# Objective functions with redundant domains

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## 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## Notes

### Acknowledgements

The generalisation of Example 5 to give Proposition 6 was suggested to us by Tony Forbes. We are pleased to acknowledge helpful discussions with Bill Jackson and Taoyang Wu. The work of Chung and Graham on universal graphs was brought to our attention by Emil Vaughan and by Francesca Merola. An anonymous referee corrected a good many errors and made some helpful suggestions on presentation.

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