A Fast, Adaptive Variant of the Goemans-Williamson Scheme for the Prize-Collecting Steiner Tree Problem

We introduce a new variant of the Goemans-Williamson (GW) scheme for the Prize-Collecting Steiner Tree Problem (PCST). Motivated by applications in signal processing, the focus of our contribution is to construct a very fast algorithm for the PCST problem that still achieves a provable approximation guarantee. Our overall algorithm runs in time O(dm logn) on a graph with m edges, where all edge costs and node prizes are specified with d bits of precision. Moreover, our algorithm maintains the Lagrangian-preserving factor-2 approximation guarantee of the GW scheme. Similar to [Cole, Hariharan, Lewenstein, and Porat, SODA 2001], we use dynamic edge splitting in order to efficiently process all cluster merge and deactivation events in the moat-growing stage of the GW scheme. Our edge splitting rules are more adaptive to the input, thereby reducing the amount of time spent on processing intermediate edge events. Numerical experiments based on the public DIMACS test instances show that our edge splitting rules are very effective in practice. In most test cases, the number of edge events processed per edge is less than 2 on average. On a laptop computer from 2010, the longest running time of our implementation on a DIMACS challenge instance is roughly 1.3 seconds (the corresponding instance has about 340,000 edges). Since the running time of our algorithm scales nearly linearly with the input size and exhibits good constant factors, we believe that our algorithm could potentially be useful in a variety of applied settings.