Dependent Randomized Rounding: The Bipartite Case

We analyze the two existing algorithms to generate dependent randomized roundings for the bipartite edge weight rounding problem together with several newly proposed variants of these algorithms. For both the edge-based approach of Gandhi, Khuller, Parthasarathy, Srinivasan (FOCS 2002) and the bit-wise approach of Doerr (STACS 2006) we give a simple derandomization (guaranteeing the same rounding errors as the randomized versions achieve with positive probability). An experimental investigation on different types of random instances show that, contrary to the randomized rounding problem with disjoint cardinality constraints, the bit-wise approach is faster than the edge-based one, while the latter still achieves the best rounding errors. We propose a hybrid approach that, in terms of running time, combines advantages of the two previous approaches; in terms of rounding errors it seems a fair compromise. In all cases, the derandomized versions yield much better rounding errors than the randomized ones. We also test how the algorithms compare when used to solve different broadcast scheduling problems (as suggested by Gandhi et al.). Since this needs more random decisions than just in the rounding process, we need to partially re-prove previous results and simplify the corresponding algorithms to finally derive a derandomized version. Again, the derandomized versions give significantly better approximations than the randomized versions. We tested the algorithms on data taken from the Wikipedia access log. For the maximum throughput version of the problem, the derandomized algorithms compute solutions that are very close to the optimum of the linear relaxation. For the minimum average delay version, Gandhi et al. gave a (2, 1)-bicriteria algorithm, i.e., an algorithm which produces a 2-speed schedule with an average delay which on expectation is no worse than that of the 1-speed optimum. For this problem variant, while the performance guarantee of the algorithms certainly holds, we find that a simple greedy heuristic generally produces superior solutions.