) and using auxiliary and logical inequalities. Associated inequalities derivable from them and from the solution procedure are discussed. Especially for mixed integer problems it is often the associated inequalities, which will determine the enumeration process. A form of logical inequalities, in turn derivable from the initial inequality system and/or the associated inequality system is discussed. A proper utilization of the logical inequalities is the best way for replacing a host of (in) feasibility tests, which are proposed for enumeration. Logical inequalities are the proper tools for a rational direction of the enumerative process. Enumerative techniques are usually confined to 0–1 problems. Alternatively, integer problems can be reduced to 0–1 problems by an expansion of the integer variables into polynomials involving binary variables only, but no extensive testing of such options appears to take place.
TL;DR: This chapter presents a survey of several more "classic” results for some of the more “standard” problem formulations, and an estimate of the worst-case running time of the algorithm is presented in the chapter.
Abstract: Publisher Summary A truly remarkable variety of discrete optimization problems can be formulated and solved as shortest path or network flow problems or can be solved by procedures that employ shortest path or network flow algorithms as subroutines. It follows that these network computations are among the most fundamental and important in the entire area of discrete optimization. This chapter presents a survey of several more “classic” results for some of the more “standard” problem formulations. In each case, an estimate of the worst-case running time of the algorithm is presented in the chapter. Bellman's equations is also discussed; virtually, all the methods for finding the length of a shortest path between two specified nodes embed the problem in the larger problem of finding the lengths of shortest paths from an origin to each of the other nodes in the network.
TL;DR: The branch and bound or search or enumerative procedures are used for solving integer or other nonconvex programs by breaking up the feasible set into subsets, calculating bounds on the objective function value over each subset, and using the bounds to discard certain subsets of solutions from consideration.
Abstract: Publisher Summary This chapter describes the class of methods known as branch and bound, or search, or enumerative procedures These methods are used for solving integer or other nonconvex programs by breaking up the feasible set into subsets, calculating bounds on the objective function value over each subset, and using the bounds to discard certain subsets of solutions from consideration However, within this general class of methods, two basic prototypes can be distinguished The chapter discusses the solution of general mixed integer programs and uses linear programming as its main vehicle It is further concerned with solving 0–1 programs, and uses as its main tool logical tests exploring the implications of the binary nature of the variables, or inequalities based on similar considerations The branch and bound, or search, or enumerative methods are also used to describe the general class of procedures under discussion and reserve the term implicit enumeration for the special subclass consisting of 0-1 programs
TL;DR: The theory of blocking and antiblocking polyhedra as discussed by the authors is an excellent framework for a joint consideration of a class of continuous problems and combinatorial problems, and it unifies several notions that are similar for both classes of problems.
Abstract: This paper describes the fundamental aspects of the theory of blocking and antiblocking polyhedra as developed by D.R. Fulkerson. The theory is an excellent framework for a joint consideration of a class of continuous problems and a class of combinatorial problems, and it unifies several notions that are similar for both classes of problems. The present paper presents shortly some of the many examples of this sort together with several references to related works.
IBM1
TL;DR: In this article, the use of linear programming duality to prove combinatorial theorems is discussed, and the role played in such investigations by ummodular matrices and related concepts.
Abstract: In this lecture we are concerned with the use of linear programming duality to prove combinatorial theorems, and the role played in such investigations by ummodular matrices and related concepts.
TL;DR: In a directed network with no negative circuit, Floyd's algorithm finds, for each pair of nodes x and y, a shortest path from x to y, here the procedure is extended to minimize more general length-functions over sets of paths that are restricted in various ways.
Abstract: In a directed network with no negative circuit, Floyd's algorithm finds, for each pair of nodes x and y, a shortest path from x to y. Here the procedure is extended to minimize more general length-functions over sets of paths that are restricted in various ways.
TL;DR: The matroid intersection problem is a special case of the linear assignment problem as mentioned in this paper, where the objective is to find a common independent set of maximum weights for two matroids on the same set.
Abstract: Publisher Summary This chapter presents the algorithms for special classes of combinatorial optimization problems. The matroid intersection problem is as follows: given two matroids on the same set and a weight for each element, find a common independent set of maximum weight. This problem embodies many discrete optimization problems as special cases. An example is the linear assignment problem; in this case, the rnatroids are defined on the edge set of the complete bipartite graph Knn, a subset of edges being independent in the first (second) matroid if no two edges have the same head (tail). In fact, it is an adaptation of the augmenting path technique familiar from network flow theory that has yielded a polynomial-time algorithm for the matroid intersection problem. The outstanding open problem in this area is the matroid parity problem, which generalizes both matroid intersection and nonbipartite matching.
TL;DR: The linear discrete optimization problem (LDOP) as discussed by the authors is the problem of maximizing a linear objective function over a finite set S over a polyhedron P, which contains S, and whose vertices are a subset of S.
Abstract: Publisher Summary The “linear discrete optimization problem” is the problem of maximizing a linear objective function cx over a finite set S. For any such finite set there exists a unique bounded polyhedron P, which contains S, and whose vertices are a subset of S. This polyhedron is called the convex hull of S. Every such polyhedron can be alternatively defined as the solution set of a finite system L of linear inequalities. Because the optimal solution to an optimization problem where one maximizes a linear function over a polyhedral set occurs at a vertex of that polyhedron, one could solve the discrete optimization problem by maximizing cx over the solution set of L using linear programming techniques. These observations motivated extensive research efforts directed to characterizing and investigating the facial structure of the polyhedron associated with a given discrete optimization problem. A systematic way is required to generate the system of linear inequalities L, or equivalently if all facets of P can be produced, then a discrete optimization problem can be replaced by its equivalent linear programming problem.
TL;DR: This chapter demonstrates that a large number of combinatorial problems, notorious for their computational intractability, are all equivalent up to a polynomial transformation, which implies that a good algorithm for any of them could be used to solve all others inPolynomial time as well, which would establish the equality of the problem classes.
Abstract: Publisher Summary Computational complexity theory as developed by theoretical computer scientists started to have a major impact on the field of discrete optimization The chapter demonstrates that a large number of combinatorial problems, notorious for their computational intractability, are all equivalent up to a polynomial transformation This implies that a good (that is, polynomial-time) algorithm for any of them could be used to solve all others in polynomial time as well, which would establish the equality of the problem classes Generally, this equality is considered to be very unlikely, and NP-completeness of a combinatorial problem has become commonly accepted as being indicative of its inherent difficulty Computer scientists again must take a lot of credit for establishing membership for many traditional problems Computer science has more to offer in terms of theoretical models for the analysis of problem complexity NP is included in the class of problems solvable in polynomial space This class is the same for deterministic and nondeterministic Turing machines