On Approximability of Steiner Tree in ℓ_p-metrics
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In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the ℓ_1-metric (and Hamming metric). Chlebík and Chlebíková [TCS'08] showed that DST is NP-hard to approximate to factor of 96/95≈ 1.01 in the graph metric (and consequently ℓ_∞-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric! In this work, we prove that DST is APX-hard in every ℓ_p-metric. We also prove that CST is APX-hard in the ℓ_∞-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in ℓ_p-metrics. As an immediate consequence, this yields a 1.39-approximation polynomial time algorithm for CST in ℓ_p-metrics.
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