Geometric Lower Bounds for Parametric Matroid Optimization

D. Eppstein
<span title="">1998</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="" style="color: black;">Discrete &amp; Computational Geometry</a> </i> &nbsp;
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
more &raquo; ... matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 /log * n) for uniform matroids were also known.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1007/pl00009396</a> <a target="_blank" rel="external noopener" href="">fatcat:ftgo2clnl5a2fcd6r6ol5xmjeu</a> </span>
<a target="_blank" rel="noopener" href="" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href=""> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> </button> </a>