Abstract
We consider the nth power metric facility location problem with linear penalties (M $$^n$$ FLPLP) in this work, extending both the nth power metric facility location problem (M $$^n$$ FLP) and the metric facility location problem with linear penalties (MFLPLP). We present an LP-rounding based approximation algorithm to the M $$^n$$ FLPLP with bi-factor approximation ratio $$(\gamma _f, \gamma _c)$$ , where $$\gamma _f$$ and $$\gamma _c$$ are the ratios corresponding to facility, and connection and penalty costs respectively. Finally we show that the bi-factor curve is close to the lower bound $$(\gamma _f, 1 + (3^n - 1) e^{-\gamma _f})$$ when the facility factor $$\gamma _f > 2$$ for the M $$^2$$ FLPLP.
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