Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between outdegree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in O(log n) flips) for general arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O(1) and O √ log n flips nearly matching degree bounds to their respective amortized solutions.

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed.First we give a non-parameterized algorithm for both problems in O * (2 n ) (where the O * (•) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemerédi-Trotter bound, we present a Curve Cover algorithm with running time O * (Ck/ log k) (d−1)k , where C is some constant.Our result improves the previous best times O * (k/1.35) k for Line Cover (where d = 2), O * k dk for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R 3 with running time O * (Ck 2 / log 1/5 k) k , improving on the previous time of O * (k 2 /1.3) k .

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