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Geometric Lower Bounds for Parametric Matroid Optimization
<span title="">1998</span>
<i title="Springer Nature">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cu3ouk4pmzhsroy4l5k2c35xtu" style="color: black;">Discrete & Computational Geometry</a>
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We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k-sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: (nr 1/3 ) for a general n-element matroid with rank r , and (mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was (n log r ) for uniform
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... matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 /log * n) for uniform matroids were also known.
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