Improved Algorithms for Scheduling Unsplittable Flows on Paths

We investigate offline and online algorithms for 𝖱𝗈𝗎𝗇𝖽-𝖴𝖥𝖯𝖯 , the problem of minimizing the number of rounds required to schedule a set of unsplittable flows of non-uniform size on a given path with heterogeneous edge capacities. 𝖱𝗈𝗎𝗇𝖽-𝖴𝖥𝖯𝖯 is known to be NP-hard and there are constant-factor approximation algorithms under the no bottleneck assumption (NBA), which stipulates that maximum size of any flow is at most the minimum global edge capacity. In this work, we present improved online and offline algorithms for 𝖱𝗈𝗎𝗇𝖽-𝖴𝖥𝖯𝖯 without the NBA . We first study offline 𝖱𝗈𝗎𝗇𝖽-𝖴𝖥𝖯𝖯 for a restricted class of instances, called α -small, where the size of each flow is at most α times the capacity of its bottleneck edge, and present an O(log (1/(1-α ))) -approximation algorithm. Next, our main result is an online O(loglog c_max) -competitive algorithm for 𝖱𝗈𝗎𝗇𝖽-𝖴𝖥𝖯𝖯 where c_max is the largest edge capacity, improving upon the previous best bound of O(log c_max) due to Epstein et al. (SIAM J Discrete Math 23(2):822–841, 2009). These new results lead to an offline O(min (log n, log m, loglog c_max)) -approximation algorithm and an online O(min (log m, loglog c_max)) -competitive algorithm for 𝖱𝗈𝗎𝗇𝖽-𝖴𝖥𝖯𝖯 , where n is the number of flows and m is the number of edges.