We design new approximation algorithms for the Multiway Cut problem,
improving the previously known factor of 1.32388 [Buchbinder et al., 2013].
We proceed in three steps. First, we analyze the rounding scheme of
Buchbinder et al., 2013 and design a modification that improves the
approximation to (3+sqrt(5))/4 (approximately 1.309017). We also present a
tight example showing that this is the best approximation one can achieve with
the type of cuts considered by Buchbinder et al., 2013: (1) partitioning by
exponential clocks, and (2) single-coordinate cuts with equal thresholds.
Then, we prove that this factor can be improved by introducing a new rounding
scheme: (3) single-coordinate cuts with descending thresholds. By combining
these three schemes, we design an algorithm that achieves a factor of (10 + 4
sqrt(3))/13 (approximately 1.30217). This is the best approximation factor that
we are able to verify by hand.
Finally, we show that by combining these three rounding schemes with the
scheme of independent thresholds from Karger et al., 2004, the approximation
factor can be further improved to 1.2965. This approximation factor has been
verified only by computer.Comment: This is an updated version and is the full version of STOC 2014 pape