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See detailQuadratization of symmetric pseudo-Boolean functions
Anthony, Martin; Boros, Endre; Crama, Yves ULg et al

E-print/Working paper (2013)

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 ${\bbr}$. For a pseudo-Boolean function $f(x)$ on $\{0,1 ... [more ▼]

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 ${\bbr}$. 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 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 \emph{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). [less ▲]

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See detailThe Mathematics of Peter L. Hammer (1936-2006): Graphs, Optimization, and Boolean Models
Boros, Endre; Crama, Yves ULg; De Werra, Dominique et al

in Boros, Endre; Crama, Yves; De Werra, Dominique (Eds.) et al The Mathematics of Peter L. Hammer (1936-2006): Graphs, Optimization, and Boolean Models (2011)

This volume of the Annals of Operations Research, contains a collection of papers published in memory of Peter L. Hammer. As we recall further down, Peter made substantial contributions to several areas ... [more ▼]

This volume of the Annals of Operations Research, contains a collection of papers published in memory of Peter L. Hammer. As we recall further down, Peter made substantial contributions to several areas of operations research and discretemathematics, including, in particular, mathematical programming (linear and quadratic 0–1 programming, pseudo-Boolean optimization, knapsack problems, etc.), combinatorial optimization (transportation problems, network flows, MAXSAT, simple plant location, etc.), graph theory (special classes of graphs, stability problems, and their applications), data mining and classification (Logical Analysis of Data), and, last but not least, Boolean theory (satisfiability, duality, Horn functions, threshold functions, and their applications). [less ▲]

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See detailThe Mathematics of Peter L. Hammer (1936-2006): Graphs, Optimization, and Boolean Models
Boros, Endre; Crama, Yves ULg; de Werra, Dominique et al

in Annals of Operations Research (2011), 188

This volume contains a collection of papers published in memory of Peter L. Hammer (1936-2006). Peter Hammer made substantial contributions to several areas of operations research and discrete mathematics ... [more ▼]

This volume contains a collection of papers published in memory of Peter L. Hammer (1936-2006). Peter Hammer made substantial contributions to several areas of operations research and discrete mathematics, including, in particular, mathematical programming (linear and quadratic 0--1 programming, pseudo-Boolean optimization, knapsack problems, etc.), combinatorial optimization (transportation problems, network flows, MAXSAT, simple plant location, etc.), graph theory (special classes of graphs, stability problems, and their applications), data mining and classification (Logical Analysis of Data), and, last but not least, Boolean theory (satisfiability, duality, Horn functions, threshold functions, and their applications). The volume contains 23 contributed papers along these lines. [less ▲]

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See detailLogical Analysis of Data: Classification with justification
Boros, Endre; Crama, Yves ULg; Hammer, Peter L. et al

in Annals of Operations Research (2011), 188

Learning from examples is a frequently arising challenge, with a large number of algorithms proposed in the classification, data mining and machine learning literature. The evaluation of the quality of ... [more ▼]

Learning from examples is a frequently arising challenge, with a large number of algorithms proposed in the classification, data mining and machine learning literature. The evaluation of the quality of such algorithms is frequently carried out ex post, on an experimental basis: their performance is measured either by cross validation on benchmark data sets, or by clinical trials. Few of these approaches evaluate the learning process ex ante, on its own merits. In this paper, we dis- cuss a property of rule-based classifiers which we call "justifiability", and which focuses on the type of information extracted from the given training set in order to classify new observations. We investigate some interesting mathematical properties of justifiable classifiers. In partic- ular, we establish the existence of justifiable classifiers, and we show that several well-known learning approaches, such as decision trees or nearest neighbor based methods, automatically provide justifiable clas- sifiers. We also identify maximal subsets of observations which must be classified in the same way by every justifiable classifier. Finally, we illustrate by a numerical example that using classifiers based on "most justifiable" rules does not seem to lead to over fitting, even though it involves an element of optimization. [less ▲]

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See detailGeršgorin variations III: On a theme of Brualdi and Varga
Boros, Endre; Brualdi, Richard; Crama, Yves ULg et al

in Journal of Linear Algebra and its Applications (2008), 428

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the ... [more ▼]

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi’s conditions are sharp. Improving Varga’s results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular. [less ▲]

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See detailPeter Ladislaw Hammer (1936-2006)
Boros, Endre; Crama, Yves ULg; Simeone, Bruno

in 4OR : Quarterly Journal of the Belgian, French and Italian Operations Research Societies (2007), 5(1), 1-4

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