Objective functions with redundant domains

Fatima Affif Chaouche, Carrie Rutherford and Robin Whitty ()
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Fatima Affif Chaouche: University of Sciences and Technology Houari Boumediene
Carrie Rutherford: London South Bank University
Robin Whitty: London South Bank University

Journal of Combinatorial Optimization, 2013, vol. 26, issue 2, No 10, 372-384

Abstract: Abstract Let $(E,{ \mathcal{A}})$ be a set system consisting of a finite collection ${ \mathcal{A}}$ of subsets of a ground set E, and suppose that we have a function ϕ which maps ${ \mathcal{A}}$ into some set S. Now removing a subset K from E gives a restriction ${ \mathcal{A}}(\bar{K})$ to those sets of ${ \mathcal{A}}$ disjoint from K, and we have a corresponding restriction $\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{K})}$ of our function ϕ. If the removal of K does not affect the image set of ϕ, that is $\mbox {Im}(\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{X})})=\mbox {Im}(\phi)$ , then we will say that K is a kernel set of ${ \mathcal{A}}$ with respect to ϕ. Such sets are potentially useful in optimisation problems defined in terms of ϕ. We will call the set of all subsets of E that are kernel sets with respect to ϕ a kernel system and denote it by $\mathrm {Ker}_{\phi}({ \mathcal{A}})$ . Motivated by the optimisation theme, we ask which kernel systems are matroids. For instance, if ${ \mathcal{A}}$ is the collection of forests in a graph G with coloured edges and ϕ counts how many edges of each colour occurs in a forest then $\mathrm {Ker}_{\phi}({ \mathcal{A}})$ is isomorphic to the disjoint sum of the cocycle matroids of the differently coloured subgraphs; on the other hand, if ${ \mathcal{A}}$ is the power set of a set of positive integers, and ϕ is the function which takes the values 1 and 0 on subsets according to whether they are sum-free or not, then we show that $\mathrm {Ker}_{\phi}({ \mathcal{A}})$ is essentially never a matroid.

Keywords: Matroid; Optimization; Objective function; Duality; Turán-type problems (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9468-9

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