Publication

From linear separability to unimodality: a hierarchy of pseudo-boolean functions

Dominique de Werra, Thomas Liebling
1988
Journal Articles

Abstract

When an injective pseudo-Boolean function f:B^n -> R is minimized, where B^n=0,1^n is the set of vertices of the unit-hypercube, it is natural to consider so-called greedy vertex-following algorithms. These algorithms construct a sequence of neighbouring (Hamming distance 1) vertices with decreasing f-value. The question arises as to when such algorithm will find the global optimum given any starting point. This paper describes a hierarchy of such classes of fonctions that are shown to strictly contain each other. These classes are, in increasing order of generality, the threshold, the saddle-free, the pseudomodular, the completely unimodal, the unimodal, and the unimin (respectively, unimax) functions. Some considerations as to the complexity of the above-mentioned class of algorithm are also made.

Official source

https://infoscience.epfl.ch/entities/publication/79d1662d-61ce-4879-94d0-3b8f71f50c26

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Dominique de Werra, Thomas Liebling
1988
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/79d1662d-61ce-4879-94d0-3b8f71f50c26

About this result

Related concepts (31)

Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

Hamming distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of substitutions required to change one string into the other, or the minimum number of errors that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences.

Greedy coloring

In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible. Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering.

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