Abstract
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 set 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 k-wise 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 \(\mathcal {O}(n^4)\) to \(\mathcal {O}(n^3)\) filter oracle calls.
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Acknowledgement
I would like to thank Ahmad Abdi, Gérard Cornuéjols and Stephan Held for helpful discussions.
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Drees, M. (2022). Intersecting and Dense Restrictions of Clutters in Polynomial Time. In: Aardal, K., Sanità , L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_16
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DOI: https://doi.org/10.1007/978-3-031-06901-7_16
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