Search for: All records

In the classical prophet inequality setting, a gambler is given a sequence of n random variables X₁, … , X_n, taken from known distributions, observes their values in adversarial order and selects one of them, immediately after it is being observed, aiming to select a value that is as high as possible. The classical prophet inequality shows a strategy that guarantees a value at least half of the value of an omniscience prophet that always picks the maximum, and this ratio is optimal. Here, we generalize the prophet inequality, allowing the gambler some additional information about the future that is otherwise privy only to the prophet. Specifically, at any point in the process, the gambler is allowed to query an oracle 𝒪. The oracle responds with a single bit answer: YES if the current realization is greater than the remaining realizations, and NO otherwise. We show that the oracle model with m oracle calls is equivalent to the Top-1-of-(m+1) model when the objective is maximizing the probability of selecting the maximum. This equivalence fails to hold when the objective is maximizing the competitive ratio, but we still show that any algorithm for the oracle model implies an equivalent competitive ratio for the Top-1-of-(m+1) model. We resolve the oracle model for any m, giving tight lower and upper bound on the best possible competitive ratio compared to an almighty adversary. As a consequence, we provide new results as well as improvements on known results for the Top-1-of-m model. 

Boob et al. [Boob et al., 2020] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Qaunrud and Torres [Chandra Chekuri et al., 2022] extended the algorithm to supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige’s quadratic program for finding a lexicographically optimal base in a (contra) polymatroid [Satoru Fujishige, 1980], and a noisy version of the Frank-Wolfe method from convex optimization [Frank and Wolfe, 1956; Jaggi, 2013]. This yields a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [Harb et al., 2022]. A second contribution of the paper is to understand Thorup’s work on ideal tree packing and greedy tree packing [Thorup, 2007; Thorup, 2008] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige’s result and convex optimization.