Online Scheduling with Interval Conflicts

In the problem of Scheduling with Interval Conflicts, there is a ground set of items indexed by integers, and the input is a collection of conflicts, each containing all the items whose index lies within some interval on the real line. Conflicts arrive in an online fashion. A scheduling algorithm must select, from each conflict, at most one survivor item, and the goal is to maximize the number (or weight) of items that survive all the conflicts they are involved in. We present a centralized deterministic online algorithm whose competitive ratio is O(lgσ), where σ is the size of the largest conflict. For the distributed setting, we present another deterministic algorithm whose competitive ratio is 2⌈lg σ⌉ in the special contiguous case, in which the item indices constitute a contiguous interval of integers. Our upper bounds are complemented by two lower bounds: one that shows that even in the contiguous case, all deterministic algorithms (centralized or distributed) have competitive ratio Ω(lgσ), and that in the non-contiguous case, no deterministic oblivious algorithm (i.e., a distributed algorithm that does not use communication) can have a bounded competitive ratio.