1 — 16:20 — ** CANCELLED ** Intersecting and Dense Restrictions of Clutters in Polynomial Time
A clutter is a family of sets, called members, such that no member contains another. It is called intersecting if every two members intersect, but not all members have a common element. Dense clutters additionally do not have a fractional packing of value 2. We are looking at certain substructures of clutters, namely minors and restrictions.
For a family of clutters we introduce a general sufficient condition such that for every clutter we can decide whether the clutter has a restriction in that set in polynomial time. It is known that the sets of intersecting and dense clutters satisfy this condition.
For intersecting clutters we generalize the statement to kwise intersecting clutters using a much simpler proof.
We also give a simplified proof that a dense clutter with no proper dense minor is either a delta or the blocker of an extended odd hole. This simplification reduces the running time of the algorithm for finding a delta or the blocker of an extended odd hole minor from previously $O(n^4)$ to $O(n^3)$ filter oracle calls.
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