Guide Combinatorial Optimization

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Graduate students and researchers in applied mathematics, optimization, engineering, computer science, and management science will find this book a useful reference which provides an introduction to applications and fundamental theories in nonlinear combinatorial optimization. Nonlinear combinatorial optimization is a new research area within combinatorial optimization and includes numerous applications to technological developments, such as wireless communication, cloud computing, data science, and social networks.

Theoretical developments including discrete Newton methods, primal-dual methods with convex relaxation, submodular optimization, discrete DC program, along with several applications are discussed and explored in this book through articles by leading experts. Springer Professional. Back to the search result list. In a wireless sensor network, a topology control problem aims to adjust power of sensors so that the topology supported by the power has some desirable property and the total power is as small as possible. In this chapter, we shall present studies on the topology control problem with the properties of containing a spanning tree, a strongly connected spanning digraph, and a broadcast tree.

Minimum spanning tree plays an important role in all these studies, serving as a linearization method for these nonlinear problems. Newton method is a classic and powerful method in continuous nonlinear optimization.

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However in this chapter, we introduce its counterpart in combinatorial optimization: discrete Newton method, and show that there exists a strong polynomial time algorithm for finding the root of a piecewise linear decreasing function, where the number of pieces is exponential. Then we show how to apply it in solving linear fractional combinatorial optimization problem, inverse combinatorial problem, and bottleneck expansion problem.

We offer an overview on submodular optimization for both single- and multiple-objectives, with the moderate goal to highlight the different angles in interpreting submodularity and associated concepts. In supply chain management and other operations management applications, various discrete convexities are important tools in modeling complementary or supplementary behaviors.

Furthermore, the discrete nature of many decision scenarios also requires optimization tools from discrete convex theory. In this chapter, we aim at introducing the classical discrete convex theory from the perspective of supply chain applications. We illustrate some direct applications and connections in supply chain applications.

Constrained submodular maximization CSM is widely used in numerous data mining and machine learning applications such as data summarization, network monitoring, exemplar-clustering, and nonparametric learning.

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The CSM can be described as: Given a ground set, a specified constraint, and a submodular set function defined on the power set of the ground set, the goal is to select a subset that satisfies the constraint such that the function value is maximized. Generally, the CSM is NP-hard, and cardinality constrained submodular maximization is well researched. The greedy algorithm and its variants have good performance guarantees for constrained submodular maximization.

  1. What is Combinatorial Optimization?.
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  4. When dealing with large input scenario, it is usually formulated as streaming constrained submodular maximization SCSM , and the classical greedy algorithm is usually inapplicable. The streaming model uses a limited memory to extract a small fraction of items at any given point of time such that the specified constraint is satisfied, and good performance guarantees are also maintained. In this chapter, we list the up-to-date popular algorithms for streaming submodular maximization with cardinality constraint and its variants, and summarize some problems in streaming submodular maximization that are still open.

    The nonsubmodular optimization is a hot research topic in the study of nonlinear combinatorial optimizations. We discuss several approaches to deal with such optimization problems, including supermodular degree, curvature, algorithms based on DS decomposition, and sandwich method. Integer programming in a variable dimension is a crucial research topic that has received a considerable attention in recent years. A series of fixed parameter tractable FPT algorithms have been developed for a variety of integer programming that has a special block structure, and such results were later applied successfully in many classical combinatorial optimization problems to derive FPT or approximation algorithms.

    From a theoretical point of view, it is important to understand the overall landscape, and distinguish the structures of integer programming that are tractable vs. From the application point of view, it is important to understand how the structure of such integer programming is related to the structure of concrete combinatorial optimization problems.

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    Combinatorial Optimization -- from Wolfram MathWorld

    Skip to main content. Packing Problems Packing Problems can be viewed as complementary to cutting problems in that the objective is to fill a larger space with specified smaller shapes in the most economical profitable way. References Textbooks Ahuja, R. Network Flows. Prentice-Hall, Inc. Bertsekas, D. Network Optimization: Continuous and Discrete Models.

    Athena Scientific, Nashua, NH. Lawler, E. Combinatorial Optimization: Networks and Matroids. Dover Publications, Inc. Murty, K.