Dealing with large hidden constants

Abstract

We present the first attempt on implementing a highly theoretical polynomial-time approximation scheme (PTAS) with huge hidden constants, namely, the PTAS for Steiner tree in planar graphs by Borradaile, Klein, and Mathieu (2009). Whereas this result, and several other PTAS results of the recent years, are of high theoretical importance, no practical applications or even implementation attempts have been known to date, due to the extremely large constants that are involved in them. We describe techniques on how to circumvent the challenges in implementing such a scheme. With today's limitations on processing power and space, we still have to sacrifice approximation guarantees for improved running times by choosing some parameters empirically. But our experiments show that with our choice of parameters, we do get the desired approximation ratios, suggesting that a much tighter analysis might be possible. Our computational experiments with benchmark instances from SteinLib and large artificial instances well exceeded our own expectations. We demonstrate that we are able to handle instances with up to a million nodes and several hundreds of terminals in 1.5 hours on a standard PC. On the rectilinear preprocessed instances from SteinLib, we observe a monotonous improvement for smaller values of ϵ, with an average gap below 1% for ϵ = 0.1. We compare our implementation against the well-known batched 1-Steiner heuristic and observe that on very large instances, we are able to produce comparable solutions much faster. We also present a thorough experimental evaluation of the influence of the various parameters of the PTAS and thus obtain a better understanding of their empirical effects.