Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper \cite{ABCG} initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to $1$).Comment: 17 page

Power indices and the measurement of control in corporate structures

This paper proposes a brief review of the use of power indices in the corporate governance literature. Without losing sight of the field of application, it places the emphasis on the game-theoretic aspects of this research and on the issues that arise in this framework

Boolean Functions: Theory, Algorithms, and Applications

This monograph provides the first comprehensive presentation of the theoretical, algorithmic and applied aspects of Boolean functions, i.e., {0,1}-valued functions of a finite number of {0,1}-valued variables. The book focuses on algebraic representations of Boolean functions, especially normal form representations. It presents the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated representations, dualization, etc.), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once, etc.), and two fruitful generalizations of the concept of Boolean functions (partially defined and pseudo-Boolean functions). It features a rich bibliography of about one thousand items. Prominent among the disciplines in which Boolean methods play a significant role are propositional logic, combinatorics, graph and hypergraph theory, complexity theory, integer programming, combinatorial optimization, game theory, reliability theory, electrical and computer engineering, artificial intelligence, etc. The book contains applications of Boolean functions in all these areas

Basel II and Operational Risk: Implications for risk measurement and management in the financial sector

This paper proposes a methodology to analyze the implications of the Advanced Measurement Approach (AMA) for the assessment of operational risk put forward by the Basel II Accord. The methodology relies on an integrated procedure for the construction of the distribution of aggregate losses, using internal and external loss data. It is illustrated on a 2x2 matrix of two selected business lines and two event types, drawn from a database of 3000 losses obtained from a large European banking institution. For each cell, the method calibrates three truncated distributions functions for the body of internal data, the tail of internal data, and external data. When the dependence structure between aggregate losses and the non-linear adjustment of external data are explicitly taken into account, the regulatory capital computed with the AMA method proves to be substantially lower than with less sophisticated approaches allowed by the Basel II Accord, although the effect is not uniform for all business lines and event types. In a second phase, our models are used to estimate the effects of operational risk management actions on bank profitability, through a measure of RAROC adapted to operational risk. The results suggest that substantial savings can be achieved through active management techniques, although the estimated effect of a reduction of the number, frequency or severity of operational losses crucially depends on the calibration of the aggregate loss distributions.operational risk management, basel II, advanced measurement approach, copulae, external data, EVT, RAROC, cost-benefit analysis.

Models for Decision Making: From Applications to Mathematics... and Back

In this inaugural lecture, I describe some facets of the interplay between mathematics and management science, economics, or engineering, as they come together in operations research models. I intend to illustrate, in particular, the complex and fruitful process through which fundamental combinatorial models find applications in management science, which in turn foster the development of new and challenging mathematical questions