In the Symmetric Rendezvous Search on the Line with Unknown Initial Distance, two identical agents are placed on the real line with their distance, the other's location, and their orientation unknown to them. Moving along the line at unit speed and executing the same randomized search strategy, the agents' goal is to meet up as early as possible. The expected meeting time obviously depends on the unknown initial distance and orientations. The quality of a randomized search strategy is thus measured by its competitive ratio, that is, the ratio of the expected meeting time and the earliest possible meeting time (half the initial distance). We present a class of successively refined randomized search strategies together with a rigorous mathematical analysis of their continuously improved competitive ratios. These strategies all rely on the basic idea of performing an infinite sequence of steps of geometrically increasing size in random directions, always returning to the agent's initial position before starting the next step. In addition, our more refined strategies use two novel ideas. First, remembering their past random choices, the agents randomly choose the direction of the next step in a Markov-chain-like manner. Second, choosing the next few random directions in advance, each agent may combine consecutive steps in the same direction into one longer step. As our main result, we show that this combination of looking into the past as well as into the future leads to a substantially improved competitive ratio of 13.93 compared to the previously best known bound of 24.85 (Ozsoyeller et al. 2013).

The Quickest Transshipment Problem is to route flow as quickly as possible from sources with supplies to sinks with demands in a network with capacities and transit times on the arcs. It is of fundamental importance for numerous applications in areas such as logistics, production, traffic, evacuation, and finance. More than 25 years ago, Hoppe and Tardos presented the first (strongly) polynomial-time algorithm for this problem. Their approach, as well as subsequently derived algorithms with strongly polynomial running time, are hardly practical as they rely on parametric submodular function minimization via Megiddo's method of parametric search. The main contribution of this paper is a considerably faster algorithm for the Quickest Transshipment Problem that instead employs a subtle extension of the Discrete Newton Method. This improves the previously best known running time of Õ(m4k14) to O(m2k5 + m3k3 + m3n), where n is the number of nodes, m the number of arcs, and k the number of sources and sinks.