Randomized Rounding for Routing and Covering Problems: Experiments and Improvements

Abstract

Following previous theoretical work by Srinivasan (FOCS 2001) and the first author (STACS 2006) and a first experimental evaluation on random instances (ALENEX 2009), we investigate how the recently developed different approaches to generate randomized roundings satisfying disjoint cardinality constraints behave when used in two classical algorithmic problems, namely low-congestion routing in networks and max-coverage problems in hypergraphs. We generally find that all randomized rounding algorithms work well, much better than what is guaranteed by existing theoretical work. The derandomized versions produce again significantly better rounding errors, with running times still negligible compared to the one for solving the corresponding LP. It thus seems worth preferring them over the randomized variants. The data created in these experiments lets us propose and investigate the following new ideas. For the low-congestion routing problems, we suggest to solve a second LP, which yields the same congestion, but aims at producing a solution that is easier to round. Experiments show that this reduces the rounding errors considerably, both in combination with randomized and derandomized rounding. For the max-coverage instances, we generally observe that the greedy heuristics also performs very good. We develop a strengthened method of derandomized rounding, and a simple greedy/rounding hybrid approach using greedy and LP-based rounding elements, and observe that both these improvements yield again better solutions than both earlier approaches on their own. For unit disk max-domination, we also develop a PTAS. Contrary to all other algorithms investigated, it performs not much better in experiments than in theory; thus, unless extremely good solutions are to be obtained with huge computational resources, greedy, LP-based rounding or hybrid approaches are preferable.