Showing papers on "Greedy algorithm published in 1985"
TL;DR: It is conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.
Abstract: We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprisingly enough, using this simple algorithm, one can get very close to the optimal solution of the problem or even find the true optimum. We demonstrate this on several examples. We conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.
3,061 citations
TL;DR: A polynomial-time algorithm is presented to transform a totally-balanced matrix into a greedy matrix as well as to recognize a totallybalanced matrix, which enables us to solve a class of integer programming problems defined on totally- balanced matrices.
Abstract: Totally-balanced and greedy matrices are $( 0,1 )$-matrices defined by excluding certain submatrices. For a $n \times m \,( 0,1)$-matrix A we show that the linear programming problem $\max \{ by \mid yA\leqq c,0\leqq y\leqq d \}$ can be solved by a greedy algorithm for all $c\geqq 0$, $d\geqq 0$ and $b_1 \geqq b_2 \geqq \cdots \geqq b_n \geqq 0$, if and only if A is a greedy matrix. Furthermore we show constructively that if b is an integer, then the corresponding primal problem $\min \{ cx + dz \mid Ax + z\geqq b,x\geqq 0,z\geqq 0 \}$ has an integer optimal solution. A polynomial-time algorithm is presented to transform a totally-balanced matrix into a greedy matrix as well as to recognize a totallybalanced matrix. This transformation algorithm together with the result on greedy matrices enables us to solve a class of integer programming problems defined on totally-balanced matrices. Two examples arising in tree location theory are presented.
157 citations
IBM1
TL;DR: In this article, the problem of partitioning and allocation of the database over the processor nodes of the network can be solved in a computationally feasible manner using a greedy heuristic.
Abstract: In a distributed database system the partitioning and allocation of the database over the processor nodes of the network can be a critical aspect of the database design effort. In this paper we develop and evaluate algorithms that perform this task in a computationally feasible manner. The network we consider is characterized by a relatively high communication bandwidth, considering the processing and input output capacities in its processors. Such a balance is typical if the processors are connected via busses or local networks. The common constraint that transactions have a specific root node no longer exists, so that there are more distribution choices. However, a poor distribution leads to less efficient computation, higher costs, and higher loads in the nodes or in the communication network so that the system may not be able to handle the required set of transactions.Our approach is to first split the database into fragments which constitute appropriate units for allocation. The fragments to be allocated are selected based on maximal benefit criteria using a greedy heuristic. The assignment to processor nodes uses a first-fit algorithm. The complete algorithm, called GFF, is stated in a procedural form.The complexity of the problem and of its candidate solutions are analyzed and several interesting relationships are proven. Alternate benefit metrics are considered, since the execution cost of the allocation procedure varies by orders of magnitude with the alternatives of benefit evaluation. A mixed benefit evaluation strategy is eventually proposed.A model for evaluation is presented. Two of the strategies are experimentally evaluated, and the reported results support the discussion. The approach should be suitable for other cases where resources have to be allocated subject to resource constraints.
114 citations
TL;DR: This work associates with each extreme point of P(f) a certain partial order, and shows that this partial order is efficiently constructible, and that it characterizes all the orderings with which the greedy algorithm can be used to generate the given extreme point.
Abstract: Given a (polymatroid) rank function f and its corresponding polymatroid P(f), we associate with each extreme point of P(f) a certain partial order. We show that this partial order is efficiently constructible, and that it characterizes all the orderings with which the greedy algorithm can be used to generate the given extreme point. We give several applications, including one to the still open problem of finding an efficient combinatorial procedure for testing membership in polymatroids. Our results can also be applied to convex games.
70 citations
TL;DR: The greedy channel router of Rivest and Fiduccia is extended into an efficient switch-box router and a good heuristic which is based on the augmented channel density distribution in finding it is proposed.
62 citations
TL;DR: It is shown that an acyclic multigraph with a single source and a single sink is series-parallel if and only if for arbitrary linear cost functions and arbitrary capacities the corresponding minimum cost flow problem can be solved by a greedy algorithm.
57 citations
01 Jan 1985
TL;DR: A comparison of WEAVER's Routing of a Channel Unroutable by the Greedy Algorithm and the Techniques of Applied AI shows that the former is more efficient and the latter is less efficient.
Abstract: 1. Introduction.- 1.1. Motivation.- 1.2. Outline.- 2. Detailed Routing.- 2.1. Problem Statement.- 2.2. Important Factors in Routing.- 2.3. Previous Approaches.- 2.3.1. Lee Algorithm.- 2.3.2. Line Routing Algorithms.- 2.3.3. Efficient Algorithms for Channel Routing.- 2.3.4. A "Greedy" Channel Router.- 2.3.5. Hierarchical Wire Routing.- 2.4. Characteristics of Previous Approaches.- 3. WEAVER Approach.- 3.1. Congestion.- 3.2. Wire Length.- 3.3. Rectilinear Steiner Tree.- 3.3.1. Steiner Tree.- 3.3.2. Minimal Rectilinear Steiner Tree for a 2xn Grid.- 3.3.3. Minimal Rectilinear Steiner Tree for A mxn Grid.- 3.4. Merging.- 3.5. Vertical/Horizontal Constraint Graph.- 3.6. Intersection.- 3.7. Conflicting Effects.- 4. Knowledge-Based Expert Systems.- 4.1. Productions Systems.- 4.2. OPS5.- 4.2.1. Working Memory.- 4.2.2. Production Memory.- 4.2.3. Interpreter.- 4.3. Applicability of Knowledge-Based Expert Systems to VLSI Design.- 4.3.1. Detailed Routing of VLSI Chips is Amenable to the Techniques of Applied AI.- 4.3.2. Detailed Routing of VLSI Chips is Important, Difficult and a High-Value Problem.- 4.4. Advantages and Disadvantages of Knowledge-Based Expert Systems.- 5. WEAVER Implementation.- 5.1. Problem State Representation.- 5.2. WEAVER Architecture.- 5.3. Blackboard Organization.- 5.4. WEAVER Experts.- 5.4.1. Wire Length Expert.- 5.4.2. Merging Expert.- 5.4.3. Congestion Expert.- 5.4.4. Vertical/Horizontal Constraint Expert.- 5.4.5. Via Expert.- 5.4.6. Common Sense Expert.- 5.4.7. Pattern Router Expert.- 5.4.8. Constraint Propagation Expert.- 5.4.9. User Expert.- 5.4.10. Minimal Rectilinear Steiner Tree Expert.- 5.5. WEAVER Control Structure.- 5.5.1. Nature of WEAVER Expertise.- 5.5.2. Generality of WEAVER Knowledge.- 5.6. Program Organization.- 6. Experiments and Results.- 6.1. Input/Output.- 6.1.1. Input.- 6.1.2. Output.- 6.2. Step by Step Trace of Routing a Channel.- 6.3. Experiments.- 6.3.1. Comparison with Efficient Algorithms for Channel Routing.- 6.3.2. Comparison with the Greedy Algorithm When Both can Route the Channel.- 6.3.3. WEAVER's Routing of a Channel Unroutable by the Greedy Algorithm.- 6.3.4. WEAVER's Solution to Provably Unroutable Channel and Switch-Box by Traditional Algorithms.- 6.3.5. Comparison with Aker's and Lee Algorithms.- 6.3.6. Comparison with the Minimum-Impact Routing Algorithm.- 6.3.7. Burstein's Difficult Switch-Box.- 6.3.8. Terminal Intensive Example.- 6.3.9. Dense Switch-Box Example.- 6.3.10. Conclusion to the Experiments.- 6.4. WEAVER's Performance Under Conditions of Disabled Experts.- 6.4.1. Merging Expert Disabled.- 6.4.2. Congestion and Merging Experts Disabled.- 6.4.3. Via Expert Disabled.- 6.4.4. Vertical/Horizontal Constraint Expert Partially Disabled.- 6.4.5. Rectilinear Steiner Tree Expert Disabled.- 6.4.6. Summary of the Results of Disabling the Experts.- 6.5. Efficiency Issues.- 6.5.1. Possible Execution Time Improvement.- 6.5.2. Writing Efficient OPS5 Programs.- 7. Conclusions and Future Work.- References.
42 citations
01 Jan 1985
TL;DR: In this paper, it is shown that the jump number of an i-free poset is equal to the cyclomatic number of its root digraph, which is the same as that of a node in the Hasse diagram.
Abstract: The purpose of this paper is to discuss the jump number problem for partially ordered sets (posets) using their arc representations, in which poset elements are assigned to arcs. We present the solution of the jump number problem restricted toN-free posets. The Hasse diagram of anN-free poset is a line digraph, so it has a unique root digraph which is its arc representation. It is shown that the jump number of an i-free poset is equal to the cyclomatic number of its root digraph. We illustrate also that several other conclusions forN-free posets can be easily proved in terms of root digraphs. In particular, we discuss the scope of the greedy algorithm and jump-critical posets. Moreover, we investigate the problem for arbitrary posets by showing that, in the general case, the iump number of a poset is eaual to the cyclomatic number of a certain digraph which can be derived from a poset arc representation. Then, we strengthen the greedy algorithm and exhibit a class of posets for which it generates optimal linear extensions. Finally, we give a short informal survey of construction methods for arc representations of posets and a list of the most important contributions to the jump number problem and construction methods for arc representations.
33 citations
TL;DR: In this article, the authors considered a more general form of the reconstruction problem where substates in general are allowed in the reconstruction and the overall system need not be known in its entirety to formulate a reconstruction.
Abstract: The aim of the reconstruction problem is to determine from among all relevant structure systems those which allow us to reconstruct a given overall system to an acceptable degree of approximation This paper considers a more general form of the reconstruction problem Instead of restricting the reconstruction to structure systems, substates in general are allowed in the reconstruction Furthermore, the overall system need not be known in its entirety to formulate a reconstruction A very effective and efficient algorithm for this generalized reconstruction problem is presented
33 citations
TL;DR: The Greedy Algorithm produces an optimal integral solution to the choice of a deployment and firing doctrine for defending separated point targets of potentially different values against an attack by an unknown number of sequentially arriving missiles.
Abstract: We study the choice of a deployment and firing doctrine for defending separated point targets of potentially different values against an attack by an unknown number of sequentially arriving missiles. We minimize the total number of defenders, subject to an upper bound on the maximum expected value damage per attacking weapon. We show that the Greedy Algorithm produces an optimal integral solution to this problem.
32 citations
TL;DR: For this class of problems, if the given bounds and cost coefficients satisfy certain conditions, an optimal solution can be found by a greedy algorithm.
Abstract: For this class of problems, if the given bounds and cost coefficients satisfy certain conditions, an optimal solution can be found by a greedy algorithm. This work was stimulated by an application of linear programming to graph partitioning.
TL;DR: It is shown that the class of N-free ordered sets is both defect optimal and strongly greedy, which means that the setup minimization problem can be solved by an efficient algorithm.
Abstract: Construction of classes of ordered sets are given for which the setup minimization problem can be solved by an efficient algorithm. Those constructions generalize series-parallel connections. Special classes of ordered sets are exhibited for which the greedy algorithm yields an optimal linear extension. In particular, it is shown that the class of N-free ordered sets is both defect optimal and strongly greedy.
TL;DR: This paper presents a treatment of max-min problems over matroids with greedy type algorithms and shows how these algorithms can be improved on the basis of data dependent performance guarantees.
Abstract: Min-max problems on matroids are NP-hard for a wide variety of matroids. However, greedy type algorithms have data independent worst case performance guarantees, andn-enumerative algorithms yielde-optimal solutions ifn is sufficiently close to the rank of the underlying matroid. Data dependent performance guarantees can be obtained for max-min problems over matroids.
TL;DR: In this paper, the setup minimization problem for linear extensions of interval orders is considered and a simple greedy heuristic is shown to be never worse than twice the optimum for any linear extension.
TL;DR: A greedy algorithm for solving a special class of convex programming problems is presented and a connection with polymatroid theory is established which yields a theoretical explanation and verification of the algorithm via some recent results of S Fujishige.
Abstract: We present a greedy algorithm for solving a special class of convex programming problems and establish a connection with polymatroid theory which yields a theoretical explanation and verification of the algorithm via some recent results of S. Fujishige.
TL;DR: The optimization problem for linear functions on finite languages is studied, and an (almost) complete characterization of those functions for which a primal and a dual greedy algorithm work well with respect to a canonically associated linear programming problem is given.
Abstract: The optimization problem for linear functions on finite languages is studied, and an (almost) complete characterization of those functions for which a primal and a dual greedy algorithm work well with respect to a canonically associated linear programming problem is given. The discussion in this paper is within the framework of ordered languages, and the characterization uses the notion of rank feasibility of a weighting with respect to an ordered language. This yields a common generalization of a sufficient condition, obtained recently by Korte and Lovasz for greedoids, and the greedy algorithm for ordered sets in Faigel's paper [6]. Ordered greedoids are considered the appropriate generalization of greedoids, and the connection is established between ordered languages, polygreedoids, and Coxeteroids. Answering a question of Bjorner, the author shows in particular that a polygreedoid is a Coxeteroid if and only if it is derived from an integral polymatroid.
01 Jan 1985
TL;DR: This dissertation is composed of three studies having a common theme: the strengthening of weak methods by improving the advice that guides them, and gives several necessary and sufficient conditions characterizing order dependent cost functions which can be optimized by greedy schemes.
Abstract: This dissertation is composed of three studies having a common theme: the strengthening of weak methods by improving the advice that guides them.
The first study examines the effects of removing some of the restrictions imposed on A* and, reexamines whether A* is computationally optimal relative to other algorithms that have access to the same heuristic information. It is shown that the wide class of algorithms which, like A*, return optimal solutions when all cost estimates are optimistic, does not contain an optimal algorithm. On the other hand A* is optimal in a somewhat more restricted sense: either (1) relative to the set of instances in which the heuristic estimates are consistent, or (2) relative to the subclass of algorithms which are Best-First.
The second and third studies examine various means of mechanically generating heuristic advice for weak methods, using the paradigm that heuristics are generated by consulting a simplified model of the task domain.
The second study generates advice to help Backtrack solve Binary Constraint Satisfaction Problems. The advice is generated automatically by consulting relaxed models known to yield a backtrack-free solutions. The information retrieved from these models induces a preference order among the choices pending in the original problem. Optimal algorithms for solving easy problems are presented and analyzed, and the utility of using the advice is evaluated experimentally, using a synthetic domain of CSP problem instances.
The last study is devoted to the analysis of optimization problems that are solvable by Greedy algorithms since such easily solved problems are natural targets for simplification. The contribution of this study is in dealing with ordering optimization problems in which a set of n elements should be ordered to minimize a certain cost function. We give several necessary and sufficient conditions characterizing order dependent cost functions which can be optimized by greedy schemes, and distinguish two types of greedily optimized ordering problems; dominant and non-dominant.
TL;DR: The greedy algorithm is shown to be optimal for problems of the form Max{ v ( S )| S (subseteq) N } that are solved optimally by the greedy algorithm, thereby providing a new algorithm different from previous approaches based on dynamic programming.
Abstract: Consider a set function v(·) defined on all subsets of a finite set N. This paper presents a class of set function maximization problems of the form Max{v(S)|S ⊆ N} that are solved optimally by the greedy algorithm. In particular, it is shown that there exists a nested sequence of subsets S*1 … S*m that increase to the optimal policy and S*k maximizes v(S) over all subsets of cardinality k. The class is characterized by conditions on the set function v(·) and related to the notion of submodularity. These results are then applied to two order-preserving machine scheduling problems. The greedy algorithm is shown to be optimal for these problems thereby providing a new algorithm different from previous approaches based on dynamic programming.
TL;DR: Weakly hereditary combinatorial optimization problems (weakly hereditary optimization problems) as discussed by the authors are a class of problems characterized by a weak hereditary property on the set of feasible solutions, that is, given a feasible solution S of size i, there always exists an element a j in S such that S-[ a j ] is a feasible solutions of size n − 1.
Abstract: Publisher Summary This chapter describes a class of combinatorial optimization problems, characterized by a weak hereditary property on the set of the feasible solutions—that is, given a feasible solution S of size i, there always exists an element a j in S such that S-[ a j ] is a feasible solution of size i-1. These problems are called “weakly hereditary optimization problems” and share some simple combinatorial properties that determine the probabilistic behavior of the whole class with respect to the greedy algorithm. The chapter presents the definition of the class of the weakly hereditary optimization problems and lists some important problems belonging to this class. The chapter focuses and compares the optimal solution and the solution given by a greedy algorithm.
TL;DR: A NP-hard problem (P) of mixed-discrete linear programming is considered which consists in the minimization of a linear objective function subject to a special non-connected subset of an unbounded polymatroid.
Abstract: A NP-hard problem (P) of mixed-discrete linear programming is considered which consists in the minimization of a linear objective function subject to a special non-connected subset of an unbounded polymatroid. For this problem we describe three polynomial approximate algorithms including a greedy algorithm and a fully polynomial approximation scheme solving a special subproblem of (P).
09 Sep 1985
TL;DR: Many discrete extremal problems on covering are formulated in the following way: given a finite set S and a certain system R, it is required to find the minimal on cost subsystem p c R such that every element of S belongs to at least one element of P.
Abstract: Many discrete extremal problems on covering are formulated in the following way. Given is a finite set S and a certain system R of subsets of S such that: a) for every element a e S there exists at least one subset b~ R such that acb ; b) every element of R has a cost an arbitrary non-negative real number. It is required to find the minimal on cost subsystem p c R such that every element of S belongs to at least one element of P (the cost of P is the sum of the costs of the elements which form the set P).