Unsplittable Flow on a Path: The Game!

The unsplittable flow on a path (UFP) problem is a well-studied optimization problem, and it has applications in various settings like bandwidth allocation, caching, and scheduling. We are given a path with capacities on its edges and a set of n tasks, each of them defined via a demand, a subpath, and a profit. The goal is to select the most profitable set of tasks that together respect the edge capacities, i.e., for each edge e the total demand of the selected tasks whose subpath contains e is at most the capacity of e. The best known polynomial time approximation algorithm for UFP is a (5/3 + ∊)-approximation [Grandoni et al., STOC 2018]. It is an important open question whether the problem admits a PTAS. Informally, a task is large if its demand is at least an ∊-fraction of the capacity of some edge on its path, and small otherwise. If all tasks are large, a PTAS can be obtained via dynamic programming: intuitively each edge e is used by only O(1) relevant tasks in the optimal solution OPT. The same approach fails for small tasks since then this number can be up to Ω(n) which would yield an exponential number of states. In this paper we introduce a novel randomized sketching technique to address this issue. We model the computation of a solution as a solitary game where tasks are presented one by one to a player, who has to decide for each task i whether to select i (hence getting its profit) or not. When a small task i is selected, with some probability its demand is rounded up to some large value (and then i behaves like a large task), and otherwise down to zero (and then i can be “forgotten” afterwards), so that in expectation the demand of i does not change. The optimal strategy to play this game can be computed using similar ideas as used in the DP for large tasks. Furthermore, the expected profit of this strategy is at least as large as the profit of OPT. One complication is that the player's solution might be infeasible, e.g., when too many tasks are rounded down. Still, via probabilistic arguments, we can use it to construct a feasible UFP solution which is 1 + + ∊ < 1.269 approximate in expectation. It is potentially possible that a more sophisticated probabilistic analysis gives a PTAS for the problem. We believe that randomized sketching might turn out to be useful to address also other problems in which “large” and “small” objects interact, for example in packing, scheduling, or resource allocation settings, in particular when dynamic programming works if there are only large objects.