H. Narayanan

Chaper 1 Introduction

Publisher Summary This chapter introduces the methods for studying the properties of electrical networks, which are independent of the device characteristic. Only topological constraints are used—namely, Krichoffs current law (KCL) and Kirchoffs voltage law (KVL). These methods are also called “network topological.” The chapter presents applications to circuit simulation and circuit partitioning and establishes the relations between the optimization problems that arise naturally, while using these methods, to the central problems in the theory of submodular functions. There are more immediate applications possible. The most popular general purpose simulator currently running—SPICE—uses the modified nodal analysis approach. In this approach, the devices are divided into two classes, generalized admittance type whose currents can be written in terms of voltages appearing somewhere in the circuit, and the remaining devices. The final variables in terms of which the solution is carried out is the set of all nodal voltages and current variables. The resulting coefficient matrix is very sparse but suffers from several defects.

Chapter 2 Mathematical Preliminaries

Publisher Summary This chapter provides an overview of sets. A set (or collection) is specified by the elements (or members) that belong to it. If element x belongs to the set X , and is written as x ∈ X (x ∉ X). Two sets are equal if they have the same members. A set is finite if it has a finite number of elements. Otherwise, it is infinite. A set is often specified by actually listing its members, for example, {e 1 , e 2 , e 3 }, is the set with members e 1 , e 2 , e 3 . The chapter also describes vectors, matrices, and related notions. The set of all vectors linearly dependent on a collection C of vectors can be shown to form a vector space, which is generated by or spanned by C . Generally, maximal and minimal members of a collection of sets may not be largest and smallest in terms of size.

Chapter 9 Submodular Functions

Publisher Summary In combinatorial mathematics, submodular functions are a relatively recent phenomenon. Submodular functions can be regarded as a generalization of matroid rank functions. The study of basic subinodular operations, such as convolution and Dilworth truncation is significant for practical algorithm designers because in addition to completely capturing the essence of many practical situations, they also allow to give acceptable approximate solutions to several intractable problems. The chapter presents simple equivalent restatements of the definition of submodularity along with a number of instances where submodular functions are found in nature. Several standard operations are discussed by which new submodular/supermodular (more compactly, semimodular) functions can be obtained. The chapter presents the important special cases of matroid and polymatroid rank functions and discusses the polyhedral approach to the study of semimodular functions. It also outlines several recent works on minimization of symmetric submodular functions.

Chapter 13 Algorithms for the PLP of a Submodular Function

Publisher Summary This chapter presents algorithms for the principal lattice of partitions (PLP) of a general submodular function and extends these to important instances of functions based on bipartite graphs. The general algorithms of the chapter are parallel to the algorithms for principal partition. Main algorithms in this context are presented. In the case of both principal partition (PP) and PLP, the problem of minimizing a submodular function is at the heart of the algorithms. The chapter explains the application of general PLP algorithms to the important cases of weighted adjacency and weighted exclusivity functions associated with a bipartite graph. In both these cases, the minimization of the basic submodular function reduces to appropriate flow problems, which can be solved extremely efficiently. Several useful techniques for improving the efficiency of the algorithms in those cases where the maximum value of the (integral) submodular function is less than the size of the underlying set are described.

Chapter 5 Electrical Networks

Publisher Summary This chapter introduces electrical network analysis. An electrical network is obtained by the interconnection of a collection of multiterminal devices. Each of the devices may have one or more terminals. The chapter discusses the multiterminal and 2-terminal of an electrical network. A list of the standard devices used in electrical networks is presented. The chapter describes the common methods of analysis—in the case of modified nodal analysis (MNA)—and its merits and demerits. It also provides a sketch of the working of a general purpose simulator. An informal account of state equations is presented for networks, with the assumption that the initial conditions of all the capacitors and inductors can be assigned independently of each other. The chapter also describes multiport decomposition for networks. The generalized Thevenin–Norton theorem is proved. Two elementary results of network theory-substitution theorem and superposition theorem are also presented.

Chapter 12 Dilworth Truncation of Submodular Functions

Publisher Summary This chapter discusses the Dilworth truncation (truncation for short) operation on submodular functions and the related notion of principal lattice of partitions. The theory of Dilworth truncation bears a strong resemblance to that of convolution. The chapter presents the properties of the truncation operation and presents a number of examples, including Dilworths own, relevant to the operation. The principal lattice of partitions (PLP) of a submodular function is discussed and the similarity in its properties with those of the principal partition (PP) and the characteristic properties are highlighted. The chapter also describes a technique for building approximation algorithms for optimum partitioning problems (including vertex partitioning for a graph, minimizing number of cut edges). It also discusses the relation of the PLP of a given submodular function to the PLP of naturally derived functions.

Chapter 3 Graphs

Publisher Summary This chapter discusses graphs and related notions. Graphs should be visualized as points joined by lines with or without arrows rather than be thought of as formal objects. A graph G is a triple (V(G), E(G), iG) where V(G) i s a finite set of vertices, E(G) is a finite set of edges, and iG is an incidence function, which associates with each edge a pair of vertices, not necessarily distinct, called its end points or end. Vertices are also called “nodes” or “junctions” while edges are also called “arcs” or “branches.” An edge may have a single end point; such edges are called “selfloops.” A vertex may have no edges incident on it; such vertices are isolated. A connected graph with each vertex having degree two is called a “circuit graph” or a “polygon graph.”

Chapter 6 Topological Hybrid Analysis

Publisher Summary This chapter discusses a method of network decomposition that is a topological generalization of hybrid analysis (analysis where unknowns involve both voltages and currents). The result states that solving a network is equivalent to solving two derived subnetworks matching certain current and voltage boundary conditions. To state the result precisely, a formal description of a general electrical network is presented. Usually an electrical network has a device characteristic in which the constraints on sets of branches can be specified independently of each other. The chapter discusses three results; two of these viz. “v-shift” and “i-shift” are well known basic results. Issues concerned with optimal application of network equations are discussed. The application of the method to the case of linear electrical networks is described.

Chapter 8 Multiport Decomposition

Publisher Summary This chapter describes the multiport decomposition using the notion of generalized minor and the implicit duality theorem. A multiport is used in two different senses: an electrical multiport is an electrical network with some devices, which are norators, specified as ports and a component multiport is a vector space with the subset specified as ports. The chapter provides necessary and sufficient conditions under which given components can be coupled to yield a given vector space. A simple lemma, which characterizes an extension of a vector space, is discussed. The chapter also discusses the decomposition of component multiports, as opposed to decomposition of vector spaces. The notion of decomposition of component multiports may be used to decompose a vector space hierarchically.

Chapter 10 Convolution of Submodular Functions

Publisher Summary The operations of convolution and Dilworth truncation are fundamental to the development of the theory of submodular functions. This chapter focuses on convolution and the related notion of principal partition. A formal description of the convolution operation and its properties is presented. Convolution is important both in terms of the resulting function and in terms of the sets arising in the course of the definition of the operation. The principal partition displays the relationships that exists between these sets when the convolved functions are scaled. The chapter discusses the principal partitions of functions derived from the original functions in simple ways, such as through restricted minor operations and dualization. It also presents efficient algorithms for its construction and extends these algorithms to an important instance based on the bipartite graph. Several examples related to the notion of convolution are presented.