An LP-rounding algorithm yields a randomized (1 + 2/e)-approximation for weighted hitting set of axis-parallel segments, with a (1 + 1/(e-1)) bound in the unweighted case and 1 + 1/e when one orientation consists of lines.
Chan and Sariel Har-Peled
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Hitting Axis-Parallel Segments with Weighted Points
An LP-rounding algorithm yields a randomized (1 + 2/e)-approximation for weighted hitting set of axis-parallel segments, with a (1 + 1/(e-1)) bound in the unweighted case and 1 + 1/e when one orientation consists of lines.